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5th-Semester-Fall-2023/EEMAGS/Notes/Chapter1.md
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# Chapter 1
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## 1.1 Vector Analysis
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#### Scalar
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A measure described by one real number. Examples include temperature, size, and mass. A scalar is a $1 \times 1$ matrix.
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#### Vector
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A measure described by more than one real number (direction and magnitude). Examples include force, velocity, and it's derivatives. Vectors are often described by $n \times 1$ or $1 \times n$ matricies.
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#### Unit Vector
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A unit (direction or normalized) vector is signified with a $\hat{ }$ symbol. Common unit vectors include $\hat{x}$, $\hat{y}$, and $\hat{z}$. The normalized version of any vector is defined as:
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$$\hat{a} = \frac{\vec{A}}{\lVert\vec{A}\rVert}$$
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#### Dot Product
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The dot product is a measure of how *parallel* two vectors are, scaled by the magnitudes of the two vectors. To compute it, find the sum of the products of the like components of two vectors. It is also defined as the product of the magnitudes of the vectors normalized by the cosine of the angle between them. It is defined as:
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$$W = \vec{F} \cdot \vec{r} = \lVert\vec{F}\rVert\lVert\vec{r}\rVert \cos{\alpha}$$
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#### Cross Product
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The cross product is a measure of how *perpendicular* two vectors are. This operation yeilds a vector quantity *orthoganal* to both original vectors. The direction vector for the cross product is $\hat{a}_c$, and its magnitude is the product of the magnitudes and the sine of the angle between them. It is defined as:
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$$\vec{C} = \vec{A} \times \vec{B} = \lVert\vec{A}\rVert \lVert\vec{B}\rVert \sin(\alpha) \hat{a}_c = \begin{vmatrix} \hat{x} & \hat{y} & \hat{z} \\ A_x & A_y & A_z \\ B_x & B_y & B_z \end{vmatrix}$$
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#### Right Hand Rule
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The right hand rule is a quick way to find $\hat{a}_c$.
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$$\overrightarrow{thumb} = \overrightarrow{pointer} \times \overrightarrow{middle}$$
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#### Cartesian to Cylindrical
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$$\vec{A} = 4\hat{x} + 4\hat{y} - 2\hat{z}$$
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$$r = \sqrt{x^2 + y^2}$$
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$$\varphi = \arctan\left(\frac{y}{x}\right)$$
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$$z = z$$
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$$\vec{A} = A_\rho \hat{\rho} + A_\varphi \hat{\varphi} + A_z \hat{z}$$
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$$A_\rho = A_x \cos(\varphi) + A_y\sin(\varphi) = 4\sqrt{2}$$
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$$A_\varphi = -A_x\sin(\varphi) + A_y\cos(\varphi) = 0$$
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$$A_z = A_z = -2$$
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## 1.2 Vector Calculus
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The differential along some path, $d\vec{l}$, is defined as:
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$$d\vec{l} = dx\hat{x} + dy\hat{y} + dz\hat{z} = d\vec{x} + d\vec{y} + d\vec{z}$$
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#### The "Del" $\left(\nabla\right)$ operator
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The gradient of the scalar field $\left(\nabla f\right)$.
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$$\nabla = \frac{\partial}{\partial x} \hat{x} + \frac{\partial}{\partial y} \hat{y} + \frac{\partial}{\partial z} \hat{z}$$
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$$df = \nabla f \cdot d\vec{l} = \left(\frac{\partial f}{\partial x}\right)dx + \left(\frac{\partial f}{\partial y}\right)dy + \left(\frac{\partial f}{\partial z}\right)dz$$
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### Directional Derivative:
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The *directional derivative* is used to find the change of a function along some infinatesimal direction and is defined as:
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$$\Delta \varphi = {\nabla}_l \varphi \cdot \Delta \vec{l}$$
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$$\therefore {\nabla}_l \varphi = \frac{\Delta \varphi}{\delta \vec{l}} = \frac{d \varphi}{d \vec{l}} = \nabla \varphi \cdot \frac{\vec{l}}{\lVert\vec{l}\rVert}$$
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#### Example 1.1
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A function, $f(x,y,z) = x^2 y^2 + xyz$. Find $\nabla f$.
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$$f(x,y,z) = f(r)$$
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$$r = \sqrt{x^2 + y^2 + z^2}$$
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$$\nabla f = (2xy^2)\hat{x} + (2x^2y + xz)\hat{y} + xy\hat{z}$$
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#### Example 1.2
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Consider a function, $w = x^2y^2 + xyz$. Find $\nabla_l w$ at $(2, -1, 0)$ in the direction, $\vec{l} = 3\hat{x} + 4\hat{y} + 12\hat{z}$.
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$$\lVert\vec{l}\rVert = \sqrt{3^2 + 4^2 + 12^2} = 13$$
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$${\nabla}_l w = \nabla w \cdot \frac{\vec{l}}{\lVert\vec{l}\rVert}$$
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Solving at $(2, -1, 0)$:
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$$\nabla w = 2(2)(-1)^2\hat{x} + 2(2)^2(-1)\hat{y} + 2(-1)\hat{z} = 4\hat{x} - 8\hat{y} - 2\hat{z}$$
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$${\nabla}_l w= (4\hat{x} - 8\hat{y} - 2\hat{z}) \cdot \left(\frac{3}{13}\hat{x} + \frac{4}{13}\hat{y} + \frac{12}{13}\hat{z}\right) = \frac{-44}{13}$$
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### Divergence and Curl
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#### Divergence of a Vector Field
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The divergence of a vector field is a measure of outward flux. It is defined as:
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$$\nabla \cdot f = \left( \frac{\partial}{\partial x} + \frac{\partial}{\partial y} + \frac{\partial}{\partial z} \right) \cdot (f_x + f_y + f_z) = \frac{\partial f_x}{\partial x} + \frac{\partial f_y}{\partial y} + \frac{\partial f_z}{\partial z}$$
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If $\nabla \cdot \vec{A} = 0$, there is no divergence.
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#### Curl of a Vector Field
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The curl of a vector field is a measure of circulation in each infinatesimally small region of the field. It is defined as:
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$$\nabla \times f = \begin{vmatrix} \hat{x} & \hat{y} & \hat{z} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ f_x & f_y & f_z \\ \end{vmatrix} = \left< \frac{\partial f_z}{\partial y} - \frac{\partial f_y}{\partial z}, \frac{\partial f_x}{\partial z} - \frac{\partial f_z}{\partial x}, \frac{\partial f_y}{\partial z} - \frac{\partial f_x}{\partial z} \right>$$
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#### Solenoidal Field
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A solenoidal field is a vector field without divergence, defined as:
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$$\nabla \cdot \vec{f} = 0$$
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#### Conservative / Rotational Field
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A conservative field is a vector field without curl, defined as:
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$$\nabla \times \vec{f} = 0$$
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#### Example 1.3
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Consider the vector field, $\vec{F} = k \hat{x}$, where both the direction and magnitude are uniform in all space.
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$$\nabla \cdot \vec{F} = 0$$
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$$\nabla \times \vec{F} = 0$$
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#### Example 1.4
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Consider the vector field, $\vec{F} = k \hat{r}$, where magnitude is constant and direction is away from a central point.
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$$\hat{r} = \sqrt{\hat{x}^2 + \hat{y}^2 + \hat{z}^2}$$
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$$\nabla \cdot \vec{F} = \sqrt{3} k$$
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$$\nabla \times \vec{F} = 0$$
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#### Example 1.5
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Consider the vector field, $\vec{F} = k \times \hat{r}$, where magnitude is uniform and the direction is perpendicular to the distance from a central point for all space.
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$$\nabla \cdot \vec{F} = 0$$
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$$\nabla \times \vec{F} = 2k$$
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### Curl and Divergence Identities
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1. The Laplacian: can operate on a scalar or vector field
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$$\nabla \cdot (\nabla f) = \nabla^2 f$$
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$$\frac{\partial^2 f}{\partial x} + \frac{\partial^2 f}{\partial y} + \frac{\partial^2 f}{\partial z}$$
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2. The curl of a gradient is $0$
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$$\nabla \times (\nabla f) = 0$$
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3. The gradient of divergence is a scalar
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$$\nabla (\nabla \cdot \vec{f})$$
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4. The divergence of curl is $0$
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$$\nabla \cdot (\nabla \times \vec{v}) = 0$$
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5. Curl of curl
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$$\nabla \times (\nabla \times \vec{A}) = \nabla(\nabla \cdot \vec{A}) - \nabla^2 \vec{A}$$
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### The Line Integral
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The line integral is the integral of the tangential component of a vector field along a path.
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The line integral is defined as:
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$$\int \vec{A} \cdot d\vec{l} = \int \lVert \vec{A} \rVert \cos(\alpha) \lVert d\vec{l} \rVert$$
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Where:
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- $\vec{A}$ is some vector field
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- $d\vec{l} = dx\hat{x} + dy\hat{y} + dz\hat{z}$
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If the path of the integral is a closed curve, it is said to be the circulation of $\vec{A}$ around $\vec{l}$, defined as:
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$$\oint_c \vec{A} \cdot d\vec{l}$$
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#### Example 1.6
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Calculate the circulation of $\vec{F}$ around the path.
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$\vec{F} = x^2 \hat{x} - xy\hat{y} - y^2\hat{z}$
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Circulation of $\vec{F}$ around the path:
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$$\oint \vec{F} \cdot d \vec{l} = \int_1 + \int_2 + \int_3 + \int_4$$
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##### Path 1
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Straight line from $(1,0,0)$ to $(0,0,0)$
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$x$ varies, $z=0$, $y=0$.
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Plug into $\vec{F}$:
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$$\vec{F}_1 = x^2\hat{x} - x(0)\hat{y} - (0)^2\hat{z} = x^2\hat{x}$$
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Since $x$ varies at some rate, $dx$ exists, and since $y$ and $z$ are constant, $dy$ and $dz$ are both $0$.
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Plug into $d\vec{l}$:
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$$d\vec{l} = dx\hat{x} + (0)\hat{y} + (0)\hat{z} = dx\hat{x}$$
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$$\int \vec{F}_1 \cdot d\vec{l}_1 = \int x^2\hat{x} \cdot dx\hat{x} = \int x^2dx$$
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For this specific path:
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$$\int_{1}^{0} x^2dx = \left. \frac{x^3}{3} \right\vert_1^0 = 0 - \frac{1}{3} = -\frac{1}{3}$$
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##### Path 2
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Straight line from $(0,0,0)$ to $(0,1,0)$
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$y$ varies, $x=0$, $z=0$
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Plug into $\vec{F}$:
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$$\vec{F}_2 = (0)^2\hat{x} - (0)y\hat{y} - y^2\hat{z} = -y^2\hat{z}$$
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Since $y$ varies at some rate, $dy$ exists, but since $x$ and $z$ are constant, $dx$ and $dz$ are both $0$.
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Plug into $d\vec{l}$:
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$$d\vec{l}_2 = (0)\hat{x} + dy\hat{y} + (0)\hat{z} = dy\hat{y}$$
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$$\int \vec{F}_2 \cdot d\vec{l} = \int -y^2\hat{z} \cdot dy\hat{y} = \int\limits_0^1 0 = 0$$
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##### Path 3
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Straight line from $(0,1,0)$ to $(1,1,1)$
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$x$ and $z$ vary at the same rate and always have the same value, $y=1$
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### Surface Integrals
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$\hat{n}$ is the unit normal vector of a surface.
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The total *flux* crossing an area, $\Delta s$, is given by the function:
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$$\Delta s \left[ \lVert \vec{F} \rVert \cos(\alpha)\right] = \vec{F} \cdot \hat{n} \Delta s$$
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Total flux is defined as:
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$$\sum_{i=1}^N \vec{F}_i \cos(\alpha_i) \Delta s_i = \int_s \vec{F} \cdot d\vec{s}$$
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#### Gradient Theorem
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The line integral through a gradient field is the difference of the values a the end points.
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$$\int\limits_{r_1}^{r_2} \nabla \varphi \cdot d\vec{l} = \varphi(r_2) - \varphi(r_1)$$
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$$\oint \nabla \varphi \cdot d\vec{l} = 0$$
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#### Divergence Theorem
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The divergence in some volume is the same as the flux through its surface.
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$$\iiint_{vol} \nabla \cdot \vec{A} d\tau = \oiint \vec{A} \cdot d\vec{s}$$
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#### Stokes Theorem
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The curl in some region is the same as the circulation of the region's border.
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$$\iint_A (\nabla \times \vec{A}) \cdot d\vec{s} = \oint \vec{A} \cdot d\vec{l}$$
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## 1.3 Coulomb's Law
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Initial observation:
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$$\vec{F} \propto q_1 q_2$$
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Vacuum Permittivity:
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$$\varepsilon_0 = 8.854 \times 10^{-12}$$
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Coulomb's Constant:
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$$k = \frac{1}{4 \pi \varepsilon_0} = 9 \times 10^{-9}$$
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Coulomb's Law:
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$$\vec{F} = k \frac{q_1 q_2}{r^2} \hat{a}_{1 2}$$
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Superposition of Coulomb's Law:
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$$\vec{F}_{Net} = \sum_{i=1}^{N} k_i \frac{Q q_i}{r_i^2} \hat{r}$$
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**Note**: $k_i$ depends on material properties. When in a vacuum, $k_i = k$.
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#### Example 1.7
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Two point charges, $q_1$ and $q_2$ are spaced 2cm apart on the x-axis. A third charge, $q_3$ is placed between the first two with a distance $x_1$ between it and $q_1$ and $x_2$ between it and $q_2$ such that $q_3$ is in static equilibrium.
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Known:
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$$\vec{F}_{1 3} + \vec{F}_{2 3} = 0$$
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By Coulomb's Law:
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$$\vec{F}_{1 3} = \frac{k_1 q_3 q_1}{r_{1 3}^2} \hat{z}$$
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$$\vec{F}_{2 3} = -\frac{k_2 q_3 q_2}{r_{1 3}^2} \hat{z}$$
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$$\therefore \vec{F}_{1 3} + \vec{F}_{2 3} = k q_3 \left( \frac{q_1}{r_{1 3}^2} - \frac{q_2}{r_{2 3}^2} \right)\hat{z} = 0$$
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$$\therefore \frac{q_1}{r_{1 3}^2} - \frac{q_2}{r_{2 3}^2} = 0$$
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Solve for $r_{1 3}$:
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$$r_{1 3} = \pm r_{2 3}\sqrt{\frac{q_1}{q_2}}$$
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Known:
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$$r_{1 3} + r_{2 3} = 2$$
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$$\therefore r_{2 3} = 2 - r_{1 3}$$
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Plug into first equation:
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$$r_{1 3} = \pm (2 - r_{1 3}) \sqrt{\frac{q_1}{q_2}}$$
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Expand:
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$$r_{1 3} = \pm \left(2 \sqrt{\frac{q_1}{q_2}} - r_{1 3}\sqrt{\frac{q_1}{q_2}} \right)$$
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$$\therefore r_{1 3} \left(1 \pm \sqrt{\frac{q_1}{q_2}} \right) = \pm 2\sqrt{\frac{q_1}{q_2}}$$
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$$\therefore r_{1 3} = \pm 2\sqrt{\frac{q_1}{q_2}} \left( 1 \pm \sqrt{\frac{q_1}{q_2}} \right)^{-1}$$
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### Electric Field Intensity
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Call a test charge at the point of measurement $Q_2$.
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Coulomb's Law:
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$$\vec{F}_{1 2} = \frac{Q_1 Q_2}{4 \pi \varepsilon_0 R^2} \hat{a}_{1 2}$$
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The electric field intensisty:
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$$\vec{E}_{1 2} = \frac{\vec{F}_{1 2}}{Q_1}$$
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$$\vec{E} = \frac{Q_1}{4 \pi \varepsilon_0 r^2} \hat{a}_{1 2}$$
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$$\vec{E}_{NET} = \sum_{i=0}^{N} \frac{k_i Q_i}{R_i^2} \hat{a}_{R_i}$$
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Force due to electric field:
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$$\vec{F} = q \vec{E}$$
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#### Electric field
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Always in the same direction as the electric field force.
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#### Electric field of a dipole
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At a point equidistant to each pole:
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$$\lVert \vec{r}_1 \rVert = \lVert \vec{r_2} \rVert = r$$
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By Coulomb's Law:
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$$\vec{E}_1 = \frac{kq}{r^2} \hat{r}_1$$
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$$\vec{E}_2 = \frac{kq}{r^2} \hat{r}_2$$
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$$\therefore \lVert \vec{E}_1 \rVert = \lVert \vec{E}_2 \rVert$$
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In terms of the component distances:
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$$r^2 = a^2 + y^2$$
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$$\therefore E = \frac{kq}{a^2 + y^2}$$
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$$\vec{E}_{NET_y} = \vec{E}_{1_y} + \vec{E}_{2_y} = 2E\cos{\theta}$$
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By definition:
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$$\cos(\theta) = \frac{a}{r}$$
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$$\therefore \vec{E}_{NET_y} = \frac{2kq}{a^2 + y^2} \frac{a}{\sqrt{a^2 + y^2}}\hat{y}$$
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$$\vec{E}_{NET_y} = 2 \frac{kqa}{(x^2 + y^2)^{3/2}} \hat{y}$$
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If $y \gg a$ (far field):
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$$\vec{E}_{NET} = 2 \frac{kqa}{y^3}$$
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**Takeaway**:
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$$\vec{E}_{monopole} \propto \frac{1}{r^2}$$
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$$\vec{E}_{dipole} \propto \frac{1}{r^3}$$
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#### Charge Densities
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$\lambda$ - Linear charge density
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$\sigma$ - Surface charge density
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$\rho$ - Volume charge density
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Electric flux density:
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$$\vec{D} = \varepsilon_0 \vec{E}$$
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Where:
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$$\vec{E} = \lim_{\Delta q \to 0} \frac{k \sum_i \Delta q_i}{\lVert \vec{r}_i \rVert ^2} \hat{r}_i = \int \frac{k}{\lVert \vec{r}_i \rVert ^2} dq$$
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#### Example 1.8
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A straight line segment of length $L$ with uniform charge density $\lambda$ extends from the origin in the $\hat{x}$ direction. Find the strength of the electric field, $\vec{E}$ at some arbitrary point, $p$ along the ray from the origin in the $\hat{z}$ direction.
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The distance along the line segment in the $\hat{x}$ direction is denoted as $x$, and the distance from the origin to point $p$ is denoted as $z$. The vector, $\vec{r}$ has length $\sqrt{x^2 + z^2}$ and makes an angle $\theta$ with the ray in the $-\hat{z}$ direction.
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The contribution $d\vec{E}$ to the total electric field, $\vec{E}$, at point $p$ is defined as:
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$$d\vec{E} = \frac{k dq}{\lVert \vec{r} \rVert ^2} \hat{r}$$
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For a linear charge distribution:
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$$dq = \lambda dl$$
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The distance, $r$, to each $x$ alond the line:
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$$\lVert \vec{r} \rVert ^2 = x^2 + z^2$$
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The components of the $\vec{E}$-field at point $p$:
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$$d\vec{E}_x = \lVert d\vec{E} \rVert \sin(\theta) \hat{x}$$
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$$d\vec{E}_z = \lVert d\vec{E} \rVert \cos(\theta) \hat{z}$$
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## Gauss's Law
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## E-Flux Density
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## EMF
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Measured in *volts*, electromotive force (EMF), is denoted by $\mathcal{E}$. The value for EMF is defined as:
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$$\mathcal{E} = \oint \vec{E} \cdot d \vec{l} = -\frac{d}{dt} \int_s B_z(t) \cdot ds = -\frac{d}{dt}\psi_m$$
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Where:
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$$\psi_m = \int_s \vec{B} \cdot d\vec{s}$$
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A perfectly conducting ring with radius, $\rho_0$, centered on the origin in the x-y plane.
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The charge distribution:
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$$\rho = \rho_0 + \rho_0 \sin(\omega t)$$
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$$\vec{B}(t) = B_0 \cos(\omega t)\hat{z}$$
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$$\mathcal{E} = \oint \vec{E} \cdot d\vec{l} = \iint\limits_{\phi R} B_0 \cos(\omega t) dr d\phi$$
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Where:
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||||
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$\varphi: [0, 2\pi]$
|
||||
|
||||
$R: [0, \rho(t)]$
|
||||
|
||||
Therefore:
|
||||
$$\psi_m = (B_0 \cos(\omega t) \hat{z})(2\pi \rho(t))$$
|
||||
|
||||
$$\mathcal{E} = -\frac{d}{dt} B_0 2\pi \cos(\omega t)(\rho_0 + \rho_0\sin(\omega t))$$
|
||||
|
||||
### Filling in some Gaps
|
||||
$$\vec{F} = -\nabla \vec{u}$$
|
||||
Where: $\vec{u}$ is potential.
|
||||
|
||||
$$\vec{E} = -\nabla \vec{v}$$
|
||||
Where: $\vec{v}$ is electric potential.
|
||||
|
||||
$$W = \vec{F} \cdot \vec{d} = q\vec{E} \cdot \vec{d}$$
|
||||
|
||||
Work done by the $\vec{E}$-field on a charge will reduce the electric potential:
|
||||
|
||||
$$-\Delta u = u_B - u_A = -\Delta W = -qEd$$
|
||||
|
||||
The total change is:
|
||||
|
||||
$$\Delta u = -q \int\limits_A^B \vec{E} \cdot d\vec{l}$$
|
||||
|
||||
$$\therefore \frac{\Delta u}{q} = \int\limits_A^B \vec{E} \cdot d\vec{l} = \Delta V$$
|
||||
|
||||
$$W = q \int \vec{E} \cdot d\vec{l}$$
|
||||
|
||||
$$V(r) = \frac{kq}{r}$$
|
||||
|
||||
For $N$ discrete charges,
|
||||
$$V = \sum\limits_{i=1}^{N} \frac{k q_i}{r_i}$$
|
||||
|
||||
In cartesian coordinates:
|
||||
|
||||
$$\vec{F} = F_x\hat{i} + F_y\hat{j} + F_z\hat{z}$$
|
||||
|
||||
Where:
|
||||
|
||||
$F_x = \frac{dw}{dx}$
|
||||
|
||||
$F_y = \frac{dw}{dy}$
|
||||
|
||||
$F_z = \frac{dw}{dz}$
|
||||
|
||||
Therefore:
|
||||
$$\vec{F} = \frac{\partial w}{\partial x}\hat{i} + \frac{\partial w}{\partial y}\hat{j} + \frac{\partial w}{\partial z}\hat{k} = \nabla \vec{w}$$
|
||||
|
||||
$$q\vec{E} = \nabla (-\vec{u})$$
|
||||
|
||||
$$\vec{F} = -\nabla \vec{u}$$
|
||||
|
||||
$$\vec{E} = -\nabla \left( \frac{\vec{u}}{q} \right) = -\nabla \vec{V}$$
|
||||
|
||||
|
||||
#### Electric Flux Density
|
||||
$$\vec{D} = \varepsilon \vec{E}$$
|
||||
Where $\varepsilon = \varepsilon_r \varepsilon_0$ (permittivity).
|
||||
|
||||
#### Magnetic Flux Density
|
||||
$$\vec{B} = \mu\vec{H}$$
|
||||
Where $\mu = \mu_r \mu_0$ (permeability).
|
||||
|
||||
### Ampere's Law
|
||||
The total current crossing an area, $s$, that is enclosed by the contour $C$:
|
||||
$$\oint_C \frac{\vec{B}}{\mu_0} \cdot d\vec{l}$$
|
||||
|
||||
The total current is the sum of the current due to charge flow and the current due to the time rate of change of the electric flux crossing an area, $s$. Maxwell was able to unify electricity and magnetism by adding the current due to the time rate of change of electric flux.
|
||||
$$\oint_C \vec{H} \cdot d\vec{l} = \int_S \vec{J} \cdot d\vec{s} + \frac{d}{dt}\int \varepsilon_0\vec{E} \cdot d\vec{s}$$
|
||||
|
||||
#### Simplified Ampere's Law
|
||||
|
||||
$$\oint_C \vec{H} \cdot d\vec{l} = I = \int_s \vec{J} \cdot d\vec{s}$$
|
||||
|
||||
The charge density, $J = \rho v$, has units $\left[ \frac{A}{m^2} \right]$.
|
||||
|
||||
#### Example
|
||||
A current, $I$, in an infinitely long cylindrical wire with radius, $R$.
|
||||
$$\int \vec{B} \cdot d\vec{l} = \mu_0 I_{enc}$$
|
||||
|
||||
Measuring the magnetic field at some distance, $r$, from the center of the conductor:
|
||||
$$B\int dl = \mu_0 I$$
|
||||
Where $\int dl$ is the circumfrece of measurement.
|
||||
$$B(2\pi r) = \mu_0 I$$
|
||||
$$B_{out} = \frac{\mu_0 I}{2\pi r} \hat{\varphi}$$
|
||||
Inside the wire:
|
||||
$$\int \vec{B} \cdot d\vec{l} = \mu_0 I$$
|
||||
$$B\int dl = \frac{\mu_0 I r^2}{R^2} = 2\pi rB$$
|
||||
$$B = \frac{\mu_0 I}{2\pi R^2} r$$
|
||||
$$\vec{B} = \frac{\mu_0 I}{2\pi R^2}r \hat{\varphi}$$
|
||||
|
||||
### Coulomb's Law
|
||||
The total displacement flux of charge:
|
||||
$$\int_s \varepsilon_0 \vec{E} \cdot d\vec{s}$$
|
||||
|
||||
The total current (charge with respect to time):
|
||||
$$I = \frac{d}{dt} \int_s \varepsilon_0 \vec{E} \cdot d\vec{s}$$
|
||||
|
||||
### Faraday's Law
|
||||
Work done in moving a unit positive test charge around a closed path, $C$:
|
||||
$$\oint_C \vec{E} \cdot d\vec{l}$$
|
||||
|
||||
Magnetic force on a poving charge and is directed perpendicular to both the direction of the motion of the charge and the magnetic field.
|
||||
$$\oint_C \vec{B} \cdot d\vec{l}$$
|
||||
|
||||
### Solenoid (Ideal)
|
||||
For an ideal solenoid with constant current, $I$, assume uniform $\vec{B}$ inside, $\vec{B} = 0$ outside, and infinite length.
|
||||
|
||||
By Ampere's Law:
|
||||
$$\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{enc}$$
|
||||
|
||||
For a square loop with one side in the solenoid, and it's parallel side outside the loop:
|
||||
$$\oint \vec{B} \cdot d\vec{l} = \int_1 + \int_2 + \int_3 + \int_4$$
|
||||
|
||||
Sides 2 and 4 are parallel, and side 3 is outside the solenoid, so:
|
||||
$$\oint \vec{B} \cdot d\vec{l} = \int_1 = Bl$$
|
||||
|
||||
Back to Ampere's Law:
|
||||
$$Bl = \mu_0 I N$$
|
||||
$$\therefore B = \frac{\mu_0 I N}{l} = \mu_0 I n$$
|
||||
Where:
|
||||
|
||||
$N$ is the total number of windings,
|
||||
|
||||
$l$ is the sidelength of the Amperian loop,
|
||||
|
||||
$n$ is the number of windings per unit length $\frac{N}{l}$
|
||||
|
||||
### Toroid (Ideal)
|
||||
From a symmetric $\vec{B}$-field, lines form concentric circles inside the toroid. For an ideal toroid, assume $\vec{B} = 0$ outside, and Ampere's law inside.
|
||||
|
||||
$$\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{enc}$$
|
||||
For some circular Amperian loop inside the toroid:
|
||||
$$\vec{B} = B\hat{\varphi}$$
|
||||
$$d\vec{l} = 2\pi r \vec{\varphi}$$
|
||||
$$B(2\pi r) = \mu_0 N I$$
|
||||
$$\therefore B = \frac{\mu_0 N I}{2\pi r}$$
|
||||
265
5th-Semester-Fall-2023/EEMAGS/Notes/Chapter2.md
Normal file
265
5th-Semester-Fall-2023/EEMAGS/Notes/Chapter2.md
Normal file
@ -0,0 +1,265 @@
|
||||
# Chapter 2
|
||||
|
||||
## Maxwell's Equations in Differential Form
|
||||
$$\int \vec{E} \cdot d\vec{s} = \frac{Q_{enc}}{\varepsilon_0} \xleftrightarrow{\text{divergence}} \nabla \cdot \vec{E} = \frac{\rho}{\varepsilon_0}$$
|
||||
$$\int \vec{B} \cdot d\vec{s} = 0 \xleftrightarrow{} \nabla \cdot \vec{B} = 0$$
|
||||
$$\int \vec{E} \cdot d\vec{l} = -\frac{d}{dt}\vec{B} \xleftrightarrow{\text{stokes}} \nabla \times \vec{E} = -\frac{d}{dt}\vec{B}$$
|
||||
$$\int \vec{B} \cdot d\vec{l} = \mu_0\left( \vec{J} + \varepsilon_0 \frac{d\vec{E}}{dt}\right) \xleftrightarrow{} \nabla \times \vec{B} = \mu_0 J + \mu_0 \varepsilon_0 \frac{\partial \vec{E}}{\partial t}$$
|
||||
|
||||
Before Maxwell (only true for magnetostatics):
|
||||
$$\nabla \times \vec{B} = \mu_0\vec{J}$$
|
||||
$$\nabla \cdot (\nabla \times \vec{B}) = \mu_0(\nabla \cdot \vec{J})$$
|
||||
$$\nabla \cdot \vec{J} = 0$$
|
||||
|
||||
For changing magnetic fields:
|
||||
$$\nabla \cdot \vec{J} = -\frac{\partial}{\partial t} \rho_v$$
|
||||
$$\nabla \cdot \vec{J} + \frac{\partial}{\partial t} \rho_v = 0$$
|
||||
|
||||
### Example 2.1
|
||||
$$\vec{J} = e^{-x^2}\hat{x}$$
|
||||
Find the time rate of change of charge densisty at $x=1$:
|
||||
$$\nabla \cdot \vec{J} = -\frac{\partial}{\partial t} \rho_v$$
|
||||
$$\frac{\partial \rho}{\partial t} = -\nabla \cdot \vec{J} = -\frac{d}{dx}e^{-x^2} = 2xe^{-x^2}$$
|
||||
$$\therefore 2 e^{-1}$$
|
||||
|
||||
### Example 2.2
|
||||
For some spherical current density:
|
||||
$$\vec{J} = \frac{J_0 e^{-t/\tau}}{\rho}\hat{\rho}$$
|
||||
Find the total current that leaves the surface of radius, $t = \tau$:
|
||||
$$I = \int \vec{J} \cdot d\vec{s} = 4\pi a^2 \left(\frac{J_0 e^{-t/\tau}}{a}\right)$$
|
||||
$$I = 4\pi a J_0 e^{-1}$$
|
||||
|
||||
Find $\rho_v(\rho, t)$:
|
||||
$$\nabla \cdot \vec{J} = -\frac{\partial \rho}{\partial t} = \frac{1}{\rho^2} \frac{\partial}{\partial \rho}(\rho^2 J)$$
|
||||
Plug in for $J$:
|
||||
$$\nabla \cdot \vec{J} = \frac{1}{\rho^2} \frac{\partial}{\partial \rho}\left(\rho^2 \frac{J_0 e^{-t/\tau}}{\rho}\right) = \frac{J_0}{\rho^2} e^{-t/\tau} = -\frac{\partial}{\partial t} \rho_v$$
|
||||
Solve for $\rho_v$:
|
||||
$$\rho_v = \int -\frac{J_0}{\rho^2}e^{-t/\tau} dt$$
|
||||
$$\rho_v = \frac{J_0}{\rho^2} e^{-t/\tau} \tau$$
|
||||
|
||||
## Wave Propagation
|
||||
$$\nabla \times \vec{E} = -\frac{\partial}{\partial t} \vec{B}$$
|
||||
$$\nabla \times \vec{B} = \mu_0 (\vec{J} + \varepsilon_0 \frac{\partial}{\partial t} \vec{E})$$
|
||||
$$\nabla \cdot \vec{E} = \frac{\rho_v}{\varepsilon_0}$$
|
||||
$$\nabla \cdot \vec{B} = 0$$
|
||||
|
||||
Take the curl of the first equation:
|
||||
$$\nabla \times (\nabla \times \vec{E}) = -\frac{\partial}{\partial t}(\nabla \times \vec{B})$$
|
||||
$$\nabla (\nabla \cdot \vec{E}) - \nabla^2\vec{E} = -\frac{\partial}{\partial t}(\nabla \times \vec{B})$$
|
||||
|
||||
Substitute the second equation into the one directly above:
|
||||
$$\nabla(\nabla \cdot \vec{E}) - \nabla^2\vec{E} = -\frac{\partial}{\partial t} \left( \mu_0 \vec{J} + \mu_0\varepsilon_0 \frac{\partial}{\partial t}\vec{E} \right)$$
|
||||
|
||||
#### Homogenious vector wave for $\vec{E}$-fields
|
||||
Lets say we are considering wave propagation in a source free region ($\rho_v =0$).
|
||||
$$\nabla \cdot \vec{E} = \frac{\rho_v}{\varepsilon_0} =0$$
|
||||
$$\vec{J} = -\frac{\partial}{\partial t} \rho_v = 0$$
|
||||
|
||||
Now we have:
|
||||
$$-\nabla^2\vec{E} = -\mu_0\varepsilon_0 \frac{\partial}{\partial t} \vec{E}$$
|
||||
$$\therefore \nabla^2\vec{E}-\mu_0\varepsilon_0 \frac{\partial}{\partial t}\vec{E} = 0$$
|
||||
|
||||
#### Homogeneous vector wave for $\vec{B}$-fields
|
||||
$$\nabla^2\vec{B} - \mu_0\varepsilon_0 \frac{\partial^2}{\partial t^2}\vec{B} = 0$$
|
||||
|
||||
### Phasor representations
|
||||
In general, all fields and their sources vary as a function of position and time. If the time variations are sinusoidal, with angular frequency, $\omega$, then each of their quantities can be represented by a time independent phasor.
|
||||
|
||||
$$\vec{E}(x,y,z,t) = \Re\{\vec{E}(x,y,z)e^{j \omega t}\}$$
|
||||
$$\vec{E}(r,t) = \Re\{\hat{E}(r) e^{j \omega t}\}$$
|
||||
$$\vec{B}(r,t) = \Re\{\hat{B}(r) e^{j \omega t}\}$$
|
||||
|
||||
$\hat{E}$ and $\hat{B}$ are the complex time variations in vector form. ($\hat{E} = \hat{\vec{E}}$)
|
||||
.
|
||||
|
||||
If $\hat{E}(r) = E_0 e^{j \theta}$,
|
||||
$$\vec{E}(r, t) = \Re\{E_0 e^{j \theta} e^{j \omega t}\} = E_0\cos(\omega t + \theta)$$
|
||||
|
||||
Consider a plane wave in the x-direction only ($E_y = E_z = 0$), and does ont vary the x or y direction ($\frac{\partial}{\partial x}\hat{E} = \frac{\partial}{\partial y}\hat{E} = 0$). Assume free space for propagation ($\hat{J} = \hat{\rho}_v = 0$).
|
||||
|
||||
$$\nabla \times \vec{E} = \begin{vmatrix} \hat{x} & \hat{y} & \hat{z} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ \hat{E}_x & \hat{E}_y & \hat{E}_z \end{vmatrix} = -j\omega (\hat{B}_x\hat{x} + \hat{B}_y \hat{y} + \hat{B}_z \hat{z}$$
|
||||
$$\therefore \begin{Bmatrix} -\frac{\partial}{\partial z}\hat{E} = -j\omega\hat{B}_x \\ \frac{\partial}{\partial z}\hat{E} = -j\omega\hat{B}_y \\ 0 = -j\omega\hat{B}_z \end{Bmatrix}$$
|
||||
|
||||
Similarly, Ampere's Law:
|
||||
$$\begin{Bmatrix} -\frac{\partial \hat{B}_y}{\partial z} = j\omega\varepsilon_0\mu_0\hat{E}_x \\ \frac{\partial \hat{B}_x}{\partial z} = j\omega\varepsilon_0\mu_0\hat{E}_y \\ 0 = j\omega\varepsilon_0\mu_0\hat{E}_z\end{Bmatrix}$$
|
||||
|
||||
Therefore, $B_x$ and $E_y$ are related to each other. $B_x$ acts as the source for generating $E_y$, and $E_y$ acts as the source for generating $B_x$. Solving for $\hat{E}_x$ and $\hat{B}_y$:
|
||||
$$\frac{\partial^2 \hat{E}_x}{\partial z^2} = -j\omega\frac{\partial \hat{B}_y}{\partial z} = -\omega^2\mu_0\varepsilon_0\hat{E}_x$$
|
||||
$$\therefore \frac{\partial^2}{\partial z^2}\hat{E}_x + \mu_0\varepsilon_0\omega^2\hat{E}_x = 0$$
|
||||
|
||||
The general solution:
|
||||
$$\hat{E}_x = \hat{C}_1 e^{-j \beta_0 z} + \hat{C}_2 e^{j \beta_0 z}$$
|
||||
Where $\hat{C}_1$ and $\hat{C}_2$ are complex.
|
||||
|
||||
$$\therefore \hat{E}_x = \hat{E}_m^+ e^{-j \beta_0 z} + \hat{E}_m^- e^{j \beta_0 z}$$
|
||||
Where $E_m^+$ and $E_m^-$ are wave amplitudes (volts per meter).
|
||||
|
||||
##### The phase constant:
|
||||
$$\beta_0 = \omega \sqrt{\mu_0 \varepsilon_0} = \frac{2\pi}{\lambda} = \frac{2\pi f}{C}$$
|
||||
|
||||
#### Real time form of the solution:
|
||||
$$E_x(z,t) = \Re\{\hat{E}_x e^{j \omega t}\} = \Re\{E_m^+ e^{j(\omega t - \beta_0 z)} + E_m^- e^{j(\omega t + \beta_0 z)}\}$$
|
||||
$$\therefore E_x(z,t) = E_m^+ \cos(\omega t - \beta_0 z) + E_m^- \cos(\omega t + \beta_0 z)$$
|
||||
|
||||
If $E_m^+ = E_m^-$, the magnitude and direction is $E_0$.
|
||||
$$\therefore E_x(z,t) = E_0\cos(\omega t \pm \beta_0 z)$$
|
||||
Where:
|
||||
|
||||
$\omega = 2\pi f$
|
||||
|
||||
$v_0$ is the phase velocity
|
||||
|
||||
##### The phase velocity
|
||||
$$v_0 = \frac{1}{\sqrt{\mu \varepsilon}}$$
|
||||
In a vacuum:
|
||||
$$v_0 = c$$
|
||||
|
||||
#### Also
|
||||
$$\hat{B}_y = \frac{\hat{E}_x}{c}$$
|
||||
|
||||
Common $\vec{E}$ and $\vec{B}$ field ratio is:
|
||||
$$\mu_0 \hat{H}_y = \frac{\hat{E}_x}{c}$$
|
||||
$$\therefore \frac{\hat{E}_x}{\hat{H}_y} = \mu_0 c = \frac{\mu_0}{\sqrt{\mu_0 \varepsilon_0}} = \sqrt{\frac{\mu_0}{\varepsilon_0}} = \eta_0 \approx 120\pi = 377\Omega$$
|
||||
|
||||
Where $\eta_0$ is the intrinsic wave impedance.
|
||||
|
||||
$$\therefore \hat{E}_x = \hat{H}_y \eta_0$$
|
||||
$$\therefore \hat{H}_y = \frac{\hat{E}_x}{\eta_0}$$
|
||||
|
||||
Therefore, both $\vec{H}$ and $\vec{E}$ fields are in phase.
|
||||
|
||||
In general:
|
||||
$$\hat{H} = \frac{\hat{k}}{\eta_0} \times \hat{E}$$
|
||||
$$\hat{E} = -\eta_0 \hat{k} \times \hat{H}$$
|
||||
|
||||
### Example 2.3
|
||||
$$\vec{E} = 50\cos(10^8t + \beta_0 x)\hat{y} \text{[v/m]}$$
|
||||
$$E_y = E_0\cos(\omega t + \beta x)$$
|
||||
$E_y$ propagates in the $-\hat{x}$ direction.
|
||||
$$\beta_0 = \omega \sqrt{\mu_0 \varepsilon_0} = \frac{\omega}{c} = \frac{10^8}{3 \times 10^8} = \frac{1}{3}$$
|
||||
What is the time it takes to travel a distance of $\frac{\lambda}{2}$?
|
||||
$$\omega = 2\pi f = \frac{2\pi}{T}$$
|
||||
$$T = \frac{2\pi}{\omega}$$
|
||||
$$\therefore t_{\lambda/2} = \frac{T}{2} = \frac{\pi}{\omega} = \frac{\pi}{10^8} \approx 31.42\text{[ns]}$$
|
||||
|
||||
The refractive index of a medium is given by the ratio of $c$ and $v_p$:
|
||||
$$c = \frac{1}{\sqrt{\mu_0 \varepsilon_0}}$$
|
||||
$$v_p = \frac{1}{\sqrt{\mu \varepsilon}}$$
|
||||
$$n = \frac{c}{v_p} = \sqrt{\frac{\mu \varepsilon}{\mu_0 \varepsilon_0}} = \sqrt{\mu_r \varepsilon_r}$$
|
||||
|
||||
For a non-magnetic material:
|
||||
$$\mu_r \approx 1$$
|
||||
$$\therefore n = \sqrt{\varepsilon_r}$$
|
||||
|
||||
## Polarization
|
||||
|
||||
$$\begin{rcases}
|
||||
\nabla \cdot \vec{E} = {\rho \over \varepsilon_0} \\
|
||||
\nabla \cdot \vec{B} = 0
|
||||
\end{rcases} \text{Gauss}$$
|
||||
|
||||
$$\begin{rcases}
|
||||
\nabla \times \vec{E} = -{\partial \over \partial t} \vec{B}
|
||||
\end{rcases} \text{Faraday}$$
|
||||
|
||||
$$\begin{rcases}
|
||||
\nabla \times B = \mu_0 \vec{J} + \mu_0 \varepsilon_0{\partial \over \partial t}\vec{E}
|
||||
\end{rcases} \text{Ampere}$$
|
||||
|
||||
Polarization of a uniform plane wave describes the shape and olcus of the tip of the $\vec{E}$-field vector in the plane orthoganal to the direction of propagation. There are two pairs of $\vec{E}$ and $\vec{B}$ fields ($\hat{E}_x$, $\hat{H}_y$) and ($\hat{E}_y$, $\hat{H}_x$). When both pairs are present, we can evaluate polarization of plane waves. The $\vec{E}$-field has components in the $\hat{x}$ and $\hat{y}$ directions and travels in $\hat{z}$.
|
||||
|
||||
$$\hat{E} = (\hat{E}_x \hat{x} + \hat{E}_y \hat{y})e^{-j \beta z}$$
|
||||
$$\hat{E}_x = \lvert \hat{E}_{x_0} \rvert e^{-j \beta a}$$
|
||||
$$\hat{E}_y = \lvert \hat{E}_{y_0} \rvert e^{-j \beta b}$$
|
||||
|
||||
#### Locus
|
||||
The shape the tip of the $\vec{E}$-field vector traces out while in motion.
|
||||
|
||||
#### Phase
|
||||
Typically defined relative to a reference point such as $z=0$ or $t=0$ or some combination.
|
||||
|
||||
### Polarization Characteristics
|
||||
#### Linear polarization
|
||||
$\hat{E}_x$ and $\hat{E}_y$ have the same phase angle: $a = b$, so the x and y components of the $\vec{E}$-field will be in phase.
|
||||
|
||||
$$\hat{E} = (\lvert \hat{E}_x \rvert \hat{x} + \lvert \hat{E}_y \rvert \hat{y} ) e^{-j( \beta z - a)}$$
|
||||
|
||||
In real time:
|
||||
$$\hat{E} = (\lvert \hat{E}_x \rvert \hat{x} + \lvert \hat{E}_y \rvert \hat{y} ) \cos(\omega t - \beta z + a)$$
|
||||
|
||||
As the wave continues to propagate in the $\hat{z}$ direction, the $\vec{E}$-field vector maintains its direction with angle, $\theta$, with respect to the y-axis.
|
||||
$$\tan(\theta) = {\lvert \hat{E}_x \rvert \over \vert \hat{E}_y \rvert}$$
|
||||
|
||||
When $z=0$, the $\vec{E}$ field is given by:
|
||||
$$\vec{E}(0, t) = (\lvert \hat{E}_x \rvert \hat{x} + \lvert \hat{E}_y \rvert \hat{y} ) \cos(\omega t + a)$$
|
||||
$$\vec{E}(0, 0) = (\lvert \hat{E}_x \rvert \hat{x} + \lvert \hat{E}_y \rvert \hat{y} ) \cos(a)$$
|
||||
|
||||
#### Elliptical Polarization
|
||||
$\hat{E}_x$ and $\hat{E}_y$ have different phase angles. $\vec{E}$ is no longer in one plate.
|
||||
|
||||
$$\hat{E} = \hat{x} \lvert \hat{E}_x \rvert e^{j (a - \beta z)} + \hat{y} \lvert \hat{E}_y \rvert e^{j(b - \beta z)}$$
|
||||
|
||||
Where:
|
||||
|
||||
$E_x = \lvert \hat{E}_x \rvert \cos(\omega t - a + \beta z)$
|
||||
|
||||
$E_y = \lvert \hat{E}_y \rvert \cos(\omega t - b + \beta z)$
|
||||
|
||||
|
||||
If $a=0$ and $b = {\pi \over 2}$:
|
||||
$$\vec{E}_x(z,t) = \lvert \hat{E}_x \rvert \cos(\omega t - a + \beta z)$$
|
||||
$$\vec{E}_y(z,t) = \lvert \hat{E}_y \rvert \cos(\omega t - b + \beta z)$$
|
||||
|
||||
#### Circular Polarization
|
||||
$\hat{E}_x$ and $\hat{E}_y$ have the same magnitude with a phase angle difference of ${\pi \over 2}$.
|
||||
|
||||
### Example
|
||||
Determine the real-valued $\vec{E}$-field.
|
||||
$$\hat{E}(z) = -3j\hat{x} e^{-j\beta z}$$
|
||||
$$\vec{E}(z, t) = -3j\hat{x} e^{-j \beta z} e^{j\omega t}$$
|
||||
$$3 \hat{x} e^{-j \beta z} e^{j \omega t} e^{-\pi \over 2}$$
|
||||
$$3 \cos\left(\omega t- \beta z - {\pi \over 2}\right)$$
|
||||
|
||||
### Example
|
||||
What are $E_x$ and $E_y$
|
||||
$$\hat{E}(z) = (3\hat{x} + 4\hat{y})e^{j\beta z}$$
|
||||
$$\vec{E}(z, t) = (3\hat{x} + 4\hat{y})e^{j\beta z}e^{j\omega t}$$
|
||||
$$E_x = 3\cos(\omega t + \beta z)$$
|
||||
$$E_y = 4\cos(\omega t + \beta z)$$
|
||||
|
||||
### Example
|
||||
$$\hat{E}(z) = (-4\hat{x} + 3\hat{y})e^{-j\beta z}$$
|
||||
$$\hat{E}(z, t) = (-4\hat{x} + 3\hat{y})e^{-j\beta z}e^{j\omega t}$$
|
||||
$$E_x = 4\cos(\omega t - \beta z + \pi)$$
|
||||
$$E_y = 3\cos(\omega t - \beta z)$$
|
||||
|
||||
## Non-Sinusoidal Waves
|
||||
Analytical solution of a 1-D traveling wave.
|
||||
$${\partial^2 \over \partial z^2}\vec{E} - {1\over c^2}{\partial^2 \over \partial t^2}\vec{E} = 0$$
|
||||
|
||||
### D'Alemberts Solution
|
||||
$$\vec{E}(z, t) = E(z - ct) + E'(z + ct)$$
|
||||
|
||||
Show that the function, $F(z - ct) = F_0 e^{-(z+ct)^2}$ is a solution of the wave equation.
|
||||
|
||||
Let $\gamma = z - ct$, and ${\partial \gamma \over \partial z} = 1$:
|
||||
$$F(z-ct) = F_0 e^{-\gamma^2}$$
|
||||
$$F'(z + ct) = 0$$
|
||||
$${\partial F \over \partial z} = {\partial F \over \partial \gamma} {\partial \gamma \over \partial z} = F_0 e^{-\gamma^2}(-2\gamma)$$
|
||||
$${\partial^2 F \over \partial z^2} = {\partial \over \partial z}\left({\partial F \over \partial \gamma}{\partial \gamma \over \partial z}\right)$$
|
||||
$$G = \left({\partial F \over \partial \gamma}{\partial \gamma \over \partial z}\right)$$
|
||||
|
||||
Don't forget chain rule:
|
||||
$${\partial G \over \partial \gamma}{\partial \gamma \over \partial z} = {\partial \over \partial z}\left({\partial F \over \partial \gamma}{\partial \gamma \over \partial z}\right){\partial \gamma \over \partial z}$$
|
||||
|
||||
$${\partial \over \partial \gamma} - 2\gamma F_0 e^{-\gamma^2} = -2F_0 {\partial \over \partial \gamma} \gamma e^{-\gamma^2}$$
|
||||
|
||||
By product rule:
|
||||
$$-2F_0 (\gamma (-2\gamma e^{-\gamma^2}) + e^{-\gamma^2})$$
|
||||
|
||||
$$F_0 (4\gamma^2 e^{-\gamma^2} - 2e^{-\gamma^2}) = F_0(-2 + 4\gamma^2)e^{-\gamma^2}$$
|
||||
$$={\partial^2 \over \partial \gamma^2}F \left({\partial \gamma \over \partial z}\right)^2$$
|
||||
$$\therefore {\partial^2 \over \partial t^2}F = {\partial^2 F \over \partial \gamma^2} \left({\partial \gamma \over \partial t}\right)^2 = F_0(-2 + 4\gamma^2)e^{-\gamma^2}(C^2)$$
|
||||
|
||||
This solution satisfies:
|
||||
$${\partial^2 \over \partial z^2}\vec{E} - {1\over c^2}{\partial^2 \over \partial t^2}\vec{E} = 0$$
|
||||
332
5th-Semester-Fall-2023/EEMAGS/Notes/Chapter2.tex
Normal file
332
5th-Semester-Fall-2023/EEMAGS/Notes/Chapter2.tex
Normal file
@ -0,0 +1,332 @@
|
||||
\hypertarget{chapter-2}{%
|
||||
\section{Chapter 2}\label{chapter-2}}
|
||||
|
||||
\hypertarget{maxwells-equations-in-differential-form}{%
|
||||
\subsection{Maxwell's Equations in Differential
|
||||
Form}\label{maxwells-equations-in-differential-form}}
|
||||
|
||||
\[\int \vec{E} \cdot d\vec{s} = \frac{Q_{enc}}{\varepsilon_0} \xleftrightarrow{\text{divergence}} \nabla \cdot \vec{E} = \frac{\rho}{\varepsilon_0}\]
|
||||
\[\int \vec{B} \cdot d\vec{s} = 0 \xleftrightarrow{} \nabla \cdot \vec{B} = 0\]
|
||||
\[\int \vec{E} \cdot d\vec{l} = -\frac{d}{dt}\vec{B} \xleftrightarrow{\text{stokes}} \nabla \times \vec{E} = -\frac{d}{dt}\vec{B}\]
|
||||
\[\int \vec{B} \cdot d\vec{l} = \mu_0\left( \vec{J} + \varepsilon_0 \frac{d\vec{E}}{dt}\right) \xleftrightarrow{} \nabla \times \vec{B} = \mu_0 J + \mu_0 \varepsilon_0 \frac{\partial \vec{E}}{\partial t}\]
|
||||
|
||||
Before Maxwell (only true for magnetostatics):
|
||||
\[\nabla \times \vec{B} = \mu_0\vec{J}\]
|
||||
\[\nabla \cdot (\nabla \times \vec{B}) = \mu_0(\nabla \cdot \vec{J})\]
|
||||
\[\nabla \cdot \vec{J} = 0\]
|
||||
|
||||
For changing magnetic fields:
|
||||
\[\nabla \cdot \vec{J} = -\frac{\partial}{\partial t} \rho_v\]
|
||||
\[\nabla \cdot \vec{J} + \frac{\partial}{\partial t} \rho_v = 0\]
|
||||
|
||||
\hypertarget{example-2.1}{%
|
||||
\subsubsection{Example 2.1}\label{example-2.1}}
|
||||
|
||||
\[\vec{J} = e^{-x^2}\hat{x}\] Find the time rate of change of charge
|
||||
densisty at \(x=1\):
|
||||
\[\nabla \cdot \vec{J} = -\frac{\partial}{\partial t} \rho_v\]
|
||||
\[\frac{\partial \rho}{\partial t} = -\nabla \cdot \vec{J} = -\frac{d}{dx}e^{-x^2} = 2xe^{-x^2}\]
|
||||
\[\therefore 2 e^{-1}\]
|
||||
|
||||
\hypertarget{example-2.2}{%
|
||||
\subsubsection{Example 2.2}\label{example-2.2}}
|
||||
|
||||
For some spherical current density:
|
||||
\[\vec{J} = \frac{J_0 e^{-t/\tau}}{\rho}\hat{\rho}\] Find the total
|
||||
current that leaves the surface of radius, \(t = \tau\):
|
||||
\[I = \int \vec{J} \cdot d\vec{s} = 4\pi a^2 \left(\frac{J_0 e^{-t/\tau}}{a}\right)\]
|
||||
\[I = 4\pi a J_0 e^{-1}\]
|
||||
|
||||
Find \(\rho_v(\rho, t)\):
|
||||
\[\nabla \cdot \vec{J} = -\frac{\partial \rho}{\partial t} = \frac{1}{\rho^2} \frac{\partial}{\partial \rho}(\rho^2 J)\]
|
||||
Plug in for \(J\):
|
||||
\[\nabla \cdot \vec{J} = \frac{1}{\rho^2} \frac{\partial}{\partial \rho}\left(\rho^2 \frac{J_0 e^{-t/\tau}}{\rho}\right) = \frac{J_0}{\rho^2} e^{-t/\tau} = -\frac{\partial}{\partial t} \rho_v\]
|
||||
Solve for \(\rho_v\):
|
||||
\[\rho_v = \int -\frac{J_0}{\rho^2}e^{-t/\tau} dt\]
|
||||
\[\rho_v = \frac{J_0}{\rho^2} e^{-t/\tau} \tau\]
|
||||
|
||||
\hypertarget{wave-propagation}{%
|
||||
\subsection{Wave Propagation}\label{wave-propagation}}
|
||||
|
||||
\[\nabla \times \vec{E} = -\frac{\partial}{\partial t} \vec{B}\]
|
||||
\[\nabla \times \vec{B} = \mu_0 (\vec{J} + \varepsilon_0 \frac{\partial}{\partial t} \vec{E})\]
|
||||
\[\nabla \cdot \vec{E} = \frac{\rho_v}{\varepsilon_0}\]
|
||||
\[\nabla \cdot \vec{B} = 0\]
|
||||
|
||||
Take the curl of the first equation:
|
||||
\[\nabla \times (\nabla \times \vec{E}) = -\frac{\partial}{\partial t}(\nabla \times \vec{B})\]
|
||||
\[\nabla (\nabla \cdot \vec{E}) - \nabla^2\vec{E} = -\frac{\partial}{\partial t}(\nabla \times \vec{B})\]
|
||||
|
||||
Substitute the second equation into the one directly above:
|
||||
\[\nabla(\nabla \cdot \vec{E}) - \nabla^2\vec{E} = -\frac{\partial}{\partial t} \left( \mu_0 \vec{J} + \mu_0\varepsilon_0 \frac{\partial}{\partial t}\vec{E} \right)\]
|
||||
|
||||
\hypertarget{homogenious-vector-wave-for-vece-fields}{%
|
||||
\paragraph{\texorpdfstring{Homogenious vector wave for
|
||||
\(\vec{E}\)-fields}{Homogenious vector wave for \textbackslash vec\{E\}-fields}}\label{homogenious-vector-wave-for-vece-fields}}
|
||||
|
||||
Lets say we are considering wave propagation in a source free region
|
||||
(\(\rho_v =0\)).
|
||||
\[\nabla \cdot \vec{E} = \frac{\rho_v}{\varepsilon_0} =0\]
|
||||
\[\vec{J} = -\frac{\partial}{\partial t} \rho_v = 0\]
|
||||
|
||||
Now we have:
|
||||
\[-\nabla^2\vec{E} = -\mu_0\varepsilon_0 \frac{\partial}{\partial t} \vec{E}\]
|
||||
\[\therefore \nabla^2\vec{E}-\mu_0\varepsilon_0 \frac{\partial}{\partial t}\vec{E} = 0\]
|
||||
|
||||
\hypertarget{homogeneous-vector-wave-for-vecb-fields}{%
|
||||
\paragraph{\texorpdfstring{Homogeneous vector wave for
|
||||
\(\vec{B}\)-fields}{Homogeneous vector wave for \textbackslash vec\{B\}-fields}}\label{homogeneous-vector-wave-for-vecb-fields}}
|
||||
|
||||
\[\nabla^2\vec{B} - \mu_0\varepsilon_0 \frac{\partial^2}{\partial t^2}\vec{B} = 0\]
|
||||
|
||||
\hypertarget{phasor-representations}{%
|
||||
\subsubsection{Phasor representations}\label{phasor-representations}}
|
||||
|
||||
In general, all fields and their sources vary as a function of position
|
||||
and time. If the time variations are sinusoidal, with angular frequency,
|
||||
\(\omega\), then each of their quantities can be represented by a time
|
||||
independent phasor.
|
||||
|
||||
\[\vec{E}(x,y,z,t) = \Re\{\vec{E}(x,y,z)e^{j \omega t}\}\]
|
||||
\[\vec{E}(r,t) = \Re\{\hat{E}(r) e^{j \omega t}\}\]
|
||||
\[\vec{B}(r,t) = \Re\{\hat{B}(r) e^{j \omega t}\}\]
|
||||
|
||||
\(\hat{E}\) and \(\hat{B}\) are the complex time variations in vector
|
||||
form. (\(\hat{E} = \hat{\vec{E}}\)) .
|
||||
|
||||
If \(\hat{E}(r) = E_0 e^{j \theta}\),
|
||||
\[\vec{E}(r, t) = \Re\{E_0 e^{j \theta} e^{j \omega t}\} = E_0\cos(\omega t + \theta)\]
|
||||
|
||||
Consider a plane wave in the x-direction only (\(E_y = E_z = 0\)), and
|
||||
does ont vary the x or y direction
|
||||
(\(\frac{\partial}{\partial x}\hat{E} = \frac{\partial}{\partial y}\hat{E} = 0\)).
|
||||
Assume free space for propagation (\(\hat{J} = \hat{\rho}_v = 0\)).
|
||||
|
||||
\[\nabla \times \vec{E} = \begin{vmatrix} \hat{x} & \hat{y} & \hat{z} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ \hat{E}_x & \hat{E}_y & \hat{E}_z \end{vmatrix} = -j\omega (\hat{B}_x\hat{x} + \hat{B}_y \hat{y} + \hat{B}_z \hat{z}\]
|
||||
\[\therefore \begin{Bmatrix} -\frac{\partial}{\partial z}\hat{E} = -j\omega\hat{B}_x \\ \frac{\partial}{\partial z}\hat{E} = -j\omega\hat{B}_y \\ 0 = -j\omega\hat{B}_z \end{Bmatrix}\]
|
||||
|
||||
Similarly, Ampere's Law:
|
||||
\[\begin{Bmatrix} -\frac{\partial \hat{B}_y}{\partial z} = j\omega\varepsilon_0\mu_0\hat{E}_x \\ \frac{\partial \hat{B}_x}{\partial z} = j\omega\varepsilon_0\mu_0\hat{E}_y \\ 0 = j\omega\varepsilon_0\mu_0\hat{E}_z\end{Bmatrix}\]
|
||||
|
||||
Therefore, \(B_x\) and \(E_y\) are related to each other. \(B_x\) acts
|
||||
as the source for generating \(E_y\), and \(E_y\) acts as the source for
|
||||
generating \(B_x\). Solving for \(\hat{E}_x\) and \(\hat{B}_y\):
|
||||
\[\frac{\partial^2 \hat{E}_x}{\partial z^2} = -j\omega\frac{\partial \hat{B}_y}{\partial z} = -\omega^2\mu_0\varepsilon_0\hat{E}_x\]
|
||||
\[\therefore \frac{\partial^2}{\partial z^2}\hat{E}_x + \mu_0\varepsilon_0\omega^2\hat{E}_x = 0\]
|
||||
|
||||
The general solution:
|
||||
\[\hat{E}_x = \hat{C}_1 e^{-j \beta_0 z} + \hat{C}_2 e^{j \beta_0 z}\]
|
||||
Where \(\hat{C}_1\) and \(\hat{C}_2\) are complex.
|
||||
|
||||
\[\therefore \hat{E}_x = \hat{E}_m^+ e^{-j \beta_0 z} + \hat{E}_m^- e^{j \beta_0 z}\]
|
||||
Where \(E_m^+\) and \(E_m^-\) are wave amplitudes (volts per meter).
|
||||
|
||||
\hypertarget{the-phase-constant}{%
|
||||
\subparagraph{The phase constant:}\label{the-phase-constant}}
|
||||
|
||||
\[\beta_0 = \omega \sqrt{\mu_0 \varepsilon_0} = \frac{2\pi}{\lambda} = \frac{2\pi f}{C}\]
|
||||
|
||||
\hypertarget{real-time-form-of-the-solution}{%
|
||||
\paragraph{Real time form of the
|
||||
solution:}\label{real-time-form-of-the-solution}}
|
||||
|
||||
\[E_x(z,t) = \Re\{\hat{E}_x e^{j \omega t}\} = \Re\{E_m^+ e^{j(\omega t - \beta_0 z)} + E_m^- e^{j(\omega t + \beta_0 z)}\}\]
|
||||
\[\therefore E_x(z,t) = E_m^+ \cos(\omega t - \beta_0 z) + E_m^- \cos(\omega t + \beta_0 z)\]
|
||||
|
||||
If \(E_m^+ = E_m^-\), the magnitude and direction is \(E_0\).
|
||||
\[\therefore E_x(z,t) = E_0\cos(\omega t \pm \beta_0 z)\] Where:
|
||||
|
||||
\(\omega = 2\pi f\)
|
||||
|
||||
\(v_0\) is the phase velocity
|
||||
|
||||
\hypertarget{the-phase-velocity}{%
|
||||
\subparagraph{The phase velocity}\label{the-phase-velocity}}
|
||||
|
||||
\[v_0 = \frac{1}{\sqrt{\mu \varepsilon}}\] In a vacuum: \[v_0 = c\]
|
||||
|
||||
\hypertarget{also}{%
|
||||
\paragraph{Also}\label{also}}
|
||||
|
||||
\[\hat{B}_y = \frac{\hat{E}_x}{c}\]
|
||||
|
||||
Common \(\vec{E}\) and \(\vec{B}\) field ratio is:
|
||||
\[\mu_0 \hat{H}_y = \frac{\hat{E}_x}{c}\]
|
||||
\[\therefore \frac{\hat{E}_x}{\hat{H}_y} = \mu_0 c = \frac{\mu_0}{\sqrt{\mu_0 \varepsilon_0}} = \sqrt{\frac{\mu_0}{\varepsilon_0}} = \eta_0 \approx 120\pi = 377\Omega\]
|
||||
|
||||
Where \(\eta_0\) is the intrinsic wave impedance.
|
||||
|
||||
\[\therefore \hat{E}_x = \hat{H}_y \eta_0\]
|
||||
\[\therefore \hat{H}_y = \frac{\hat{E}_x}{\eta_0}\]
|
||||
|
||||
Therefore, both \(\vec{H}\) and \(\vec{E}\) fields are in phase.
|
||||
|
||||
In general: \[\hat{H} = \frac{\hat{k}}{\eta_0} \times \hat{E}\]
|
||||
\[\hat{E} = -\eta_0 \hat{k} \times \hat{H}\]
|
||||
|
||||
\hypertarget{example-2.3}{%
|
||||
\subsubsection{Example 2.3}\label{example-2.3}}
|
||||
|
||||
\[\vec{E} = 50\cos(10^8t + \beta_0 x)\hat{y} \text{[v/m]}\]
|
||||
\[E_y = E_0\cos(\omega t + \beta x)\] \(E_y\) propagates in the
|
||||
\(-\hat{x}\) direction.
|
||||
\[\beta_0 = \omega \sqrt{\mu_0 \varepsilon_0} = \frac{\omega}{c} = \frac{10^8}{3 \times 10^8} = \frac{1}{3}\]
|
||||
What is the time it takes to travel a distance of \(\frac{\lambda}{2}\)?
|
||||
\[\omega = 2\pi f = \frac{2\pi}{T}\] \[T = \frac{2\pi}{\omega}\]
|
||||
\[\therefore t_{\lambda/2} = \frac{T}{2} = \frac{\pi}{\omega} = \frac{\pi}{10^8} \approx 31.42\text{[ns]}\]
|
||||
|
||||
The refractive index of a medium is given by the ratio of \(c\) and
|
||||
\(v_p\): \[c = \frac{1}{\sqrt{\mu_0 \varepsilon_0}}\]
|
||||
\[v_p = \frac{1}{\sqrt{\mu \varepsilon}}\]
|
||||
\[n = \frac{c}{v_p} = \sqrt{\frac{\mu \varepsilon}{\mu_0 \varepsilon_0}} = \sqrt{\mu_r \varepsilon_r}\]
|
||||
|
||||
For a non-magnetic material: \[\mu_r \approx 1\]
|
||||
\[\therefore n = \sqrt{\varepsilon_r}\]
|
||||
|
||||
\hypertarget{polarization}{%
|
||||
\subsection{Polarization}\label{polarization}}
|
||||
|
||||
\[\begin{rcases}
|
||||
\nabla \cdot \vec{E} = {\rho \over \varepsilon_0} \\
|
||||
\nabla \cdot \vec{B} = 0
|
||||
\end{rcases} \text{Gauss}\]
|
||||
|
||||
\[\begin{rcases}
|
||||
\nabla \times \vec{E} = -{\partial \over \partial t} \vec{B}
|
||||
\end{rcases} \text{Faraday}\]
|
||||
|
||||
\[\begin{rcases}
|
||||
\nabla \times B = \mu_0 \vec{J} + \mu_0 \varepsilon_0{\partial \over \partial t}\vec{E}
|
||||
\end{rcases} \text{Ampere}\]
|
||||
|
||||
Polarization of a uniform plane wave describes the shape and olcus of
|
||||
the tip of the \(\vec{E}\)-field vector in the plane orthoganal to the
|
||||
direction of propagation. There are two pairs of \(\vec{E}\) and
|
||||
\(\vec{B}\) fields (\(\hat{E}_x\), \(\hat{H}_y\)) and (\(\hat{E}_y\),
|
||||
\(\hat{H}_x\)). When both pairs are present, we can evaluate
|
||||
polarization of plane waves. The \(\vec{E}\)-field has components in the
|
||||
\(\hat{x}\) and \(\hat{y}\) directions and travels in \(\hat{z}\).
|
||||
|
||||
\[\hat{E} = (\hat{E}_x \hat{x} + \hat{E}_y \hat{y})e^{-j \beta z}\]
|
||||
\[\hat{E}_x = \lvert \hat{E}_{x_0} \rvert e^{-j \beta a}\]
|
||||
\[\hat{E}_y = \lvert \hat{E}_{y_0} \rvert e^{-j \beta b}\]
|
||||
|
||||
\hypertarget{locus}{%
|
||||
\paragraph{Locus}\label{locus}}
|
||||
|
||||
The shape the tip of the \(\vec{E}\)-field vector traces out while in
|
||||
motion.
|
||||
|
||||
\hypertarget{phase}{%
|
||||
\paragraph{Phase}\label{phase}}
|
||||
|
||||
Typically defined relative to a reference point such as \(z=0\) or
|
||||
\(t=0\) or some combination.
|
||||
|
||||
\hypertarget{polarization-characteristics}{%
|
||||
\subsubsection{Polarization
|
||||
Characteristics}\label{polarization-characteristics}}
|
||||
|
||||
\hypertarget{linear-polarization}{%
|
||||
\paragraph{Linear polarization}\label{linear-polarization}}
|
||||
|
||||
\(\hat{E}_x\) and \(\hat{E}_y\) have the same phase angle: \(a = b\), so
|
||||
the x and y components of the \(\vec{E}\)-field will be in phase.
|
||||
|
||||
\[\hat{E} = (\lvert \hat{E}_x \rvert \hat{x} + \lvert \hat{E}_y \rvert \hat{y} ) e^{-j( \beta z - a)}\]
|
||||
|
||||
In real time:
|
||||
\[\hat{E} = (\lvert \hat{E}_x \rvert \hat{x} + \lvert \hat{E}_y \rvert \hat{y} ) \cos(\omega t - \beta z + a)\]
|
||||
|
||||
As the wave continues to propagate in the \(\hat{z}\) direction, the
|
||||
\(\vec{E}\)-field vector maintains its direction with angle, \(\theta\),
|
||||
with respect to the y-axis.
|
||||
\[\tan(\theta) = {\lvert \hat{E}_x \rvert \over \vert \hat{E}_y \rvert}\]
|
||||
|
||||
When \(z=0\), the \(\vec{E}\) field is given by:
|
||||
\[\vec{E}(0, t) = (\lvert \hat{E}_x \rvert \hat{x} + \lvert \hat{E}_y \rvert \hat{y} ) \cos(\omega t + a)\]
|
||||
\[\vec{E}(0, 0) = (\lvert \hat{E}_x \rvert \hat{x} + \lvert \hat{E}_y \rvert \hat{y} ) \cos(a)\]
|
||||
|
||||
\hypertarget{elliptical-polarization}{%
|
||||
\paragraph{Elliptical Polarization}\label{elliptical-polarization}}
|
||||
|
||||
\(\hat{E}_x\) and \(\hat{E}_y\) have different phase angles. \(\vec{E}\)
|
||||
is no longer in one plate.
|
||||
|
||||
\[\hat{E} = \hat{x} \lvert \hat{E}_x \rvert e^{j (a - \beta z)} + \hat{y} \lvert \hat{E}_y \rvert e^{j(b - \beta z)}\]
|
||||
|
||||
Where:
|
||||
|
||||
\(E_x = \lvert \hat{E}_x \rvert \cos(\omega t - a + \beta z)\)
|
||||
|
||||
\(E_y = \lvert \hat{E}_y \rvert \cos(\omega t - b + \beta z)\)
|
||||
|
||||
If \(a=0\) and \(b = {\pi \over 2}\):
|
||||
\[\vec{E}_x(z,t) = \lvert \hat{E}_x \rvert \cos(\omega t - a + \beta z)\]
|
||||
\[\vec{E}_y(z,t) = \lvert \hat{E}_y \rvert \cos(\omega t - b + \beta z)\]
|
||||
|
||||
\hypertarget{circular-polarization}{%
|
||||
\paragraph{Circular Polarization}\label{circular-polarization}}
|
||||
|
||||
\(\hat{E}_x\) and \(\hat{E}_y\) have the same magnitude with a phase
|
||||
angle difference of \({\pi \over 2}\).
|
||||
|
||||
\hypertarget{example}{%
|
||||
\subsubsection{Example}\label{example}}
|
||||
|
||||
Determine the real-valued \(\vec{E}\)-field.
|
||||
\[\hat{E}(z) = -3j\hat{x} e^{-j\beta z}\]
|
||||
\[\vec{E}(z, t) = -3j\hat{x} e^{-j \beta z} e^{j\omega t}\]
|
||||
\[3 \hat{x} e^{-j \beta z} e^{j \omega t} e^{-\pi \over 2}\]
|
||||
\[3 \cos\left(\omega t- \beta z - {\pi \over 2}\right)\]
|
||||
|
||||
\hypertarget{example-1}{%
|
||||
\subsubsection{Example}\label{example-1}}
|
||||
|
||||
What are \(E_x\) and \(E_y\)
|
||||
\[\hat{E}(z) = (3\hat{x} + 4\hat{y})e^{j\beta z}\]
|
||||
\[\vec{E}(z, t) = (3\hat{x} + 4\hat{y})e^{j\beta z}e^{j\omega t}\]
|
||||
\[E_x = 3\cos(\omega t + \beta z)\] \[E_y = 4\cos(\omega t + \beta z)\]
|
||||
|
||||
\hypertarget{example-2}{%
|
||||
\subsubsection{Example}\label{example-2}}
|
||||
|
||||
\[\hat{E}(z) = (-4\hat{x} + 3\hat{y})e^{-j\beta z}\]
|
||||
\[\hat{E}(z, t) = (-4\hat{x} + 3\hat{y})e^{-j\beta z}e^{j\omega t}\]
|
||||
\[E_x = 4\cos(\omega t - \beta z + \pi)\]
|
||||
\[E_y = 3\cos(\omega t - \beta z)\]
|
||||
|
||||
\hypertarget{non-sinusoidal-waves}{%
|
||||
\subsection{Non-Sinusoidal Waves}\label{non-sinusoidal-waves}}
|
||||
|
||||
Analytical solution of a 1-D traveling wave.
|
||||
\[{\partial^2 \over \partial z^2}\vec{E} - {1\over c^2}{\partial^2 \over \partial t^2}\vec{E} = 0\]
|
||||
|
||||
\hypertarget{dalemberts-solution}{%
|
||||
\subsubsection{D'Alemberts Solution}\label{dalemberts-solution}}
|
||||
|
||||
\[\vec{E}(z, t) = E(z - ct) + E'(z + ct)\]
|
||||
|
||||
Show that the function, \(F(z - ct) = F_0 e^{-(z+ct)^2}\) is a solution
|
||||
of the wave equation.
|
||||
|
||||
Let \(\gamma = z - ct\), and \({\partial \gamma \over \partial z} = 1\):
|
||||
\[F(z-ct) = F_0 e^{-\gamma^2}\] \[F'(z + ct) = 0\]
|
||||
\[{\partial F \over \partial z} = {\partial F \over \partial \gamma} {\partial \gamma \over \partial z} = F_0 e^{-\gamma^2}(-2\gamma)\]
|
||||
\[{\partial^2 F \over \partial z^2} = {\partial \over \partial z}\left({\partial F \over \partial \gamma}{\partial \gamma \over \partial z}\right)\]
|
||||
\[G = \left({\partial F \over \partial \gamma}{\partial \gamma \over \partial z}\right)\]
|
||||
|
||||
Don't forget chain rule:
|
||||
\[{\partial G \over \partial \gamma}{\partial \gamma \over \partial z} = {\partial \over \partial z}\left({\partial F \over \partial \gamma}{\partial \gamma \over \partial z}\right){\partial \gamma \over \partial z}\]
|
||||
|
||||
\[{\partial \over \partial \gamma} - 2\gamma F_0 e^{-\gamma^2} = -2F_0 {\partial \over \partial \gamma} \gamma e^{-\gamma^2}\]
|
||||
|
||||
By product rule:
|
||||
\[-2F_0 (\gamma (-2\gamma e^{-\gamma^2}) + e^{-\gamma^2})\]
|
||||
|
||||
\[F_0 (4\gamma^2 e^{-\gamma^2} - 2e^{-\gamma^2}) = F_0(-2 + 4\gamma^2)e^{-\gamma^2}\]
|
||||
\[={\partial^2 \over \partial \gamma^2}F \left({\partial \gamma \over \partial z}\right)^2\]
|
||||
\[\therefore {\partial^2 \over \partial t^2}F = {\partial^2 F \over \partial \gamma^2} \left({\partial \gamma \over \partial t}\right)^2 = F_0(-2 + 4\gamma^2)e^{-\gamma^2}(C^2)\]
|
||||
|
||||
This solution satisfies:
|
||||
\[{\partial^2 \over \partial z^2}\vec{E} - {1\over c^2}{\partial^2 \over \partial t^2}\vec{E} = 0\]
|
||||
151
5th-Semester-Fall-2023/EEMAGS/Notes/EEMAGS-Notes.aux
Normal file
151
5th-Semester-Fall-2023/EEMAGS/Notes/EEMAGS-Notes.aux
Normal file
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309
5th-Semester-Fall-2023/EEMAGS/Notes/EEMAGS-Notes.fdb_latexmk
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(rewritten before read)
|
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import numpy as np
|
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from matplotlib import pyplot as plt
|
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from copy import deepcopy
|
||||
|
||||
tau = 0.25 # Size of the time step
|
||||
N = 4 # Number of grid points
|
||||
c = 1 # Normalized speed of light
|
||||
L = 1 # Size of the system
|
||||
h = L / N # Spacing of grids (0.25)
|
||||
hct = h / (c*tau) # 1 here
|
||||
coef = 1 / 2*(hct) # 1/2 here
|
||||
sig = 1 # Width of the pulse
|
||||
|
||||
z = np.arange(-L/2 + h/2, L/2 + h/2, h)
|
||||
|
||||
phi = [0, 1, 1, 0]
|
||||
|
||||
phinew = [0]*N
|
||||
|
||||
steps = 5
|
||||
for i in range(steps):
|
||||
for k in range(N):
|
||||
if k != 0 and k != N-1:
|
||||
phinew[k] = (0.5 - coef)*phi[k+1] + (0.5 + coef)*phi[k-1]
|
||||
else:
|
||||
phinew[0] = (0.5 - coef)*phi[1] + (0.5 + coef)*phi[N-1]
|
||||
phinew[N-1] = (0.5 - coef)*phi[0] + (0.5 + coef)*phi[N-2]
|
||||
phi = deepcopy(phinew)
|
||||
print(phi)
|
||||
BIN
5th-Semester-Fall-2023/EEMAGS/Notes/TransmissionLine.png
Normal file
BIN
5th-Semester-Fall-2023/EEMAGS/Notes/TransmissionLine.png
Normal file
Binary file not shown.
|
After Width: | Height: | Size: 4.3 KiB |
178
5th-Semester-Fall-2023/EEMAGS/Notes/letterfonts.tex
Normal file
178
5th-Semester-Fall-2023/EEMAGS/Notes/letterfonts.tex
Normal file
@ -0,0 +1,178 @@
|
||||
% Things Lie
|
||||
\newcommand{\kb}{\mathfrak b}
|
||||
\newcommand{\kg}{\mathfrak g}
|
||||
\newcommand{\kh}{\mathfrak h}
|
||||
\newcommand{\kn}{\mathfrak n}
|
||||
\newcommand{\ku}{\mathfrak u}
|
||||
\newcommand{\kz}{\mathfrak z}
|
||||
\DeclareMathOperator{\Ext}{Ext} % Ext functor
|
||||
\DeclareMathOperator{\Tor}{Tor} % Tor functor
|
||||
\newcommand{\gl}{\opname{\mathfrak{gl}}} % frak gl group
|
||||
\renewcommand{\sl}{\opname{\mathfrak{sl}}} % frak sl group chktex 6
|
||||
|
||||
% More script letters etc.
|
||||
\newcommand{\SA}{\mathcal A}
|
||||
\newcommand{\SB}{\mathcal B}
|
||||
\newcommand{\SC}{\mathcal C}
|
||||
\newcommand{\SF}{\mathcal F}
|
||||
\newcommand{\SG}{\mathcal G}
|
||||
\newcommand{\SH}{\mathcal H}
|
||||
\newcommand{\OO}{\mathcal O}
|
||||
|
||||
\newcommand{\SCA}{\mathscr A}
|
||||
\newcommand{\SCB}{\mathscr B}
|
||||
\newcommand{\SCC}{\mathscr C}
|
||||
\newcommand{\SCD}{\mathscr D}
|
||||
\newcommand{\SCE}{\mathscr E}
|
||||
\newcommand{\SCF}{\mathscr F}
|
||||
\newcommand{\SCG}{\mathscr G}
|
||||
\newcommand{\SCH}{\mathscr H}
|
||||
|
||||
% Mathfrak primes
|
||||
\newcommand{\km}{\mathfrak m}
|
||||
\newcommand{\kp}{\mathfrak p}
|
||||
\newcommand{\kq}{\mathfrak q}
|
||||
|
||||
% number sets
|
||||
\newcommand{\RR}[1][]{\ensuremath{\ifstrempty{#1}{\mathbb{R}}{\mathbb{R}^{#1}}}}
|
||||
\newcommand{\NN}[1][]{\ensuremath{\ifstrempty{#1}{\mathbb{N}}{\mathbb{N}^{#1}}}}
|
||||
\newcommand{\ZZ}[1][]{\ensuremath{\ifstrempty{#1}{\mathbb{Z}}{\mathbb{Z}^{#1}}}}
|
||||
\newcommand{\QQ}[1][]{\ensuremath{\ifstrempty{#1}{\mathbb{Q}}{\mathbb{Q}^{#1}}}}
|
||||
\newcommand{\CC}[1][]{\ensuremath{\ifstrempty{#1}{\mathbb{C}}{\mathbb{C}^{#1}}}}
|
||||
\newcommand{\PP}[1][]{\ensuremath{\ifstrempty{#1}{\mathbb{P}}{\mathbb{P}^{#1}}}}
|
||||
\newcommand{\HH}[1][]{\ensuremath{\ifstrempty{#1}{\mathbb{H}}{\mathbb{H}^{#1}}}}
|
||||
\newcommand{\FF}[1][]{\ensuremath{\ifstrempty{#1}{\mathbb{F}}{\mathbb{F}^{#1}}}}
|
||||
% expected value
|
||||
\newcommand{\EE}{\ensuremath{\mathbb{E}}}
|
||||
\newcommand{\charin}{\text{ char }}
|
||||
\DeclareMathOperator{\sign}{sign}
|
||||
\DeclareMathOperator{\Aut}{Aut}
|
||||
\DeclareMathOperator{\Inn}{Inn}
|
||||
\DeclareMathOperator{\Syl}{Syl}
|
||||
\DeclareMathOperator{\Gal}{Gal}
|
||||
\DeclareMathOperator{\GL}{GL} % General linear group
|
||||
\DeclareMathOperator{\SL}{SL} % Special linear group
|
||||
|
||||
%---------------------------------------
|
||||
% BlackBoard Math Fonts :-
|
||||
%---------------------------------------
|
||||
|
||||
%Captital Letters
|
||||
\newcommand{\bbA}{\mathbb{A}} \newcommand{\bbB}{\mathbb{B}}
|
||||
\newcommand{\bbC}{\mathbb{C}} \newcommand{\bbD}{\mathbb{D}}
|
||||
\newcommand{\bbE}{\mathbb{E}} \newcommand{\bbF}{\mathbb{F}}
|
||||
\newcommand{\bbG}{\mathbb{G}} \newcommand{\bbH}{\mathbb{H}}
|
||||
\newcommand{\bbI}{\mathbb{I}} \newcommand{\bbJ}{\mathbb{J}}
|
||||
\newcommand{\bbK}{\mathbb{K}} \newcommand{\bbL}{\mathbb{L}}
|
||||
\newcommand{\bbM}{\mathbb{M}} \newcommand{\bbN}{\mathbb{N}}
|
||||
\newcommand{\bbO}{\mathbb{O}} \newcommand{\bbP}{\mathbb{P}}
|
||||
\newcommand{\bbQ}{\mathbb{Q}} \newcommand{\bbR}{\mathbb{R}}
|
||||
\newcommand{\bbS}{\mathbb{S}} \newcommand{\bbT}{\mathbb{T}}
|
||||
\newcommand{\bbU}{\mathbb{U}} \newcommand{\bbV}{\mathbb{V}}
|
||||
\newcommand{\bbW}{\mathbb{W}} \newcommand{\bbX}{\mathbb{X}}
|
||||
\newcommand{\bbY}{\mathbb{Y}} \newcommand{\bbZ}{\mathbb{Z}}
|
||||
|
||||
%---------------------------------------
|
||||
% MathCal Fonts :-
|
||||
%---------------------------------------
|
||||
|
||||
%Captital Letters
|
||||
\newcommand{\mcA}{\mathcal{A}} \newcommand{\mcB}{\mathcal{B}}
|
||||
\newcommand{\mcC}{\mathcal{C}} \newcommand{\mcD}{\mathcal{D}}
|
||||
\newcommand{\mcE}{\mathcal{E}} \newcommand{\mcF}{\mathcal{F}}
|
||||
\newcommand{\mcG}{\mathcal{G}} \newcommand{\mcH}{\mathcal{H}}
|
||||
\newcommand{\mcI}{\mathcal{I}} \newcommand{\mcJ}{\mathcal{J}}
|
||||
\newcommand{\mcK}{\mathcal{K}} \newcommand{\mcL}{\mathcal{L}}
|
||||
\newcommand{\mcM}{\mathcal{M}} \newcommand{\mcN}{\mathcal{N}}
|
||||
\newcommand{\mcO}{\mathcal{O}} \newcommand{\mcP}{\mathcal{P}}
|
||||
\newcommand{\mcQ}{\mathcal{Q}} \newcommand{\mcR}{\mathcal{R}}
|
||||
\newcommand{\mcS}{\mathcal{S}} \newcommand{\mcT}{\mathcal{T}}
|
||||
\newcommand{\mcU}{\mathcal{U}} \newcommand{\mcV}{\mathcal{V}}
|
||||
\newcommand{\mcW}{\mathcal{W}} \newcommand{\mcX}{\mathcal{X}}
|
||||
\newcommand{\mcY}{\mathcal{Y}} \newcommand{\mcZ}{\mathcal{Z}}
|
||||
|
||||
|
||||
%---------------------------------------
|
||||
% Bold Math Fonts :-
|
||||
%---------------------------------------
|
||||
|
||||
%Captital Letters
|
||||
\newcommand{\bmA}{\boldsymbol{A}} \newcommand{\bmB}{\boldsymbol{B}}
|
||||
\newcommand{\bmC}{\boldsymbol{C}} \newcommand{\bmD}{\boldsymbol{D}}
|
||||
\newcommand{\bmE}{\boldsymbol{E}} \newcommand{\bmF}{\boldsymbol{F}}
|
||||
\newcommand{\bmG}{\boldsymbol{G}} \newcommand{\bmH}{\boldsymbol{H}}
|
||||
\newcommand{\bmI}{\boldsymbol{I}} \newcommand{\bmJ}{\boldsymbol{J}}
|
||||
\newcommand{\bmK}{\boldsymbol{K}} \newcommand{\bmL}{\boldsymbol{L}}
|
||||
\newcommand{\bmM}{\boldsymbol{M}} \newcommand{\bmN}{\boldsymbol{N}}
|
||||
\newcommand{\bmO}{\boldsymbol{O}} \newcommand{\bmP}{\boldsymbol{P}}
|
||||
\newcommand{\bmQ}{\boldsymbol{Q}} \newcommand{\bmR}{\boldsymbol{R}}
|
||||
\newcommand{\bmS}{\boldsymbol{S}} \newcommand{\bmT}{\boldsymbol{T}}
|
||||
\newcommand{\bmU}{\boldsymbol{U}} \newcommand{\bmV}{\boldsymbol{V}}
|
||||
\newcommand{\bmW}{\boldsymbol{W}} \newcommand{\bmX}{\boldsymbol{X}}
|
||||
\newcommand{\bmY}{\boldsymbol{Y}} \newcommand{\bmZ}{\boldsymbol{Z}}
|
||||
%Small Letters
|
||||
\newcommand{\bma}{\boldsymbol{a}} \newcommand{\bmb}{\boldsymbol{b}}
|
||||
\newcommand{\bmc}{\boldsymbol{c}} \newcommand{\bmd}{\boldsymbol{d}}
|
||||
\newcommand{\bme}{\boldsymbol{e}} \newcommand{\bmf}{\boldsymbol{f}}
|
||||
\newcommand{\bmg}{\boldsymbol{g}} \newcommand{\bmh}{\boldsymbol{h}}
|
||||
\newcommand{\bmi}{\boldsymbol{i}} \newcommand{\bmj}{\boldsymbol{j}}
|
||||
\newcommand{\bmk}{\boldsymbol{k}} \newcommand{\bml}{\boldsymbol{l}}
|
||||
\newcommand{\bmm}{\boldsymbol{m}} \newcommand{\bmn}{\boldsymbol{n}}
|
||||
\newcommand{\bmo}{\boldsymbol{o}} \newcommand{\bmp}{\boldsymbol{p}}
|
||||
\newcommand{\bmq}{\boldsymbol{q}} \newcommand{\bmr}{\boldsymbol{r}}
|
||||
\newcommand{\bms}{\boldsymbol{s}} \newcommand{\bmt}{\boldsymbol{t}}
|
||||
\newcommand{\bmu}{\boldsymbol{u}} \newcommand{\bmv}{\boldsymbol{v}}
|
||||
\newcommand{\bmw}{\boldsymbol{w}} \newcommand{\bmx}{\boldsymbol{x}}
|
||||
\newcommand{\bmy}{\boldsymbol{y}} \newcommand{\bmz}{\boldsymbol{z}}
|
||||
|
||||
%---------------------------------------
|
||||
% Scr Math Fonts :-
|
||||
%---------------------------------------
|
||||
|
||||
\newcommand{\sA}{{\mathscr{A}}} \newcommand{\sB}{{\mathscr{B}}}
|
||||
\newcommand{\sC}{{\mathscr{C}}} \newcommand{\sD}{{\mathscr{D}}}
|
||||
\newcommand{\sE}{{\mathscr{E}}} \newcommand{\sF}{{\mathscr{F}}}
|
||||
\newcommand{\sG}{{\mathscr{G}}} \newcommand{\sH}{{\mathscr{H}}}
|
||||
\newcommand{\sI}{{\mathscr{I}}} \newcommand{\sJ}{{\mathscr{J}}}
|
||||
\newcommand{\sK}{{\mathscr{K}}} \newcommand{\sL}{{\mathscr{L}}}
|
||||
\newcommand{\sM}{{\mathscr{M}}} \newcommand{\sN}{{\mathscr{N}}}
|
||||
\newcommand{\sO}{{\mathscr{O}}} \newcommand{\sP}{{\mathscr{P}}}
|
||||
\newcommand{\sQ}{{\mathscr{Q}}} \newcommand{\sR}{{\mathscr{R}}}
|
||||
\newcommand{\sS}{{\mathscr{S}}} \newcommand{\sT}{{\mathscr{T}}}
|
||||
\newcommand{\sU}{{\mathscr{U}}} \newcommand{\sV}{{\mathscr{V}}}
|
||||
\newcommand{\sW}{{\mathscr{W}}} \newcommand{\sX}{{\mathscr{X}}}
|
||||
\newcommand{\sY}{{\mathscr{Y}}} \newcommand{\sZ}{{\mathscr{Z}}}
|
||||
|
||||
|
||||
%---------------------------------------
|
||||
% Math Fraktur Font
|
||||
%---------------------------------------
|
||||
|
||||
%Captital Letters
|
||||
\newcommand{\mfA}{\mathfrak{A}} \newcommand{\mfB}{\mathfrak{B}}
|
||||
\newcommand{\mfC}{\mathfrak{C}} \newcommand{\mfD}{\mathfrak{D}}
|
||||
\newcommand{\mfE}{\mathfrak{E}} \newcommand{\mfF}{\mathfrak{F}}
|
||||
\newcommand{\mfG}{\mathfrak{G}} \newcommand{\mfH}{\mathfrak{H}}
|
||||
\newcommand{\mfI}{\mathfrak{I}} \newcommand{\mfJ}{\mathfrak{J}}
|
||||
\newcommand{\mfK}{\mathfrak{K}} \newcommand{\mfL}{\mathfrak{L}}
|
||||
\newcommand{\mfM}{\mathfrak{M}} \newcommand{\mfN}{\mathfrak{N}}
|
||||
\newcommand{\mfO}{\mathfrak{O}} \newcommand{\mfP}{\mathfrak{P}}
|
||||
\newcommand{\mfQ}{\mathfrak{Q}} \newcommand{\mfR}{\mathfrak{R}}
|
||||
\newcommand{\mfS}{\mathfrak{S}} \newcommand{\mfT}{\mathfrak{T}}
|
||||
\newcommand{\mfU}{\mathfrak{U}} \newcommand{\mfV}{\mathfrak{V}}
|
||||
\newcommand{\mfW}{\mathfrak{W}} \newcommand{\mfX}{\mathfrak{X}}
|
||||
\newcommand{\mfY}{\mathfrak{Y}} \newcommand{\mfZ}{\mathfrak{Z}}
|
||||
%Small Letters
|
||||
\newcommand{\mfa}{\mathfrak{a}} \newcommand{\mfb}{\mathfrak{b}}
|
||||
\newcommand{\mfc}{\mathfrak{c}} \newcommand{\mfd}{\mathfrak{d}}
|
||||
\newcommand{\mfe}{\mathfrak{e}} \newcommand{\mff}{\mathfrak{f}}
|
||||
\newcommand{\mfg}{\mathfrak{g}} \newcommand{\mfh}{\mathfrak{h}}
|
||||
\newcommand{\mfi}{\mathfrak{i}} \newcommand{\mfj}{\mathfrak{j}}
|
||||
\newcommand{\mfk}{\mathfrak{k}} \newcommand{\mfl}{\mathfrak{l}}
|
||||
\newcommand{\mfm}{\mathfrak{m}} \newcommand{\mfn}{\mathfrak{n}}
|
||||
\newcommand{\mfo}{\mathfrak{o}} \newcommand{\mfp}{\mathfrak{p}}
|
||||
\newcommand{\mfq}{\mathfrak{q}} \newcommand{\mfr}{\mathfrak{r}}
|
||||
\newcommand{\mfs}{\mathfrak{s}} \newcommand{\mft}{\mathfrak{t}}
|
||||
\newcommand{\mfu}{\mathfrak{u}} \newcommand{\mfv}{\mathfrak{v}}
|
||||
\newcommand{\mfw}{\mathfrak{w}} \newcommand{\mfx}{\mathfrak{x}}
|
||||
\newcommand{\mfy}{\mathfrak{y}} \newcommand{\mfz}{\mathfrak{z}}
|
||||
101
5th-Semester-Fall-2023/EEMAGS/Notes/macros.tex
Normal file
101
5th-Semester-Fall-2023/EEMAGS/Notes/macros.tex
Normal file
@ -0,0 +1,101 @@
|
||||
%From M275 "Topology" at SJSU
|
||||
\newcommand{\id}{\mathrm{id}}
|
||||
\newcommand{\taking}[1]{\xrightarrow{#1}}
|
||||
\newcommand{\inv}{^{-1}}
|
||||
|
||||
%From M170 "Introduction to Graph Theory" at SJSU
|
||||
\DeclareMathOperator{\diam}{diam}
|
||||
\DeclareMathOperator{\ord}{ord}
|
||||
\newcommand{\defeq}{\overset{\mathrm{def}}{=}}
|
||||
|
||||
%From the USAMO .tex files
|
||||
\newcommand{\ts}{\textsuperscript}
|
||||
\newcommand{\dg}{^\circ}
|
||||
\newcommand{\ii}{\item}
|
||||
|
||||
% % From Math 55 and Math 145 at Harvard
|
||||
% \newenvironment{subproof}[1][Proof]{%
|
||||
% \begin{proof}[#1] \renewcommand{\qedsymbol}{$\blacksquare$}}%
|
||||
% {\end{proof}}
|
||||
|
||||
\newcommand{\liff}{\leftrightarrow}
|
||||
\newcommand{\lthen}{\rightarrow}
|
||||
\newcommand{\opname}{\operatorname}
|
||||
\newcommand{\surjto}{\twoheadrightarrow}
|
||||
\newcommand{\injto}{\hookrightarrow}
|
||||
\newcommand{\On}{\mathrm{On}} % ordinals
|
||||
\DeclareMathOperator{\img}{im} % Image
|
||||
\DeclareMathOperator{\Img}{Im} % Image
|
||||
\DeclareMathOperator{\coker}{coker} % Cokernel
|
||||
\DeclareMathOperator{\Coker}{Coker} % Cokernel
|
||||
\DeclareMathOperator{\Ker}{Ker} % Kernel
|
||||
\DeclareMathOperator{\rank}{rank}
|
||||
\DeclareMathOperator{\Spec}{Spec} % spectrum
|
||||
\DeclareMathOperator{\Tr}{Tr} % trace
|
||||
\DeclareMathOperator{\pr}{pr} % projection
|
||||
\DeclareMathOperator{\ext}{ext} % extension
|
||||
\DeclareMathOperator{\pred}{pred} % predecessor
|
||||
\DeclareMathOperator{\dom}{dom} % domain
|
||||
\DeclareMathOperator{\ran}{ran} % range
|
||||
\DeclareMathOperator{\Hom}{Hom} % homomorphism
|
||||
\DeclareMathOperator{\Mor}{Mor} % morphisms
|
||||
\DeclareMathOperator{\End}{End} % endomorphism
|
||||
|
||||
% Trig stuff
|
||||
\DeclareMathOperator{\sech}{sech}
|
||||
\DeclareMathOperator{\csch}{csch}
|
||||
\DeclareMathOperator{\arcsec}{arcsec}
|
||||
\DeclareMathOperator{\arccot}{arcCot}
|
||||
\DeclareMathOperator{\arccsc}{arcCsc}
|
||||
\DeclareMathOperator{\arccosh}{arcCosh}
|
||||
\DeclareMathOperator{\arcsinh}{arcsinh}
|
||||
\DeclareMathOperator{\arctanh}{arctanh}
|
||||
\DeclareMathOperator{\arcsech}{arcsech}
|
||||
\DeclareMathOperator{\arccsch}{arcCsch}
|
||||
\DeclareMathOperator{\arccoth}{arcCoth}
|
||||
|
||||
\newcommand{\eps}{\epsilon}
|
||||
\newcommand{\veps}{\varepsilon}
|
||||
\newcommand{\ol}{\overline}
|
||||
\newcommand{\ul}{\underline}
|
||||
\newcommand{\wt}{\widetilde}
|
||||
\newcommand{\wh}{\widehat}
|
||||
\newcommand{\vocab}[1]{\textbf{\color{blue} #1}}
|
||||
\providecommand{\half}{\frac{1}{2}}
|
||||
\newcommand{\dang}{\measuredangle} %% Directed angle
|
||||
\newcommand{\ray}[1]{\overrightarrow{#1}}
|
||||
\newcommand{\seg}[1]{\overline{#1}}
|
||||
\newcommand{\arc}[1]{\wideparen{#1}}
|
||||
\DeclareMathOperator{\cis}{cis}
|
||||
\DeclareMathOperator*{\lcm}{lcm}
|
||||
\DeclareMathOperator*{\argmin}{arg min}
|
||||
\DeclareMathOperator*{\argmax}{arg max}
|
||||
\newcommand{\cycsum}{\sum_{\mathrm{cyc}}}
|
||||
\newcommand{\symsum}{\sum_{\mathrm{sym}}}
|
||||
\newcommand{\cycprod}{\prod_{\mathrm{cyc}}}
|
||||
\newcommand{\symprod}{\prod_{\mathrm{sym}}}
|
||||
\newcommand{\Qed}{\begin{flushright}\qed\end{flushright}}
|
||||
\newcommand{\parinn}{\setlength{\parindent}{1cm}}
|
||||
\newcommand{\parinf}{\setlength{\parindent}{0cm}}
|
||||
% \newcommand{\norm}{\|\cdot\|}
|
||||
\newcommand{\inorm}{\norm_{\infty}}
|
||||
\newcommand{\opensets}{\{V_{\alpha}\}_{\alpha\in I}}
|
||||
\newcommand{\oset}{V_{\alpha}}
|
||||
\newcommand{\opset}[1]{V_{\alpha_{#1}}}
|
||||
\newcommand{\lub}{\text{lub}}
|
||||
\newcommand{\del}[2]{\frac{\partial #1}{\partial #2}}
|
||||
\newcommand{\Del}[3]{\frac{\partial^{#1} #2}{\partial #3^{#1}}}
|
||||
\newcommand{\deld}[2]{\dfrac{\partial #1}{\partial #2}}
|
||||
\newcommand{\Deld}[3]{\dfrac{\partial^{#1} #2}{\partial^{#1} #3}}
|
||||
\newcommand{\lm}{\lambda}
|
||||
\newcommand{\uin}{\mathbin{\rotatebox[origin=c]{90}{$\in$}}}
|
||||
\newcommand{\usubset}{\mathbin{\rotatebox[origin=c]{90}{$\subset$}}}
|
||||
\newcommand{\lt}{\left}
|
||||
\newcommand{\rt}{\right}
|
||||
\newcommand{\bs}[1]{\boldsymbol{#1}}
|
||||
\newcommand{\exs}{\exists}
|
||||
\newcommand{\st}{\strut}
|
||||
\newcommand{\dps}[1]{\displaystyle{#1}}
|
||||
|
||||
\newcommand{\sol}{\setlength{\parindent}{0cm}\textbf{\textit{Solution:}}\setlength{\parindent}{1cm} }
|
||||
\newcommand{\solve}[1]{\setlength{\parindent}{0cm}\textbf{\textit{Solution: }}\setlength{\parindent}{1cm}#1 \Qed}
|
||||
608
5th-Semester-Fall-2023/EEMAGS/Notes/old-chapter-1.tex
Normal file
608
5th-Semester-Fall-2023/EEMAGS/Notes/old-chapter-1.tex
Normal file
@ -0,0 +1,608 @@
|
||||
\hypertarget{vector-analysis}{%
|
||||
\subsection{1.1 Vector Analysis}\label{vector-analysis}}
|
||||
|
||||
\hypertarget{scalar}{%
|
||||
\paragraph{Scalar}\label{scalar}}
|
||||
|
||||
A measure described by one real number. Examples include temperature,
|
||||
size, and mass. A scalar is a \(1 \times 1\) matrix.
|
||||
|
||||
\hypertarget{vector}{%
|
||||
\paragraph{Vector}\label{vector}}
|
||||
|
||||
A measure described by more than one real number (direction and
|
||||
magnitude). Examples include force, velocity, and it's derivatives.
|
||||
Vectors are often described by \(n \times 1\) or \(1 \times n\)
|
||||
matricies.
|
||||
|
||||
\hypertarget{unit-vector}{%
|
||||
\paragraph{Unit Vector}\label{unit-vector}}
|
||||
|
||||
A unit (direction or normalized) vector is signified with a \(\hat{ }\)
|
||||
symbol. Common unit vectors include \(\hat{x}\), \(\hat{y}\), and
|
||||
\(\hat{z}\). The normalized version of any vector is defined as:
|
||||
\[\hat{a} = \frac{\vec{A}}{\lVert\vec{A}\rVert}\]
|
||||
|
||||
\hypertarget{dot-product}{%
|
||||
\paragraph{Dot Product}\label{dot-product}}
|
||||
|
||||
The dot product is a measure of how \emph{parallel} two vectors are,
|
||||
scaled by the magnitudes of the two vectors. To compute it, find the sum
|
||||
of the products of the like components of two vectors. It is also
|
||||
defined as the product of the magnitudes of the vectors normalized by
|
||||
the cosine of the angle between them. It is defined as:
|
||||
\[W = \vec{F} \cdot \vec{r} = \lVert\vec{F}\rVert\lVert\vec{r}\rVert \cos{\alpha}\]
|
||||
|
||||
\hypertarget{cross-product}{%
|
||||
\paragraph{Cross Product}\label{cross-product}}
|
||||
|
||||
The cross product is a measure of how \emph{perpendicular} two vectors
|
||||
are. This operation yeilds a vector quantity \emph{orthoganal} to both
|
||||
original vectors. The direction vector for the cross product is
|
||||
\(\hat{a}_c\), and its magnitude is the product of the magnitudes and
|
||||
the sine of the angle between them. It is defined as:
|
||||
\[\vec{C} = \vec{A} \times \vec{B} = \lVert\vec{A}\rVert \lVert\vec{B}\rVert \sin(\alpha) \hat{a}_c = \begin{vmatrix} \hat{x} & \hat{y} & \hat{z} \\ A_x & A_y & A_z \\ B_x & B_y & B_z \end{vmatrix}\]
|
||||
|
||||
\hypertarget{right-hand-rule}{%
|
||||
\paragraph{Right Hand Rule}\label{right-hand-rule}}
|
||||
|
||||
The right hand rule is a quick way to find \(\hat{a}_c\).
|
||||
\[\overrightarrow{thumb} = \overrightarrow{pointer} \times \overrightarrow{middle}\]
|
||||
|
||||
\hypertarget{cartesian-to-cylindrical}{%
|
||||
\paragraph{Cartesian to Cylindrical}\label{cartesian-to-cylindrical}}
|
||||
|
||||
\[\vec{A} = 4\hat{x} + 4\hat{y} - 2\hat{z}\] \[r = \sqrt{x^2 + y^2}\]
|
||||
\[\varphi = \arctan\left(\frac{y}{x}\right)\] \[z = z\]
|
||||
\[\vec{A} = A_\rho \hat{\rho} + A_\varphi \hat{\varphi} + A_z \hat{z}\]
|
||||
\[A_\rho = A_x \cos(\varphi) + A_y\sin(\varphi) = 4\sqrt{2}\]
|
||||
\[A_\varphi = -A_x\sin(\varphi) + A_y\cos(\varphi) = 0\]
|
||||
\[A_z = A_z = -2\]
|
||||
|
||||
\hypertarget{vector-calculus}{%
|
||||
\subsection{1.2 Vector Calculus}\label{vector-calculus}}
|
||||
|
||||
The differential along some path, \(d\vec{l}\), is defined as:
|
||||
\[d\vec{l} = dx\hat{x} + dy\hat{y} + dz\hat{z} = d\vec{x} + d\vec{y} + d\vec{z}\]
|
||||
|
||||
\hypertarget{the-del-leftnablaright-operator}{%
|
||||
\paragraph{\texorpdfstring{The ``Del'' \(\left(\nabla\right)\)
|
||||
operator}{The ``Del'' \textbackslash left(\textbackslash nabla\textbackslash right) operator}}\label{the-del-leftnablaright-operator}}
|
||||
|
||||
The gradient of the scalar field \(\left(\nabla f\right)\).
|
||||
\[\nabla = \frac{\partial}{\partial x} \hat{x} + \frac{\partial}{\partial y} \hat{y} + \frac{\partial}{\partial z} \hat{z}\]
|
||||
\[df = \nabla f \cdot d\vec{l} = \left(\frac{\partial f}{\partial x}\right)dx + \left(\frac{\partial f}{\partial y}\right)dy + \left(\frac{\partial f}{\partial z}\right)dz\]
|
||||
|
||||
\hypertarget{directional-derivative}{%
|
||||
\subsubsection{Directional Derivative:}\label{directional-derivative}}
|
||||
|
||||
The \emph{directional derivative} is used to find the change of a
|
||||
function along some infinatesimal direction and is defined as:
|
||||
\[\Delta \varphi = {\nabla}_l \varphi \cdot \Delta \vec{l}\]
|
||||
\[\therefore {\nabla}_l \varphi = \frac{\Delta \varphi}{\delta \vec{l}} = \frac{d \varphi}{d \vec{l}} = \nabla \varphi \cdot \frac{\vec{l}}{\lVert\vec{l}\rVert}\]
|
||||
|
||||
\hypertarget{example-1.1}{%
|
||||
\paragraph{Example 1.1}\label{example-1.1}}
|
||||
|
||||
A function, \(f(x,y,z) = x^2 y^2 + xyz\). Find \(\nabla f\).
|
||||
\[f(x,y,z) = f(r)\] \[r = \sqrt{x^2 + y^2 + z^2}\]
|
||||
\[\nabla f = (2xy^2)\hat{x} + (2x^2y + xz)\hat{y} + xy\hat{z}\]
|
||||
|
||||
\hypertarget{example-1.2}{%
|
||||
\paragraph{Example 1.2}\label{example-1.2}}
|
||||
|
||||
Consider a function, \(w = x^2y^2 + xyz\). Find \(\nabla_l w\) at
|
||||
\((2, -1, 0)\) in the direction,
|
||||
\(\vec{l} = 3\hat{x} + 4\hat{y} + 12\hat{z}\).
|
||||
\[\lVert\vec{l}\rVert = \sqrt{3^2 + 4^2 + 12^2} = 13\]
|
||||
\[{\nabla}_l w = \nabla w \cdot \frac{\vec{l}}{\lVert\vec{l}\rVert}\]
|
||||
Solving at \((2, -1, 0)\):
|
||||
\[\nabla w = 2(2)(-1)^2\hat{x} + 2(2)^2(-1)\hat{y} + 2(-1)\hat{z} = 4\hat{x} - 8\hat{y} - 2\hat{z}\]
|
||||
\[{\nabla}_l w= (4\hat{x} - 8\hat{y} - 2\hat{z}) \cdot \left(\frac{3}{13}\hat{x} + \frac{4}{13}\hat{y} + \frac{12}{13}\hat{z}\right) = \frac{-44}{13}\]
|
||||
|
||||
\hypertarget{divergence-and-curl}{%
|
||||
\subsubsection{Divergence and Curl}\label{divergence-and-curl}}
|
||||
|
||||
\hypertarget{divergence-of-a-vector-field}{%
|
||||
\paragraph{Divergence of a Vector
|
||||
Field}\label{divergence-of-a-vector-field}}
|
||||
|
||||
The divergence of a vector field is a measure of outward flux. It is
|
||||
defined as:
|
||||
\[\nabla \cdot f = \left( \frac{\partial}{\partial x} + \frac{\partial}{\partial y} + \frac{\partial}{\partial z} \right) \cdot (f_x + f_y + f_z) = \frac{\partial f_x}{\partial x} + \frac{\partial f_y}{\partial y} + \frac{\partial f_z}{\partial z}\]
|
||||
|
||||
If \(\nabla \cdot \vec{A} = 0\), there is no divergence.
|
||||
|
||||
\hypertarget{curl-of-a-vector-field}{%
|
||||
\paragraph{Curl of a Vector Field}\label{curl-of-a-vector-field}}
|
||||
|
||||
The curl of a vector field is a measure of circulation in each
|
||||
infinatesimally small region of the field. It is defined as:
|
||||
\[\nabla \times f = \begin{vmatrix} \hat{x} & \hat{y} & \hat{z} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ f_x & f_y & f_z \\ \end{vmatrix} = \left< \frac{\partial f_z}{\partial y} - \frac{\partial f_y}{\partial z}, \frac{\partial f_x}{\partial z} - \frac{\partial f_z}{\partial x}, \frac{\partial f_y}{\partial z} - \frac{\partial f_x}{\partial z} \right>\]
|
||||
|
||||
\hypertarget{solenoidal-field}{%
|
||||
\paragraph{Solenoidal Field}\label{solenoidal-field}}
|
||||
|
||||
A solenoidal field is a vector field without divergence, defined as:
|
||||
\[\nabla \cdot \vec{f} = 0\]
|
||||
|
||||
\hypertarget{conservative-rotational-field}{%
|
||||
\paragraph{Conservative / Rotational
|
||||
Field}\label{conservative-rotational-field}}
|
||||
|
||||
A conservative field is a vector field without curl, defined as:
|
||||
\[\nabla \times \vec{f} = 0\]
|
||||
|
||||
\hypertarget{example-1.3}{%
|
||||
\paragraph{Example 1.3}\label{example-1.3}}
|
||||
|
||||
Consider the vector field, \(\vec{F} = k \hat{x}\), where both the
|
||||
direction and magnitude are uniform in all space.
|
||||
\[\nabla \cdot \vec{F} = 0\] \[\nabla \times \vec{F} = 0\]
|
||||
|
||||
\hypertarget{example-1.4}{%
|
||||
\paragraph{Example 1.4}\label{example-1.4}}
|
||||
|
||||
Consider the vector field, \(\vec{F} = k \hat{r}\), where magnitude is
|
||||
constant and direction is away from a central point.
|
||||
\[\hat{r} = \sqrt{\hat{x}^2 + \hat{y}^2 + \hat{z}^2}\]
|
||||
\[\nabla \cdot \vec{F} = \sqrt{3} k\] \[\nabla \times \vec{F} = 0\]
|
||||
|
||||
\hypertarget{example-1.5}{%
|
||||
\paragraph{Example 1.5}\label{example-1.5}}
|
||||
|
||||
Consider the vector field, \(\vec{F} = k \times \hat{r}\), where
|
||||
magnitude is uniform and the direction is perpendicular to the distance
|
||||
from a central point for all space. \[\nabla \cdot \vec{F} = 0\]
|
||||
\[\nabla \times \vec{F} = 2k\]
|
||||
|
||||
\hypertarget{curl-and-divergence-identities}{%
|
||||
\subsubsection{Curl and Divergence
|
||||
Identities}\label{curl-and-divergence-identities}}
|
||||
|
||||
\begin{enumerate}
|
||||
\def\labelenumi{\arabic{enumi}.}
|
||||
\tightlist
|
||||
\item
|
||||
The Laplacian: can operate on a scalar or vector field
|
||||
\[\nabla \cdot (\nabla f) = \nabla^2 f\]
|
||||
\[\frac{\partial^2 f}{\partial x} + \frac{\partial^2 f}{\partial y} + \frac{\partial^2 f}{\partial z}\]
|
||||
\item
|
||||
The curl of a gradient is \(0\) \[\nabla \times (\nabla f) = 0\]
|
||||
\item
|
||||
The gradient of divergence is a scalar
|
||||
\[\nabla (\nabla \cdot \vec{f})\]
|
||||
\item
|
||||
The divergence of curl is \(0\)
|
||||
\[\nabla \cdot (\nabla \times \vec{v}) = 0\]
|
||||
\item
|
||||
Curl of curl
|
||||
\[\nabla \times (\nabla \times \vec{A}) = \nabla(\nabla \cdot \vec{A}) - \nabla^2 \vec{A}\]
|
||||
\end{enumerate}
|
||||
|
||||
\hypertarget{the-line-integral}{%
|
||||
\subsubsection{The Line Integral}\label{the-line-integral}}
|
||||
|
||||
The line integral is the integral of the tangential component of a
|
||||
vector field along a path.
|
||||
|
||||
The line integral is defined as:
|
||||
\[\int \vec{A} \cdot d\vec{l} = \int \lVert \vec{A} \rVert \cos(\alpha) \lVert d\vec{l} \rVert\]
|
||||
|
||||
Where:
|
||||
|
||||
\begin{itemize}
|
||||
\item
|
||||
\(\vec{A}\) is some vector field
|
||||
\item
|
||||
\(d\vec{l} = dx\hat{x} + dy\hat{y} + dz\hat{z}\)
|
||||
\end{itemize}
|
||||
|
||||
If the path of the integral is a closed curve, it is said to be the
|
||||
circulation of \(\vec{A}\) around \(\vec{l}\), defined as:
|
||||
|
||||
\[\oint_c \vec{A} \cdot d\vec{l}\]
|
||||
|
||||
\hypertarget{example-1.6}{%
|
||||
\paragraph{Example 1.6}\label{example-1.6}}
|
||||
|
||||
Calculate the circulation of \(\vec{F}\) around the path.
|
||||
|
||||
\(\vec{F} = x^2 \hat{x} - xy\hat{y} - y^2\hat{z}\)
|
||||
|
||||
Circulation of \(\vec{F}\) around the path:
|
||||
\[\oint \vec{F} \cdot d \vec{l} = \int_1 + \int_2 + \int_3 + \int_4\]
|
||||
|
||||
\hypertarget{path-1}{%
|
||||
\subparagraph{Path 1}\label{path-1}}
|
||||
|
||||
Straight line from \((1,0,0)\) to \((0,0,0)\)
|
||||
|
||||
\(x\) varies, \(z=0\), \(y=0\).
|
||||
|
||||
Plug into \(\vec{F}\):
|
||||
\[\vec{F}_1 = x^2\hat{x} - x(0)\hat{y} - (0)^2\hat{z} = x^2\hat{x}\]
|
||||
|
||||
Since \(x\) varies at some rate, \(dx\) exists, and since \(y\) and
|
||||
\(z\) are constant, \(dy\) and \(dz\) are both \(0\).
|
||||
|
||||
Plug into \(d\vec{l}\):
|
||||
\[d\vec{l} = dx\hat{x} + (0)\hat{y} + (0)\hat{z} = dx\hat{x}\]
|
||||
\[\int \vec{F}_1 \cdot d\vec{l}_1 = \int x^2\hat{x} \cdot dx\hat{x} = \int x^2dx\]
|
||||
|
||||
For this specific path:
|
||||
\[\int_{1}^{0} x^2dx = \left. \frac{x^3}{3} \right\vert_1^0 = 0 - \frac{1}{3} = -\frac{1}{3}\]
|
||||
|
||||
\hypertarget{path-2}{%
|
||||
\subparagraph{Path 2}\label{path-2}}
|
||||
|
||||
Straight line from \((0,0,0)\) to \((0,1,0)\)
|
||||
|
||||
\(y\) varies, \(x=0\), \(z=0\)
|
||||
|
||||
Plug into \(\vec{F}\):
|
||||
\[\vec{F}_2 = (0)^2\hat{x} - (0)y\hat{y} - y^2\hat{z} = -y^2\hat{z}\]
|
||||
|
||||
Since \(y\) varies at some rate, \(dy\) exists, but since \(x\) and
|
||||
\(z\) are constant, \(dx\) and \(dz\) are both \(0\).
|
||||
|
||||
Plug into \(d\vec{l}\):
|
||||
\[d\vec{l}_2 = (0)\hat{x} + dy\hat{y} + (0)\hat{z} = dy\hat{y}\]
|
||||
\[\int \vec{F}_2 \cdot d\vec{l} = \int -y^2\hat{z} \cdot dy\hat{y} = \int\limits_0^1 0 = 0\]
|
||||
|
||||
\hypertarget{path-3}{%
|
||||
\subparagraph{Path 3}\label{path-3}}
|
||||
|
||||
Straight line from \((0,1,0)\) to \((1,1,1)\)
|
||||
|
||||
\(x\) and \(z\) vary at the same rate and always have the same value,
|
||||
\(y=1\)
|
||||
|
||||
\hypertarget{surface-integrals}{%
|
||||
\subsubsection{Surface Integrals}\label{surface-integrals}}
|
||||
|
||||
\(\hat{n}\) is the unit normal vector of a surface.
|
||||
|
||||
The total \emph{flux} crossing an area, \(\Delta s\), is given by the
|
||||
function:
|
||||
\[\Delta s \left[ \lVert \vec{F} \rVert \cos(\alpha)\right] = \vec{F} \cdot \hat{n} \Delta s\]
|
||||
|
||||
Total flux is defined as:
|
||||
\[\sum_{i=1}^N \vec{F}_i \cos(\alpha_i) \Delta s_i = \int_s \vec{F} \cdot d\vec{s}\]
|
||||
|
||||
\hypertarget{gradient-theorem}{%
|
||||
\paragraph{Gradient Theorem}\label{gradient-theorem}}
|
||||
|
||||
The line integral through a gradient field is the difference of the
|
||||
values a the end points.
|
||||
|
||||
\[\int\limits_{r_1}^{r_2} \nabla \varphi \cdot d\vec{l} = \varphi(r_2) - \varphi(r_1)\]
|
||||
|
||||
\[\oint \nabla \varphi \cdot d\vec{l} = 0\]
|
||||
|
||||
\hypertarget{divergence-theorem}{%
|
||||
\paragraph{Divergence Theorem}\label{divergence-theorem}}
|
||||
|
||||
The divergence in some volume is the same as the flux through its
|
||||
surface.
|
||||
|
||||
\[\iiint_{vol} \nabla \cdot \vec{A} d\tau = \oiint \vec{A} \cdot d\vec{s}\]
|
||||
|
||||
\hypertarget{stokes-theorem}{%
|
||||
\paragraph{Stokes Theorem}\label{stokes-theorem}}
|
||||
|
||||
The curl in some region is the same as the circulation of the region's
|
||||
border.
|
||||
\[\iint_A (\nabla \times \vec{A}) \cdot d\vec{s} = \oint \vec{A} \cdot d\vec{l}\]
|
||||
|
||||
\hypertarget{coulombs-law}{%
|
||||
\subsection{1.3 Coulomb's Law}\label{coulombs-law}}
|
||||
|
||||
Initial observation: \[\vec{F} \propto q_1 q_2\]
|
||||
|
||||
Vacuum Permittivity: \[\varepsilon_0 = 8.854 \times 10^{-12}\]
|
||||
|
||||
Coulomb's Constant:
|
||||
\[k = \frac{1}{4 \pi \varepsilon_0} = 9 \times 10^{-9}\]
|
||||
|
||||
Coulomb's Law: \[\vec{F} = k \frac{q_1 q_2}{r^2} \hat{a}_{1 2}\]
|
||||
|
||||
Superposition of Coulomb's Law:
|
||||
\[\vec{F}_{Net} = \sum_{i=1}^{N} k_i \frac{Q q_i}{r_i^2} \hat{r}\]
|
||||
|
||||
\textbf{Note}: \(k_i\) depends on material properties. When in a vacuum,
|
||||
\(k_i = k\).
|
||||
|
||||
\hypertarget{example-1.7}{%
|
||||
\paragraph{Example 1.7}\label{example-1.7}}
|
||||
|
||||
Two point charges, \(q_1\) and \(q_2\) are spaced 2cm apart on the
|
||||
x-axis. A third charge, \(q_3\) is placed between the first two with a
|
||||
distance \(x_1\) between it and \(q_1\) and \(x_2\) between it and
|
||||
\(q_2\) such that \(q_3\) is in static equilibrium.
|
||||
|
||||
Known: \[\vec{F}_{1 3} + \vec{F}_{2 3} = 0\] By Coulomb's Law:
|
||||
\[\vec{F}_{1 3} = \frac{k_1 q_3 q_1}{r_{1 3}^2} \hat{z}\]
|
||||
\[\vec{F}_{2 3} = -\frac{k_2 q_3 q_2}{r_{1 3}^2} \hat{z}\]
|
||||
\[\therefore \vec{F}_{1 3} + \vec{F}_{2 3} = k q_3 \left( \frac{q_1}{r_{1 3}^2} - \frac{q_2}{r_{2 3}^2} \right)\hat{z} = 0\]
|
||||
\[\therefore \frac{q_1}{r_{1 3}^2} - \frac{q_2}{r_{2 3}^2} = 0\]
|
||||
|
||||
Solve for \(r_{1 3}\): \[r_{1 3} = \pm r_{2 3}\sqrt{\frac{q_1}{q_2}}\]
|
||||
|
||||
Known: \[r_{1 3} + r_{2 3} = 2\] \[\therefore r_{2 3} = 2 - r_{1 3}\]
|
||||
|
||||
Plug into first equation:
|
||||
\[r_{1 3} = \pm (2 - r_{1 3}) \sqrt{\frac{q_1}{q_2}}\]
|
||||
|
||||
Expand:
|
||||
\[r_{1 3} = \pm \left(2 \sqrt{\frac{q_1}{q_2}} - r_{1 3}\sqrt{\frac{q_1}{q_2}} \right)\]
|
||||
\[\therefore r_{1 3} \left(1 \pm \sqrt{\frac{q_1}{q_2}} \right) = \pm 2\sqrt{\frac{q_1}{q_2}}\]
|
||||
\[\therefore r_{1 3} = \pm 2\sqrt{\frac{q_1}{q_2}} \left( 1 \pm \sqrt{\frac{q_1}{q_2}} \right)^{-1}\]
|
||||
|
||||
\hypertarget{electric-field-intensity}{%
|
||||
\subsubsection{Electric Field
|
||||
Intensity}\label{electric-field-intensity}}
|
||||
|
||||
Call a test charge at the point of measurement \(Q_2\).
|
||||
|
||||
Coulomb's Law:
|
||||
\[\vec{F}_{1 2} = \frac{Q_1 Q_2}{4 \pi \varepsilon_0 R^2} \hat{a}_{1 2}\]
|
||||
|
||||
The electric field intensisty:
|
||||
\[\vec{E}_{1 2} = \frac{\vec{F}_{1 2}}{Q_1}\]
|
||||
\[\vec{E} = \frac{Q_1}{4 \pi \varepsilon_0 r^2} \hat{a}_{1 2}\]
|
||||
\[\vec{E}_{NET} = \sum_{i=0}^{N} \frac{k_i Q_i}{R_i^2} \hat{a}_{R_i}\]
|
||||
|
||||
Force due to electric field: \[\vec{F} = q \vec{E}\]
|
||||
|
||||
\hypertarget{electric-field}{%
|
||||
\paragraph{Electric field}\label{electric-field}}
|
||||
|
||||
Always in the same direction as the electric field force.
|
||||
|
||||
\hypertarget{electric-field-of-a-dipole}{%
|
||||
\paragraph{Electric field of a
|
||||
dipole}\label{electric-field-of-a-dipole}}
|
||||
|
||||
At a point equidistant to each pole:
|
||||
\[\lVert \vec{r}_1 \rVert = \lVert \vec{r_2} \rVert = r\]
|
||||
|
||||
By Coulomb's Law: \[\vec{E}_1 = \frac{kq}{r^2} \hat{r}_1\]
|
||||
\[\vec{E}_2 = \frac{kq}{r^2} \hat{r}_2\]
|
||||
\[\therefore \lVert \vec{E}_1 \rVert = \lVert \vec{E}_2 \rVert\]
|
||||
|
||||
In terms of the component distances: \[r^2 = a^2 + y^2\]
|
||||
\[\therefore E = \frac{kq}{a^2 + y^2}\]
|
||||
|
||||
\[\vec{E}_{NET_y} = \vec{E}_{1_y} + \vec{E}_{2_y} = 2E\cos{\theta}\]
|
||||
|
||||
By definition: \[\cos(\theta) = \frac{a}{r}\]
|
||||
\[\therefore \vec{E}_{NET_y} = \frac{2kq}{a^2 + y^2} \frac{a}{\sqrt{a^2 + y^2}}\hat{y}\]
|
||||
\[\vec{E}_{NET_y} = 2 \frac{kqa}{(x^2 + y^2)^{3/2}} \hat{y}\]
|
||||
|
||||
If \(y \gg a\) (far field): \[\vec{E}_{NET} = 2 \frac{kqa}{y^3}\]
|
||||
|
||||
\textbf{Takeaway}: \[\vec{E}_{monopole} \propto \frac{1}{r^2}\]
|
||||
\[\vec{E}_{dipole} \propto \frac{1}{r^3}\]
|
||||
|
||||
\hypertarget{charge-densities}{%
|
||||
\paragraph{Charge Densities}\label{charge-densities}}
|
||||
|
||||
\(\lambda\) - Linear charge density
|
||||
|
||||
\(\sigma\) - Surface charge density
|
||||
|
||||
\(\rho\) - Volume charge density
|
||||
|
||||
Electric flux density: \[\vec{D} = \varepsilon_0 \vec{E}\]
|
||||
|
||||
Where:
|
||||
\[\vec{E} = \lim_{\Delta q \to 0} \frac{k \sum_i \Delta q_i}{\lVert \vec{r}_i \rVert ^2} \hat{r}_i = \int \frac{k}{\lVert \vec{r}_i \rVert ^2} dq\]
|
||||
|
||||
\hypertarget{example-1.8}{%
|
||||
\paragraph{Example 1.8}\label{example-1.8}}
|
||||
|
||||
A straight line segment of length \(L\) with uniform charge density
|
||||
\(\lambda\) extends from the origin in the \(\hat{x}\) direction. Find
|
||||
the strength of the electric field, \(\vec{E}\) at some arbitrary point,
|
||||
\(p\) along the ray from the origin in the \(\hat{z}\) direction.
|
||||
|
||||
The distance along the line segment in the \(\hat{x}\) direction is
|
||||
denoted as \(x\), and the distance from the origin to point \(p\) is
|
||||
denoted as \(z\). The vector, \(\vec{r}\) has length
|
||||
\(\sqrt{x^2 + z^2}\) and makes an angle \(\theta\) with the ray in the
|
||||
\(-\hat{z}\) direction.
|
||||
|
||||
The contribution \(d\vec{E}\) to the total electric field, \(\vec{E}\),
|
||||
at point \(p\) is defined as:
|
||||
\[d\vec{E} = \frac{k dq}{\lVert \vec{r} \rVert ^2} \hat{r}\]
|
||||
|
||||
For a linear charge distribution: \[dq = \lambda dl\] The distance,
|
||||
\(r\), to each \(x\) alond the line:
|
||||
\[\lVert \vec{r} \rVert ^2 = x^2 + z^2\] The components of the
|
||||
\(\vec{E}\)-field at point \(p\):
|
||||
\[d\vec{E}_x = \lVert d\vec{E} \rVert \sin(\theta) \hat{x}\]
|
||||
\[d\vec{E}_z = \lVert d\vec{E} \rVert \cos(\theta) \hat{z}\]
|
||||
|
||||
\hypertarget{gausss-law}{%
|
||||
\subsection{Gauss's Law}\label{gausss-law}}
|
||||
|
||||
\hypertarget{e-flux-density}{%
|
||||
\subsection{E-Flux Density}\label{e-flux-density}}
|
||||
|
||||
\hypertarget{emf}{%
|
||||
\subsection{EMF}\label{emf}}
|
||||
|
||||
Measured in \emph{volts}, electromotive force (EMF), is denoted by
|
||||
\(\mathcal{E}\). The value for EMF is defined as:
|
||||
\[\mathcal{E} = \oint \vec{E} \cdot d \vec{l} = -\frac{d}{dt} \int_s B_z(t) \cdot ds = -\frac{d}{dt}\psi_m\]
|
||||
|
||||
Where:
|
||||
|
||||
\[\psi_m = \int_s \vec{B} \cdot d\vec{s}\]
|
||||
|
||||
A perfectly conducting ring with radius, \(\rho_0\), centered on the
|
||||
origin in the x-y plane.
|
||||
|
||||
The charge distribution: \[\rho = \rho_0 + \rho_0 \sin(\omega t)\]
|
||||
\[\vec{B}(t) = B_0 \cos(\omega t)\hat{z}\]
|
||||
\[\mathcal{E} = \oint \vec{E} \cdot d\vec{l} = \iint\limits_{\phi R} B_0 \cos(\omega t) dr d\phi\]
|
||||
|
||||
Where:
|
||||
|
||||
\(\varphi: [0, 2\pi]\)
|
||||
|
||||
\(R: [0, \rho(t)]\)
|
||||
|
||||
Therefore: \[\psi_m = (B_0 \cos(\omega t) \hat{z})(2\pi \rho(t))\]
|
||||
|
||||
\[\mathcal{E} = -\frac{d}{dt} B_0 2\pi \cos(\omega t)(\rho_0 + \rho_0\sin(\omega t))\]
|
||||
|
||||
\hypertarget{filling-in-some-gaps}{%
|
||||
\subsubsection{Filling in some Gaps}\label{filling-in-some-gaps}}
|
||||
|
||||
\[\vec{F} = -\nabla \vec{u}\] Where: \(\vec{u}\) is potential.
|
||||
|
||||
\[\vec{E} = -\nabla \vec{v}\] Where: \(\vec{v}\) is electric potential.
|
||||
|
||||
\[W = \vec{F} \cdot \vec{d} = q\vec{E} \cdot \vec{d}\]
|
||||
|
||||
Work done by the \(\vec{E}\)-field on a charge will reduce the electric
|
||||
potential:
|
||||
|
||||
\[-\Delta u = u_B - u_A = -\Delta W = -qEd\]
|
||||
|
||||
The total change is:
|
||||
|
||||
\[\Delta u = -q \int\limits_A^B \vec{E} \cdot d\vec{l}\]
|
||||
|
||||
\[\therefore \frac{\Delta u}{q} = \int\limits_A^B \vec{E} \cdot d\vec{l} = \Delta V\]
|
||||
|
||||
\[W = q \int \vec{E} \cdot d\vec{l}\]
|
||||
|
||||
\[V(r) = \frac{kq}{r}\]
|
||||
|
||||
For \(N\) discrete charges,
|
||||
\[V = \sum\limits_{i=1}^{N} \frac{k q_i}{r_i}\]
|
||||
|
||||
In cartesian coordinates:
|
||||
|
||||
\[\vec{F} = F_x\hat{i} + F_y\hat{j} + F_z\hat{z}\]
|
||||
|
||||
Where:
|
||||
|
||||
\(F_x = \frac{dw}{dx}\)
|
||||
|
||||
\(F_y = \frac{dw}{dy}\)
|
||||
|
||||
\(F_z = \frac{dw}{dz}\)
|
||||
|
||||
Therefore:
|
||||
\[\vec{F} = \frac{\partial w}{\partial x}\hat{i} + \frac{\partial w}{\partial y}\hat{j} + \frac{\partial w}{\partial z}\hat{k} = \nabla \vec{w}\]
|
||||
|
||||
\[q\vec{E} = \nabla (-\vec{u})\]
|
||||
|
||||
\[\vec{F} = -\nabla \vec{u}\]
|
||||
|
||||
\[\vec{E} = -\nabla \left( \frac{\vec{u}}{q} \right) = -\nabla \vec{V}\]
|
||||
|
||||
\hypertarget{electric-flux-density}{%
|
||||
\paragraph{Electric Flux Density}\label{electric-flux-density}}
|
||||
|
||||
\[\vec{D} = \varepsilon \vec{E}\] Where
|
||||
\(\varepsilon = \varepsilon_r \varepsilon_0\) (permittivity).
|
||||
|
||||
\hypertarget{magnetic-flux-density}{%
|
||||
\paragraph{Magnetic Flux Density}\label{magnetic-flux-density}}
|
||||
|
||||
\[\vec{B} = \mu\vec{H}\] Where \(\mu = \mu_r \mu_0\) (permeability).
|
||||
|
||||
\hypertarget{amperes-law}{%
|
||||
\subsubsection{Ampere's Law}\label{amperes-law}}
|
||||
|
||||
The total current crossing an area, \(s\), that is enclosed by the
|
||||
contour \(C\): \[\oint_C \frac{\vec{B}}{\mu_0} \cdot d\vec{l}\]
|
||||
|
||||
The total current is the sum of the current due to charge flow and the
|
||||
current due to the time rate of change of the electric flux crossing an
|
||||
area, \(s\). Maxwell was able to unify electricity and magnetism by
|
||||
adding the current due to the time rate of change of electric flux.
|
||||
\[\oint_C \vec{H} \cdot d\vec{l} = \int_S \vec{J} \cdot d\vec{s} + \frac{d}{dt}\int \varepsilon_0\vec{E} \cdot d\vec{s}\]
|
||||
|
||||
\hypertarget{simplified-amperes-law}{%
|
||||
\paragraph{Simplified Ampere's Law}\label{simplified-amperes-law}}
|
||||
|
||||
\[\oint_C \vec{H} \cdot d\vec{l} = I = \int_s \vec{J} \cdot d\vec{s}\]
|
||||
|
||||
The charge density, \(J = \rho v\), has units
|
||||
\(\left[ \frac{A}{m^2} \right]\).
|
||||
|
||||
\hypertarget{example}{%
|
||||
\paragraph{Example}\label{example}}
|
||||
|
||||
A current, \(I\), in an infinitely long cylindrical wire with radius,
|
||||
\(R\). \[\int \vec{B} \cdot d\vec{l} = \mu_0 I_{enc}\]
|
||||
|
||||
Measuring the magnetic field at some distance, \(r\), from the center of
|
||||
the conductor: \[B\int dl = \mu_0 I\] Where \(\int dl\) is the
|
||||
circumfrece of measurement. \[B(2\pi r) = \mu_0 I\]
|
||||
\[B_{out} = \frac{\mu_0 I}{2\pi r} \hat{\varphi}\] Inside the wire:
|
||||
\[\int \vec{B} \cdot d\vec{l} = \mu_0 I\]
|
||||
\[B\int dl = \frac{\mu_0 I r^2}{R^2} = 2\pi rB\]
|
||||
\[B = \frac{\mu_0 I}{2\pi R^2} r\]
|
||||
\[\vec{B} = \frac{\mu_0 I}{2\pi R^2}r \hat{\varphi}\]
|
||||
|
||||
\hypertarget{coulombs-law-1}{%
|
||||
\subsubsection{Coulomb's Law}\label{coulombs-law-1}}
|
||||
|
||||
The total displacement flux of charge:
|
||||
\[\int_s \varepsilon_0 \vec{E} \cdot d\vec{s}\]
|
||||
|
||||
The total current (charge with respect to time):
|
||||
\[I = \frac{d}{dt} \int_s \varepsilon_0 \vec{E} \cdot d\vec{s}\]
|
||||
|
||||
\hypertarget{faradays-law}{%
|
||||
\subsubsection{Faraday's Law}\label{faradays-law}}
|
||||
|
||||
Work done in moving a unit positive test charge around a closed path,
|
||||
\(C\): \[\oint_C \vec{E} \cdot d\vec{l}\]
|
||||
|
||||
Magnetic force on a poving charge and is directed perpendicular to both
|
||||
the direction of the motion of the charge and the magnetic field.
|
||||
\[\oint_C \vec{B} \cdot d\vec{l}\]
|
||||
|
||||
\hypertarget{solenoid-ideal}{%
|
||||
\subsubsection{Solenoid (Ideal)}\label{solenoid-ideal}}
|
||||
|
||||
For an ideal solenoid with constant current, \(I\), assume uniform
|
||||
\(\vec{B}\) inside, \(\vec{B} = 0\) outside, and infinite length.
|
||||
|
||||
By Ampere's Law: \[\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{enc}\]
|
||||
|
||||
For a square loop with one side in the solenoid, and it's parallel side
|
||||
outside the loop:
|
||||
\[\oint \vec{B} \cdot d\vec{l} = \int_1 + \int_2 + \int_3 + \int_4\]
|
||||
|
||||
Sides 2 and 4 are parallel, and side 3 is outside the solenoid, so:
|
||||
\[\oint \vec{B} \cdot d\vec{l} = \int_1 = Bl\]
|
||||
|
||||
Back to Ampere's Law: \[Bl = \mu_0 I N\]
|
||||
\[\therefore B = \frac{\mu_0 I N}{l} = \mu_0 I n\] Where:
|
||||
|
||||
\(N\) is the total number of windings,
|
||||
|
||||
\(l\) is the sidelength of the Amperian loop,
|
||||
|
||||
\(n\) is the number of windings per unit length \(\frac{N}{l}\)
|
||||
|
||||
\hypertarget{toroid-ideal}{%
|
||||
\subsubsection{Toroid (Ideal)}\label{toroid-ideal}}
|
||||
|
||||
From a symmetric \(\vec{B}\)-field, lines form concentric circles inside
|
||||
the toroid. For an ideal toroid, assume \(\vec{B} = 0\) outside, and
|
||||
Ampere's law inside.
|
||||
|
||||
\[\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{enc}\] For some circular
|
||||
Amperian loop inside the toroid: \[\vec{B} = B\hat{\varphi}\]
|
||||
\[d\vec{l} = 2\pi r \vec{\varphi}\] \[B(2\pi r) = \mu_0 N I\]
|
||||
\[\therefore B = \frac{\mu_0 N I}{2\pi r}\]
|
||||
778
5th-Semester-Fall-2023/EEMAGS/Notes/preamble.tex
Normal file
778
5th-Semester-Fall-2023/EEMAGS/Notes/preamble.tex
Normal file
@ -0,0 +1,778 @@
|
||||
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|
||||
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|
||||
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|
||||
|
||||
|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
\usepackage{varwidth}
|
||||
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|
||||
%\usepackage{authblk}
|
||||
\usepackage{nameref}
|
||||
\usepackage{multicol,array}
|
||||
\usepackage{tikz-cd}
|
||||
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|
||||
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|
||||
\usepackage{import}
|
||||
\usepackage{xifthen}
|
||||
\usepackage{pdfpages}
|
||||
\usepackage{transparent}
|
||||
|
||||
\newcommand\mycommfont[1]{\footnotesize\ttfamily\textcolor{blue}{#1}}
|
||||
\SetCommentSty{mycommfont}
|
||||
\newcommand{\incfig}[1]{%
|
||||
\def\svgwidth{\columnwidth}
|
||||
\import{./figures/}{#1.pdf_tex}
|
||||
}
|
||||
|
||||
\usepackage{tikzsymbols}
|
||||
\renewcommand\qedsymbol{$\Laughey$}
|
||||
|
||||
|
||||
%\usepackage{import}
|
||||
%\usepackage{xifthen}
|
||||
%\usepackage{pdfpages}
|
||||
%\usepackage{transparent}
|
||||
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
% SELF MADE COLORS
|
||||
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|
||||
|
||||
|
||||
|
||||
\definecolor{myg}{RGB}{56, 140, 70}
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|
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|
||||
|
||||
|
||||
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|
||||
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|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
|
||||
\setlength{\parindent}{1cm}
|
||||
%================================
|
||||
% THEOREM BOX
|
||||
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|
||||
|
||||
\tcbuselibrary{theorems,skins,hooks}
|
||||
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||||
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|
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|
||||
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||||
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||||
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||||
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|
||||
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||||
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||||
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||||
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|
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|
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|
||||
|
||||
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|
||||
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|
||||
%================================
|
||||
|
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|
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|
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||||
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|
||||
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|
||||
%================================
|
||||
|
||||
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|
||||
\newtcbtheorem[number within=section]{Method}{Method}
|
||||
{%
|
||||
enhanced,
|
||||
breakable,
|
||||
colback = mypropbg,
|
||||
frame hidden,
|
||||
boxrule = 0sp,
|
||||
borderline west = {2pt}{0pt}{mypropfr},
|
||||
sharp corners,
|
||||
detach title,
|
||||
before upper = \tcbtitle\par\smallskip,
|
||||
coltitle = mypropfr,
|
||||
fonttitle = \bfseries\sffamily,
|
||||
description font = \mdseries,
|
||||
separator sign none,
|
||||
segmentation style={solid, mypropfr},
|
||||
}
|
||||
{th}
|
||||
|
||||
\tcbuselibrary{theorems,skins,hooks}
|
||||
\newtcbtheorem[number within=chapter]{method}{Method}
|
||||
{%
|
||||
enhanced,
|
||||
breakable,
|
||||
colback = mypropbg,
|
||||
frame hidden,
|
||||
boxrule = 0sp,
|
||||
borderline west = {2pt}{0pt}{mypropfr},
|
||||
sharp corners,
|
||||
detach title,
|
||||
before upper = \tcbtitle\par\smallskip,
|
||||
coltitle = mypropfr,
|
||||
fonttitle = \bfseries\sffamily,
|
||||
description font = \mdseries,
|
||||
separator sign none,
|
||||
segmentation style={solid, mypropfr},
|
||||
}
|
||||
{th}
|
||||
|
||||
|
||||
%================================
|
||||
% CLAIM
|
||||
%================================
|
||||
|
||||
\tcbuselibrary{theorems,skins,hooks}
|
||||
\newtcbtheorem[number within=section]{claim}{Claim}
|
||||
{%
|
||||
enhanced
|
||||
,breakable
|
||||
,colback = myg!10
|
||||
,frame hidden
|
||||
,boxrule = 0sp
|
||||
,borderline west = {2pt}{0pt}{myg}
|
||||
,sharp corners
|
||||
,detach title
|
||||
,before upper = \tcbtitle\par\smallskip
|
||||
,coltitle = myg!85!black
|
||||
,fonttitle = \bfseries\sffamily
|
||||
,description font = \mdseries
|
||||
,separator sign none
|
||||
,segmentation style={solid, myg!85!black}
|
||||
}
|
||||
{th}
|
||||
|
||||
|
||||
|
||||
%================================
|
||||
% Exercise
|
||||
%================================
|
||||
|
||||
\tcbuselibrary{theorems,skins,hooks}
|
||||
\newtcbtheorem[number within=section]{Exercise}{Exercise}
|
||||
{%
|
||||
enhanced,
|
||||
breakable,
|
||||
colback = myexercisebg,
|
||||
frame hidden,
|
||||
boxrule = 0sp,
|
||||
borderline west = {2pt}{0pt}{myexercisefg},
|
||||
sharp corners,
|
||||
detach title,
|
||||
before upper = \tcbtitle\par\smallskip,
|
||||
coltitle = myexercisefg,
|
||||
fonttitle = \bfseries\sffamily,
|
||||
description font = \mdseries,
|
||||
separator sign none,
|
||||
segmentation style={solid, myexercisefg},
|
||||
}
|
||||
{th}
|
||||
|
||||
\tcbuselibrary{theorems,skins,hooks}
|
||||
\newtcbtheorem[number within=chapter]{exercise}{Exercise}
|
||||
{%
|
||||
enhanced,
|
||||
breakable,
|
||||
colback = myexercisebg,
|
||||
frame hidden,
|
||||
boxrule = 0sp,
|
||||
borderline west = {2pt}{0pt}{myexercisefg},
|
||||
sharp corners,
|
||||
detach title,
|
||||
before upper = \tcbtitle\par\smallskip,
|
||||
coltitle = myexercisefg,
|
||||
fonttitle = \bfseries\sffamily,
|
||||
description font = \mdseries,
|
||||
separator sign none,
|
||||
segmentation style={solid, myexercisefg},
|
||||
}
|
||||
{th}
|
||||
|
||||
%================================
|
||||
% EXAMPLE BOX
|
||||
%================================
|
||||
|
||||
\newtcbtheorem[number within=section]{Example}{Example}
|
||||
{%
|
||||
colback = myexamplebg
|
||||
,breakable
|
||||
,colframe = myexamplefr
|
||||
,coltitle = myexampleti
|
||||
,boxrule = 1pt
|
||||
,sharp corners
|
||||
,detach title
|
||||
,before upper=\tcbtitle\par\smallskip
|
||||
,fonttitle = \bfseries
|
||||
,description font = \mdseries
|
||||
,separator sign none
|
||||
,description delimiters parenthesis
|
||||
}
|
||||
{ex}
|
||||
|
||||
\newtcbtheorem[number within=chapter]{example}{Example}
|
||||
{%
|
||||
colback = myexamplebg
|
||||
,breakable
|
||||
,colframe = myexamplefr
|
||||
,coltitle = myexampleti
|
||||
,boxrule = 1pt
|
||||
,sharp corners
|
||||
,detach title
|
||||
,before upper=\tcbtitle\par\smallskip
|
||||
,fonttitle = \bfseries
|
||||
,description font = \mdseries
|
||||
,separator sign none
|
||||
,description delimiters parenthesis
|
||||
}
|
||||
{ex}
|
||||
|
||||
%================================
|
||||
% DEFINITION BOX
|
||||
%================================
|
||||
|
||||
\newtcbtheorem[number within=section]{Definition}{Definition}{enhanced,
|
||||
before skip=2mm,after skip=2mm, colback=red!5,colframe=red!80!black,boxrule=0.5mm,
|
||||
attach boxed title to top left={xshift=1cm,yshift*=1mm-\tcboxedtitleheight}, varwidth boxed title*=-3cm,
|
||||
boxed title style={frame code={
|
||||
\path[fill=tcbcolback]
|
||||
([yshift=-1mm,xshift=-1mm]frame.north west)
|
||||
arc[start angle=0,end angle=180,radius=1mm]
|
||||
([yshift=-1mm,xshift=1mm]frame.north east)
|
||||
arc[start angle=180,end angle=0,radius=1mm];
|
||||
\path[left color=tcbcolback!60!black,right color=tcbcolback!60!black,
|
||||
middle color=tcbcolback!80!black]
|
||||
([xshift=-2mm]frame.north west) -- ([xshift=2mm]frame.north east)
|
||||
[rounded corners=1mm]-- ([xshift=1mm,yshift=-1mm]frame.north east)
|
||||
-- (frame.south east) -- (frame.south west)
|
||||
-- ([xshift=-1mm,yshift=-1mm]frame.north west)
|
||||
[sharp corners]-- cycle;
|
||||
},interior engine=empty,
|
||||
},
|
||||
fonttitle=\bfseries,
|
||||
title={#2},#1}{def}
|
||||
\newtcbtheorem[number within=chapter]{definition}{Definition}{enhanced,
|
||||
before skip=2mm,after skip=2mm, colback=red!5,colframe=red!80!black,boxrule=0.5mm,
|
||||
attach boxed title to top left={xshift=1cm,yshift*=1mm-\tcboxedtitleheight}, varwidth boxed title*=-3cm,
|
||||
boxed title style={frame code={
|
||||
\path[fill=tcbcolback]
|
||||
([yshift=-1mm,xshift=-1mm]frame.north west)
|
||||
arc[start angle=0,end angle=180,radius=1mm]
|
||||
([yshift=-1mm,xshift=1mm]frame.north east)
|
||||
arc[start angle=180,end angle=0,radius=1mm];
|
||||
\path[left color=tcbcolback!60!black,right color=tcbcolback!60!black,
|
||||
middle color=tcbcolback!80!black]
|
||||
([xshift=-2mm]frame.north west) -- ([xshift=2mm]frame.north east)
|
||||
[rounded corners=1mm]-- ([xshift=1mm,yshift=-1mm]frame.north east)
|
||||
-- (frame.south east) -- (frame.south west)
|
||||
-- ([xshift=-1mm,yshift=-1mm]frame.north west)
|
||||
[sharp corners]-- cycle;
|
||||
},interior engine=empty,
|
||||
},
|
||||
fonttitle=\bfseries,
|
||||
title={#2},#1}{def}
|
||||
|
||||
|
||||
|
||||
%================================
|
||||
% Solution BOX
|
||||
%================================
|
||||
|
||||
\makeatletter
|
||||
\newtcbtheorem{question}{Question}{enhanced,
|
||||
breakable,
|
||||
colback=white,
|
||||
colframe=myb!80!black,
|
||||
attach boxed title to top left={yshift*=-\tcboxedtitleheight},
|
||||
fonttitle=\bfseries,
|
||||
title={#2},
|
||||
boxed title size=title,
|
||||
boxed title style={%
|
||||
sharp corners,
|
||||
rounded corners=northwest,
|
||||
colback=tcbcolframe,
|
||||
boxrule=0pt,
|
||||
},
|
||||
underlay boxed title={%
|
||||
\path[fill=tcbcolframe] (title.south west)--(title.south east)
|
||||
to[out=0, in=180] ([xshift=5mm]title.east)--
|
||||
(title.center-|frame.east)
|
||||
[rounded corners=\kvtcb@arc] |-
|
||||
(frame.north) -| cycle;
|
||||
},
|
||||
#1
|
||||
}{def}
|
||||
\makeatother
|
||||
|
||||
%================================
|
||||
% SOLUTION BOX
|
||||
%================================
|
||||
|
||||
\makeatletter
|
||||
\newtcolorbox{solution}{enhanced,
|
||||
breakable,
|
||||
colback=white,
|
||||
colframe=myg!80!black,
|
||||
attach boxed title to top left={yshift*=-\tcboxedtitleheight},
|
||||
title=Solution,
|
||||
boxed title size=title,
|
||||
boxed title style={%
|
||||
sharp corners,
|
||||
rounded corners=northwest,
|
||||
colback=tcbcolframe,
|
||||
boxrule=0pt,
|
||||
},
|
||||
underlay boxed title={%
|
||||
\path[fill=tcbcolframe] (title.south west)--(title.south east)
|
||||
to[out=0, in=180] ([xshift=5mm]title.east)--
|
||||
(title.center-|frame.east)
|
||||
[rounded corners=\kvtcb@arc] |-
|
||||
(frame.north) -| cycle;
|
||||
},
|
||||
}
|
||||
\makeatother
|
||||
|
||||
%================================
|
||||
% Question BOX
|
||||
%================================
|
||||
|
||||
\makeatletter
|
||||
\newtcbtheorem{qstion}{Question}{enhanced,
|
||||
breakable,
|
||||
colback=white,
|
||||
colframe=mygr,
|
||||
attach boxed title to top left={yshift*=-\tcboxedtitleheight},
|
||||
fonttitle=\bfseries,
|
||||
title={#2},
|
||||
boxed title size=title,
|
||||
boxed title style={%
|
||||
sharp corners,
|
||||
rounded corners=northwest,
|
||||
colback=tcbcolframe,
|
||||
boxrule=0pt,
|
||||
},
|
||||
underlay boxed title={%
|
||||
\path[fill=tcbcolframe] (title.south west)--(title.south east)
|
||||
to[out=0, in=180] ([xshift=5mm]title.east)--
|
||||
(title.center-|frame.east)
|
||||
[rounded corners=\kvtcb@arc] |-
|
||||
(frame.north) -| cycle;
|
||||
},
|
||||
#1
|
||||
}{def}
|
||||
\makeatother
|
||||
|
||||
\newtcbtheorem[number within=chapter]{wconc}{Wrong Concept}{
|
||||
breakable,
|
||||
enhanced,
|
||||
colback=white,
|
||||
colframe=myr,
|
||||
arc=0pt,
|
||||
outer arc=0pt,
|
||||
fonttitle=\bfseries\sffamily\large,
|
||||
colbacktitle=myr,
|
||||
attach boxed title to top left={},
|
||||
boxed title style={
|
||||
enhanced,
|
||||
skin=enhancedfirst jigsaw,
|
||||
arc=3pt,
|
||||
bottom=0pt,
|
||||
interior style={fill=myr}
|
||||
},
|
||||
#1
|
||||
}{def}
|
||||
|
||||
|
||||
|
||||
%================================
|
||||
% NOTE BOX
|
||||
%================================
|
||||
|
||||
\usetikzlibrary{arrows,calc,shadows.blur}
|
||||
\tcbuselibrary{skins}
|
||||
\newtcolorbox{note}[1][]{%
|
||||
enhanced jigsaw,
|
||||
colback=gray!20!white,%
|
||||
colframe=gray!80!black,
|
||||
size=small,
|
||||
boxrule=1pt,
|
||||
title=\textbf{Note:},
|
||||
halign title=flush center,
|
||||
coltitle=black,
|
||||
breakable,
|
||||
drop shadow=black!50!white,
|
||||
attach boxed title to top left={xshift=1cm,yshift=-\tcboxedtitleheight/2,yshifttext=-\tcboxedtitleheight/2},
|
||||
minipage boxed title=1.5cm,
|
||||
boxed title style={%
|
||||
colback=white,
|
||||
size=fbox,
|
||||
boxrule=1pt,
|
||||
boxsep=2pt,
|
||||
underlay={%
|
||||
\coordinate (dotA) at ($(interior.west) + (-0.5pt,0)$);
|
||||
\coordinate (dotB) at ($(interior.east) + (0.5pt,0)$);
|
||||
\begin{scope}
|
||||
\clip (interior.north west) rectangle ([xshift=3ex]interior.east);
|
||||
\filldraw [white, blur shadow={shadow opacity=60, shadow yshift=-.75ex}, rounded corners=2pt] (interior.north west) rectangle (interior.south east);
|
||||
\end{scope}
|
||||
\begin{scope}[gray!80!black]
|
||||
\fill (dotA) circle (2pt);
|
||||
\fill (dotB) circle (2pt);
|
||||
\end{scope}
|
||||
},
|
||||
},
|
||||
#1,
|
||||
}
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
% SELF MADE COMMANDS
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
|
||||
|
||||
\newcommand{\thm}[2]{\begin{Theorem}{#1}{}#2\end{Theorem}}
|
||||
\newcommand{\cor}[2]{\begin{Corollary}{#1}{}#2\end{Corollary}}
|
||||
\newcommand{\rvw}[2]{\begin{Review}{#1}{}#2\end{Review}}
|
||||
\newcommand{\mthd}[2]{\begin{Method}{#1}{}#2\end{Method}}
|
||||
\newcommand{\clm}[3]{\begin{claim}{#1}{#2}#3\end{claim}}
|
||||
\newcommand{\wc}[2]{\begin{wconc}{#1}{}\setlength{\parindent}{1cm}#2\end{wconc}}
|
||||
\newcommand{\thmcon}[1]{\begin{Theoremcon}{#1}\end{Theoremcon}}
|
||||
\newcommand{\ex}[2]{\begin{Example}{#1}{#2}\end{Example}}
|
||||
\newcommand{\dfn}[2]{\begin{Definition}[colbacktitle=red!75!black]{#1}{}#2\end{Definition}}
|
||||
\newcommand{\dfnc}[2]{\begin{definition}[colbacktitle=red!75!black]{#1}{}#2\end{definition}}
|
||||
\newcommand{\qs}[2]{\begin{question}{#1}{}#2\end{question}}
|
||||
\newcommand{\pf}[2]{\begin{myproof}[#1]#2\end{myproof}}
|
||||
\newcommand{\nt}[1]{\begin{note}#1\end{note}}
|
||||
|
||||
\newcommand*\circled[1]{\tikz[baseline=(char.base)]{
|
||||
\node[shape=circle,draw,inner sep=1pt] (char) {#1};}}
|
||||
\newcommand\getcurrentref[1]{%
|
||||
\ifnumequal{\value{#1}}{0}
|
||||
{??}
|
||||
{\the\value{#1}}%
|
||||
}
|
||||
\newcommand{\getCurrentSectionNumber}{\getcurrentref{section}}
|
||||
\newenvironment{myproof}[1][\proofname]{%
|
||||
\proof[\bfseries #1: ]%
|
||||
}{\endproof}
|
||||
|
||||
\newcommand{\mclm}[2]{\begin{myclaim}[#1]#2\end{myclaim}}
|
||||
\newenvironment{myclaim}[1][\claimname]{\proof[\bfseries #1: ]}{}
|
||||
|
||||
\newcounter{mylabelcounter}
|
||||
|
||||
\makeatletter
|
||||
\newcommand{\setword}[2]{%
|
||||
\phantomsection
|
||||
#1\def\@currentlabel{\unexpanded{#1}}\label{#2}%
|
||||
}
|
||||
\makeatother
|
||||
|
||||
|
||||
|
||||
|
||||
\tikzset{
|
||||
symbol/.style={
|
||||
draw=none,
|
||||
every to/.append style={
|
||||
edge node={node [sloped, allow upside down, auto=false]{$#1$}}}
|
||||
}
|
||||
}
|
||||
|
||||
|
||||
% deliminators
|
||||
\DeclarePairedDelimiter{\abs}{\lvert}{\rvert}
|
||||
\DeclarePairedDelimiter{\norm}{\lVert}{\rVert}
|
||||
|
||||
\DeclarePairedDelimiter{\ceil}{\lceil}{\rceil}
|
||||
\DeclarePairedDelimiter{\floor}{\lfloor}{\rfloor}
|
||||
\DeclarePairedDelimiter{\round}{\lfloor}{\rceil}
|
||||
|
||||
\newsavebox\diffdbox
|
||||
\newcommand{\slantedromand}{{\mathpalette\makesl{d}}}
|
||||
\newcommand{\makesl}[2]{%
|
||||
\begingroup
|
||||
\sbox{\diffdbox}{$\mathsurround=0pt#1\mathrm{#2}$}%
|
||||
\pdfsave
|
||||
\pdfsetmatrix{1 0 0.2 1}%
|
||||
\rlap{\usebox{\diffdbox}}%
|
||||
\pdfrestore
|
||||
\hskip\wd\diffdbox
|
||||
\endgroup
|
||||
}
|
||||
\newcommand{\dd}[1][]{\ensuremath{\mathop{}\!\ifstrempty{#1}{%
|
||||
\slantedromand\@ifnextchar^{\hspace{0.2ex}}{\hspace{0.1ex}}}%
|
||||
{\slantedromand\hspace{0.2ex}^{#1}}}}
|
||||
\ProvideDocumentCommand\dv{o m g}{%
|
||||
\ensuremath{%
|
||||
\IfValueTF{#3}{%
|
||||
\IfNoValueTF{#1}{%
|
||||
\frac{\dd #2}{\dd #3}%
|
||||
}{%
|
||||
\frac{\dd^{#1} #2}{\dd #3^{#1}}%
|
||||
}%
|
||||
}{%
|
||||
\IfNoValueTF{#1}{%
|
||||
\frac{\dd}{\dd #2}%
|
||||
}{%
|
||||
\frac{\dd^{#1}}{\dd #2^{#1}}%
|
||||
}%
|
||||
}%
|
||||
}%
|
||||
}
|
||||
\providecommand*{\pdv}[3][]{\frac{\partial^{#1}#2}{\partial#3^{#1}}}
|
||||
% - others
|
||||
\DeclareMathOperator{\Lap}{\mathcal{L}}
|
||||
\DeclareMathOperator{\Var}{Var} % varience
|
||||
\DeclareMathOperator{\Cov}{Cov} % covarience
|
||||
\DeclareMathOperator{\E}{E} % expected
|
||||
|
||||
% Since the amsthm package isn't loaded
|
||||
|
||||
% I prefer the slanted \leq
|
||||
\let\oldleq\leq % save them in case they're every wanted
|
||||
\let\oldgeq\geq
|
||||
\renewcommand{\leq}{\leqslant}
|
||||
\renewcommand{\geq}{\geqslant}
|
||||
|
||||
% % redefine matrix env to allow for alignment, use r as default
|
||||
% \renewcommand*\env@matrix[1][r]{\hskip -\arraycolsep
|
||||
% \let\@ifnextchar\new@ifnextchar
|
||||
% \array{*\c@MaxMatrixCols #1}}
|
||||
|
||||
|
||||
%\usepackage{framed}
|
||||
%\usepackage{titletoc}
|
||||
%\usepackage{etoolbox}
|
||||
%\usepackage{lmodern}
|
||||
|
||||
|
||||
%\patchcmd{\tableofcontents}{\contentsname}{\sffamily\contentsname}{}{}
|
||||
|
||||
%\renewenvironment{leftbar}
|
||||
%{\def\FrameCommand{\hspace{6em}%
|
||||
% {\color{myyellow}\vrule width 2pt depth 6pt}\hspace{1em}}%
|
||||
% \MakeFramed{\parshape 1 0cm \dimexpr\textwidth-6em\relax\FrameRestore}\vskip2pt%
|
||||
%}
|
||||
%{\endMakeFramed}
|
||||
|
||||
%\titlecontents{chapter}
|
||||
%[0em]{\vspace*{2\baselineskip}}
|
||||
%{\parbox{4.5em}{%
|
||||
% \hfill\Huge\sffamily\bfseries\color{myred}\thecontentspage}%
|
||||
% \vspace*{-2.3\baselineskip}\leftbar\textsc{\small\chaptername~\thecontentslabel}\\\sffamily}
|
||||
%{}{\endleftbar}
|
||||
%\titlecontents{section}
|
||||
%[8.4em]
|
||||
%{\sffamily\contentslabel{3em}}{}{}
|
||||
%{\hspace{0.5em}\nobreak\itshape\color{myred}\contentspage}
|
||||
%\titlecontents{subsection}
|
||||
%[8.4em]
|
||||
%{\sffamily\contentslabel{3em}}{}{}
|
||||
%{\hspace{0.5em}\nobreak\itshape\color{myred}\contentspage}
|
||||
|
||||
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
% TABLE OF CONTENTS
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
|
||||
\usepackage{tikz}
|
||||
\definecolor{doc}{RGB}{0,60,110}
|
||||
\usepackage{titletoc}
|
||||
\contentsmargin{0cm}
|
||||
\titlecontents{chapter}[3.7pc]
|
||||
{\addvspace{30pt}%
|
||||
\begin{tikzpicture}[remember picture, overlay]%
|
||||
\draw[fill=doc!60,draw=doc!60] (-7,-.1) rectangle (-0.9,.5);%
|
||||
\pgftext[left,x=-3.5cm,y=0.2cm]{\color{white}\Large\sc\bfseries Chapter\ \thecontentslabel};%
|
||||
\end{tikzpicture}\color{doc!60}\large\sc\bfseries}%
|
||||
{}
|
||||
{}
|
||||
{\;\titlerule\;\large\sc\bfseries Page \thecontentspage
|
||||
\begin{tikzpicture}[remember picture, overlay]
|
||||
\draw[fill=doc!60,draw=doc!60] (2pt,0) rectangle (4,0.1pt);
|
||||
\end{tikzpicture}}%
|
||||
\titlecontents{section}[3.7pc]
|
||||
{\addvspace{2pt}}
|
||||
{\contentslabel[\thecontentslabel]{2pc}}
|
||||
{}
|
||||
{\hfill\small \thecontentspage}
|
||||
[]
|
||||
\titlecontents*{subsection}[3.7pc]
|
||||
{\addvspace{-1pt}\small}
|
||||
{}
|
||||
{}
|
||||
{\ --- \small\thecontentspage}
|
||||
[ \textbullet\ ][]
|
||||
|
||||
\makeatletter
|
||||
\renewcommand{\tableofcontents}{%
|
||||
\chapter*{%
|
||||
\vspace*{-20\p@}%
|
||||
\begin{tikzpicture}[remember picture, overlay]%
|
||||
\pgftext[right,x=15cm,y=0.2cm]{\color{doc!60}\Huge\sc\bfseries \contentsname};%
|
||||
\draw[fill=doc!60,draw=doc!60] (13,-.75) rectangle (20,1);%
|
||||
\clip (13,-.75) rectangle (20,1);
|
||||
\pgftext[right,x=15cm,y=0.2cm]{\color{white}\Huge\sc\bfseries \contentsname};%
|
||||
\end{tikzpicture}}%
|
||||
\@starttoc{toc}}
|
||||
\makeatother
|
||||
|
||||
Reference in New Issue
Block a user