5th semester files

5th-Semester-Fall-2023/EEMAGS/EquationSheet/Equation-Sheet.md

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+# EEMAGS Equation Sheet
+
+## Curl and Divergence Identities
+1. The Laplacian: can operate on a scalar or vector field
+$$\nabla \cdot (\nabla f) = \nabla^2 f$$
+$$\frac{\partial^2 f}{\partial x} + \frac{\partial^2 f}{\partial y} + \frac{\partial^2 f}{\partial z}$$
+2. The curl of a gradient is $0$
+$$\nabla \times (\nabla f) = 0$$
+3. The gradient of divergence is a scalar
+$$\nabla (\nabla \cdot \vec{f})$$
+4. The divergence of curl is $0$
+$$\nabla \cdot (\nabla \times \vec{v}) = 0$$
+5. Curl of curl
+$$\nabla \times (\nabla \times \vec{A}) = \nabla(\nabla \cdot \vec{A}) - \nabla^2 \vec{A}$$
+
+## Maxwell's Equations
+Gauss's Law for $\vec{E}$-fields:
+$$\Phi_E =\int \vec{E} \cdot d\vec{s} = \frac{Q_{enc}}{\varepsilon_0}$$
+$$\nabla \cdot \vec{E} = \frac{\rho}{\varepsilon_0}$$
+
+Gauss's Law for $\vec{B}$-fields:
+$$\int \vec{B} \cdot d\vec{s} = 0 $$
+$$ \nabla \cdot \vec{B} = 0$$
+
+Faraday's Law
+$$\int \vec{E} \cdot d\vec{l} = -\frac{d}{dt}\vec{B} $$
+$$ \nabla \times \vec{E} = -\frac{d}{dt}\vec{B}$$
+
+Ampere's Law
+$$\int \vec{B} \cdot d\vec{l} = \mu_0\left( \vec{J} + \varepsilon_0 \frac{d\vec{E}}{dt}\right) = \mu_0 I_{enc}$$
+$$ \nabla \times \vec{B} = \mu_0 J + \mu_0 \varepsilon_0 \frac{\partial \vec{E}}{\partial t}$$
+
+## Work and Voltage
+$$\Delta V = -\int \vec{E} \cdot d\vec{l}$$
+$$-\nabla V = \vec{E}$$