# Homework 2 - Aidan Sharpe
## 1
Consider a long cylindrical wire of radius, $a$, carrying a current $I = I_0 \cos(\omega t)\hat{z}$.

### a)
Write an expression for the magnetic field strength, $B$, outside the wire ($\rho > a$):

By Ampere's Law:
$$\int \vec{B} \cdot d\vec{l} = \mu_0 I_{enc}$$

For a closed loop of radius, $\rho$:
$$\int \vec{B} \cdot d\vec{l} = B(2\pi\rho)$$
$$\therefore B(2\pi\rho) = \mu_0 I_{enc}$$

Since ($\rho > a$):
$$I_{enc} = I = I_0 \cos(\omega t)$$

Recombining:
$$B(2\pi\rho) = \mu_0 I_0 \cos(\omega t)$$
$$\therefore B = {\mu_0 I_0 \cos(\omega t) \over 2\pi\rho} \text{[T]}$$

### b)
Consider a rectangular loop a distance, $d$, from the wire with sidelengths $\alpha$ in the $\hat{x}$ direction, and $\beta$ in the $\hat{z}$ direction.

#### i)
Calculate the magnetic flux, $\Phi_B$, through the loop.

By Gauss's Law for magnetism:
$$\Phi_B = \iint \vec{B} \cdot d\vec{s}$$

Since the $\beta$ does not vary:
$$\Phi_B = \beta \int\limits_{d}^{d+\alpha} {\mu_0 I_0 \cos(\omega t) \over 2\pi\rho} d\rho$$

Taking out constants:
$$\Phi_B = {\beta \mu_0 I_0 \cos(\omega t) \over 2 \pi} \int\limits_d^{d+\alpha} {1 \over \rho}d\rho$$

Evaluate:
$$\Phi_B = {\beta \mu_0 I_0 \cos(\omega t) \over 2\pi} \ln\left( {\lvert d + \alpha \rvert \over \lvert d \rvert} \right) \text{[Vs]}$$

#### ii)
Find the induced EMF, $\cal{E}$:

$$\mathcal{E} = -{d \over dt}\Phi_B$$

Plug in:
$$\mathcal{E} = -{d \over dt} {\beta \mu_0 I_0 \cos(\omega t) \over 2\pi} \ln\left({\lvert d + \alpha \rvert \over \lvert d \rvert}\right)$$

Evaluate:
$$\mathcal{E} = {\beta \mu_0 I_0 \sin(\omega t) \over 2\pi} \ln\left({\lvert d + \alpha \rvert \over \lvert d \rvert}\right) \text{[V]}$$

## 2
A quarter circle loop of wire with inner radius, $a$, and outer radius, $b$, has current, $I$. Find the magnetic field strength at the center of the circle.

By superposition:
$$B = B_1 + B_2 + B_3 + B_4$$

By Ampere's law:
$$\int \vec{B_1} \cdot d\vec{l} = \mu_0 I_{enc_1}$$
$$\int \vec{B_2} \cdot d\vec{l} = \mu_0 I_{enc_2}$$
$$\int \vec{B_3} \cdot d\vec{l} = \mu_0 I_{enc_3}$$
$$\int \vec{B_4} \cdot d\vec{l} = \mu_0 I_{enc_4}$$

Since the current in the first and third segments are either parallel or antiparallel:
$$I_{enc_1} = 0$$
$$I_{enc_3} = 0$$

Since all of $I$ is enclosed in a loop from either segment two or four:
$$I_{enc_2} = I_{enc_4} = I$$

Back to Ampere's Law:
$$\int \vec{B_1} \cdot d\vec{l} = 0 \therefore B_1 = 0$$
$$\int \vec{B_2} \cdot d\vec{l} = \mu_0 I$$
$$\int \vec{B_3} \cdot d\vec{l} = 0 \therefore B_2 = 0$$
$$\int \vec{B_4} \cdot d\vec{l} = \mu_0 I$$

Determining $d\vec{l}$ for segments two and four:
$$\int \vec{B_2} \cdot \vec{a}d\varphi = \mu_0 I$$
$$\int \vec{B_4} \cdot \vec{b}d\varphi = \mu_0 I$$

Determining the bounds:
$$\int\limits_0^{\pi \over 2} \vec{B_2} \cdot \vec{a} d\varphi = \mu_0 I$$
$$\int\limits^0_{\pi \over 2} \vec{B_4} \cdot \vec{b} d\varphi = \mu_0 I$$

Evaluate:
$$B_2 \left({\pi a \over 2}\right) = \mu_0 I$$
$$B_4 \left({-\pi b \over 2}\right) = \mu_0 I$$

Solve for $B$:
$$B_2 = {2 \mu_0 I \over \pi a}$$
$$B_4 = -{2 \mu_0 I \over \pi b}$$
$$B = {2 \mu_0 I \over \pi a} - {2 \mu_0 I \over \pi b} = {2 \mu_0 I (a-b) \over \pi^2 a b} \text{[T]}$$

## 3
Consider a cylinder of radius, $\rho_0 = 0.5\text{[m]}$, with a current density, $\vec{J} = 4.5e^{-2\rho}\hat{z}[A/m^2]$.

By Ampere's Law:
$$\int \vec{B} \cdot d\vec{l} = \mu_0 I_{enc}$$

For a radial current density:
$$I_{enc} = \int\limits_{0}^{2\pi} \int\limits_{0}^{\rho} J sdsd\varphi$$

Plugging in $J$ with $s$ as the integrating variable:
$$I_{enc} = \int\limits_{0}^{2\pi} \int\limits_{0}^{\rho} 4.5e^{-2s} sdsd\varphi$$
$$I_{enc} = 4.5\int\limits_{0}^{2\pi} \int\limits_{0}^{\rho} e^{-2s} sdsd\varphi$$
$$I_{enc} = 4.5\int\limits_{0}^{2\pi} \left[ \left( {-\rho \over 2} - {1 \over 4}\right)e^{-2\rho} + {1 \over 4}\right]d\varphi$$
$$\therefore I_{enc} = {9 \pi e^{-2\rho}(e^{2\rho} -2\rho -1) \over 4}$$

Back to Ampere's Law:
$$\int \vec{B} \cdot d\vec{l} = \mu_0 I_{enc}$$

For a circular loop:
$$\int\limits_{0}^{2\pi} B\rho d\varphi = \mu_0 I_{enc}$$
$$B(2\pi\rho) = {9 \pi e^{-2\rho}(e^{2\rho} -2\rho -1) \over 4}$$

For $\rho \le \rho_0$
$$B = {9 e^{-2\rho}(e^{2\rho} -2\rho -1) \over 8\rho}$$

For $\rho \ge \rho_0$:
$$I_{enc} = 4.5\int\limits_{0}^{2\pi} \int\limits_{0}^{\rho_0} e^{-2s} sdsd\varphi = {9 (e-2) \pi \over 4 e}$$

$$B(2\pi\rho) = {9(e-2)\pi \over 4e}$$
$$\therefore B = {9(e-2) \over 8e\rho}\text{[T]}$$

## 4
Consider a solenoidal wire with $n$ coils per unit length. The core becomes magnetized when a current $I=10[A]$ is put into the wire coil, and this causes a bound current to flow around the cylindrical surface of the core as shown in the side view diagram. This bound core surface current density has magnitude, $J = 20n [A/m]$

$$B = B_s + B_c$$
$$B_s = \mu_0 I n = 10\mu_0n$$
Pretend the core current is just another solenoid:
$$B_c = \mu_0 I n = 20\mu_0n$$
$$B = 30\mu_0 n \text{[T]}$$

## 5
A satellite travelling at $5\text{[km/s]}$ enters a current filled curtain. From $t=1\text{[s]}$ to $t=3\text{[s]}$, the satellite's magnetometer increases from $-95\hat{x}\text{[nT]}$ to $95\hat{x}\text{[nT]}$. If the current flows in the $\hat{z}$ direction, find the current density, $J$.

By Ampere's Law:
$$\int \vec{B} \cdot d\vec{l} = \mu_0 \iint \vec{J} \cdot d\vec{s}$$

The total increase in $B$ is $195 \text{[nT]}$, so:
$$195 \times 10^{-9} = \mu_0 \iint \vec{J} \cdot d\vec{s}$$

Plug in bounds to get the total distance through the curtain:
$$195 \times 10^{-9} = \mu_0 \int\limits_{1}^{3} \int\limits_{0}^{-5000} J dvdt$$
$$195 \times 10^{-9} = -10000J\mu_0$$
$$J = {-195 \times 10^{-13} \over \mu_0} \text{[A/m}]$$