# 2019-2 전기역학 연습문제

## Contents

### 2019. 9. 6

Due by 2019. 9. 16

1. Write down the expressions of gradient, divergence, curl, and Laplacian in cylindrical and spherical coordinates. (Practice this so you can write them without helping hands)
2. Calculate the divergence and curl of following vector field $\mathbf{a}$
• $\mathbf{a}=(x,0,0)$
• $\mathbf{a}=(-y,x,0)$
3. Show that Laplacian of $\Psi(r)$ can be expressed as any of the following forms:
• $\frac1{r^2}\frac{\partial}{\partial r}\left(r^2 \frac{\partial \Psi}{\partial r} \right)$
• $\frac1{r} \frac{\partial^2}{\partial r^2} \left( r\Psi \right)$
4. Prove Green's 2nd identity (shown below) by using divergence theorem.
• $\oint_S \left[\left(\psi\nabla\varphi-\varphi\nabla\psi\right)\cdot\hat{\mathbf{n}}\right]da=\,\!\int_S \left[\psi\frac{\partial\varphi}{\partial n}-\varphi\frac{\partial\psi}{\partial n}\right]da = \int_V\left(\psi\nabla^{2}\varphi-\varphi\nabla^{2}\psi\right) d\tau\,\! \qquad$
5. Mathematically prove the following: ( refer to [1] )
• Curl is solenoidal

### 2019. 9. 17

Due by 23:59, 2019. 9. 25

1. A spherical shell of radius R is uniformly charged with surface charge density $\sigma$. Calculate the potential at all positions by following the two methods described below, Set the potential reference at infinity, i.e., $V(r=\infty)=0$.
• Use Gauss' law to find electric field first, and line-integrate it from the reference point $r=\infty$
• Directly obtain the potential by surface-integration on the spherical surface, without using Gauss' law.
2. Two infinitely long wires running parallel to the x axis carry uniform charge densities $+\lambda$ and $-\lambda$ as shown on your right. Find the potential at any point (x,y,z), using the origin as your reference.
3. We can obtain the potential at any point if we know the charge distribution throughout the whole space, by $V(\mathbf{r})=\frac1{4\pi\epsilon_0}\int_{whole space}\frac{\rho(\mathbf{r'})}{|\mathbf{r}-\mathbf{r'}|}d\tau'$. Prove, then, that it is sufficient to know the charge distribution within the volume surrounded by a closed surface S and the potential $V$ and its normal derivative $\frac{\partial V}{\partial n}$ on the surface S to obtain the potential. (Hint: use Green's 2nd identity and let $\phi$ be the electrostatic potential and let $\psi=1/r$ )

### 2019. 9. 26

1. Multipole expansion
• Obtain the Legendre polynomial $P_0(x), P_1(x), ... , P_5(x)$ by using Rodrigues formula $P_n(x) = {1 \over 2^nn!} {d^n \over dx^n } \left[ (x^2 -1)^n \right].$
• We want to see if $\frac{1}{\sqrt{1-2xt+t^2}} = \sum_{n=0}^\infty P_n(x) t^n$ holds true. Perform Taylor expansion of $\frac{1}{\sqrt{1-2xt+t^2}}$ for the lowest 4 orders in $t$ and verify the coefficients correspond to Legendre polynomial $P_0(x), P_1(x), ... P_3(x)$.
2. Calculate the quadrupole moment, defined as $Q_{ij}=\int_V (3x'_i x'_j-\delta_{ij}r'^2)\rho(\mathbf{r}')d\tau'$, of the following charge distributions.
• Four point charges +q at $(0,d,0)$, -q at $(d,0,0)$, +q at $(0,-d,0)$, -q at $(-d,0,0)$
• Three point charges +q at $(0,d,0)$, -2q at $(0,0,0)$, and +q at $(0,-d,0)$

### 2019. 10. 4

Due by 23:59 on Oct 10

1. An uncharged metal sphere of radius R is placed in an otherwise uniform electric field $\mathbf{E}=E_0 \hat{\mathbf{z}}$. Some polarization will be induced because of the electric field, and the induced charges, in turn, will distort the field around the sphere. Find the potential in the region outside the sphere. Also, obtain the induced charge density on the metal surface. (Hint: Use separation of variables in spherical coordinate)
2. A point charge q is situated at a distance D from the center of a grounded conducting sphere of radius R. Find the potential outside the sphere, and the force acting on the charge q. (Hint: place an image charge inside the conducting sphere)

### 2019. 11. 7

Due by 23:59 on Nov 15

1. Suppose you have two infinite straight line charges $\lambda$, a distance d apart, moving along at a constant speed v as shown below. How great would v have to be in order for the magnetic attraction to balance the electrical repulsion? (Griffiths Prob 13, Magnetostatics)
2. If $\mathbf{B}$ is uniform, show that $\mathbf{A}(\mathbf{r})=\frac12 \mathbf{B}\times\mathbf{r}$ satisties both $\nabla\cdot\mathbf{A}=0$ and $\nabla\times\mathbf{A}=\mathbf{B}$. (Griffiths Prob 25, Magnetostatics)
3. Show that the magnetic field of a dipole can be written in coordinate-free form as below (Griffiths Prob 34, Magnetostatics)
$\mathbf{B}(\mathbf{r})=\frac{\mu_0}{4\pi}\frac1{r^3}[3(\mathbf{m}\cdot\mathbf{\hat{r}})\mathbf{\hat{r}}-\mathbf{m}]$
4. If $\mathbf{J}_f=\mathbf{0}$ everywhere, the curl of $\mathbf{H}$ vanishes, and we can express $\mathbf{H}$ as the gradient of a scalar potential W:
$\mathbf{H}=-\nabla W$
Then, W obeys Poisson's equation as shown below, with $\nabla\cdot\mathbf{M}$ as the source.
$\nabla^2 W = (\nabla\cdot\mathbf{M})$
Now, find the $\mathbf{H}$ field and $\mathbf{B}$ field inside a uniformly magnetized sphere of radius R by separation of variables. (Hint: W satisfies Laplace's equation in the region $r\lt R$ and $r\gt R$. You should figure out the appropriate boundary condition on W at the boundary.)

### 2019. 11. 18

Maxwell Equation의 대칭성 음미하기 (Due by 23:59 on Nov 24)

• Write down the microscopic Maxwell's equation in source-free region, and show that a change of variable ($\mathbf{E}\rightarrow c\mathbf{B}, \mathbf{B}\rightarrow -\mathbf{E}/c$) leads to the same set of equations.
• The dual symmetry described above doesn't hold when the source $\rho$ and $\mathbf{J}$ comes in. In order to recover the symmetry, let us assume we have magnetic charge density $\rho_m$ and magnetic current density $\mathbf{J}_m$ as well as electric charge density $\rho_e$ and electric current density $\mathbf{J}_e$, satistying following set of equations:
 $\nabla \cdot \mathbf{E} = \frac{\rho_e}{\varepsilon_0}$ $\nabla \cdot \mathbf{B} = \mu_0\rho_m$ $\nabla \times \mathbf{E} = -\mu_0\mathbf{J}_m-\frac{\partial \mathbf{B}} {\partial t}$ $\nabla \times \mathbf{B} = \mu_0\mathbf{J}_e + \frac1{c^2}\frac{\partial \mathbf{E}}{\partial t}$
Show that magnetic charge, as well as electric charge, satisfies the continuity equations, i.e.,
 $\nabla\cdot\mathbf{J}_m = -\frac{\partial\rho_m}{\partial t}$ $\nabla\cdot\mathbf{J}_e = -\frac{\partial\rho_e}{\partial t}$
• Show that above Maxwell's equations are invariant under the duality transformation written below:
$\left(\begin{array}{c} \mathbf{E}' \\ c\mathbf{B}'\end{array}\right) = \left(\begin{array}{cc} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{array}\right)\left(\begin{array}{c} \mathbf{E} \\ c\mathbf{B}\end{array}\right)$
$\left(\begin{array}{c} c\rho'_e \\ \rho'_m\end{array}\right) = \left(\begin{array}{cc} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{array}\right)\left(\begin{array}{c} c\rho_e \\ \rho_m \end{array}\right)$
$\left(\begin{array}{c} c\mathbf{J}'_e \\ \mathbf{J}'_m\end{array}\right) = \left(\begin{array}{cc} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{array}\right)\left(\begin{array}{c} c\mathbf{J}_e \\ \mathbf{J}_m\end{array}\right)$
• Show that generalized Lorentz force density given below is also invariant under duality transformation.
$\mathbf{f}=(\rho_e\mathbf{E}+\mathbf{J}_e\times\mathbf{B})+(\rho_m\mathbf{B}-\mathbf{J}_m\times\mathbf{E}/c^2)$

### 2019. 12. 4

Due by 9:00am on Dec 16

• Coaxial cable A long coaxial cable ($l\gg b$) is connected between a battery with electromotive force of $\mathcal{E}$ and a resistor of R. When it reached a steady state, calculate the E field and B field inside the coaxial cable, and find the Poynting vector as a function of position in the crosssection. Discuss the energy flow in this system.
• Fresnel's equations Derive the following equations, starting from the boundary conditions of E field and H field.
$r_s\equiv\left[\frac{E'}{E}\right]_{TE} = \frac{\cos\theta-n\cos\phi}{\cos\theta+n\cos\phi} = \frac{\cos\theta-\sqrt{n^2-\sin^2\theta}}{\cos\theta+\sqrt{n^2-\sin^2\theta}}=-\frac{\sin(\theta-\phi)}{\sin(\theta+\phi)}$
$t_s\equiv\left[\frac{E''}{E}\right]_{TE} = \frac{2\cos\theta\sin\phi}{\sin(\theta+\phi)}$
$r_p\equiv\left[\frac{E'}{E}\right]_{TM} = \frac{-n\cos\theta+\cos\phi}{n\cos\theta+\cos\phi} = \frac{-n^2\cos\theta+\sqrt{n^2-\sin^2\theta}}{n^2\cos\theta+\sqrt{n^2-\sin^2\theta}}=-\frac{\tan(\theta-\phi)}{\tan(\theta+\phi)}$
$t_p\equiv\left[\frac{E''}{E}\right]_{TM} = \frac{2\cos\theta\sin\phi}{\sin(\theta+\phi)\cos(\theta-\phi)}$
• Lorentz transformation The coordinate transformation between stationary observer and moving observer in x-direction is given by $\bar{x} = \gamma(x-vt), \bar{y}=y, \bar{z}=z, \bar{t} = \gamma(t-\frac{v}{c^2}x)$ where $\gamma = 1/(1-v^2/c^2)^{1/2}$.
1. Derive the inverse transformation, i.e., find $x, y, z,$ and $t$ in terms of $\bar{x}, \bar{y}, \bar{z}$ and $\bar{t}$.
2. Prove the relativistic velocity addition rule $u_x = \frac{\bar{u}_x+v}{1+v\bar{u}_x/c^2}$ where $u_x=\frac{dx}{dt}$ and $\bar{u}_x = \frac{d\bar{x}}{d\bar{t}}$
3. What is the relationship between $u_y$ and $\bar{u}_y$ where $u_y=\frac{dy}{dt}$ and $\bar{u}_y = \frac{d\bar{y}}{d\bar{t}}$?
• Lorentz invariants Show the following quantities are invariant under the Lorentz transformation. The field transformation rule appears here.
• $\mathbf{E}\cdot\mathbf{B}$
• $E^2 - c^2B^2$