Electrodynamics

The main topics here are electro- and magnetostatics, their interplay in electromagnetism and wave phenomena that lead to optics.

Electrostatics

Coulomb's law and the first Maxwell equation.

Electric Field of Point Charges

Two point charges are fixed in a coordinate system. Charge $q_1$ is at position $\textbf r_1 = (x_1,y_1)$ and charge $q_2$ lies at $\textbf r_2 = (x_2,y_2)$.

1. Calculate the resulting electric field $\textbf E$ for this system at an arbitrary point $\textbf r \neq \{\textbf r_1, \textbf r_2\}$!

Since Maxwell's equations are linear equations, you can use superposition.

2. Find the point $\textbf r_0=(x_0,y_0)$ where the electric field vanishes!

Electric Field of Point Charges (2)

Two point-like charges $q_1$ and $q_2$ are located at $\textbf r_1=(0,0,a)$ and $\textbf r_2=(0,0,-a)$.

1. Write down the charge density $\rho(\textbf r)$ of this arrangement and calculate the total charge by integrating over it!

$Q = \int \text d^3\textbf r\, \rho(\textbf r)$.

2. Calculate the electric field $\textbf E(\textbf r)$ at an arbitrary point $\textbf r \neq \{\textbf r_1, \textbf r_2\}$! What is the electric field along the $x$-axis ($y=z=0$) for (i) $q_1=q_2$ (ii) $q_1=-q_2$?

Use superposition.

1. Calculate the force that $q_1$ is applying on $q_2$!

Two Spheres

A sphere of radius $R_0$ lies at the center of a coordinate system. At a point $\textbf a$ within the sphere lies another sphere of radius $r_0$ ($r_0<R_0$), such that the smaller sphere is completely within the greater sphere ($|\textbf a|+r_0\le R_0$). The volume within the two spheres is filled with a constant charge density $\rho_0$.

Calculate the electrostatic potential $\phi(\textbf r)$ and the corresponding electric field $\textbf E(\textbf r)$ for the volume within the smaller sphere, where there is no charge.

Use superposition: one sphere with charge density $\rho_0$ and one with charge density $-\rho_0$.

Electric Field of a Point Charge

For a point charge $q$ at the origin of the coordinate system, the electric field is given by $\textbf R(\textbf r) = \frac{1}{4\pi\epsilon_0} \frac{q}{r^2}\frac{\textbf r}{|\textbf r|}$.

1. Show that $\nabla \cdot \textbf E = \frac{q}{\epsilon_0}\delta^{(3)}(\textbf r)$!

Treat $r=0$ and $r\neq 0$ separately.

2. Show that $\nabla\times \textbf E = 0$!

For $r=0$, write the curl as $$\nabla\times \textbf E = \lim\limits_{V\to 0}\int\limits_{\partial V}\text d\textbf f \times E,$$where $\partial V$ is the boundary of the volume $V$ and $\text d\textbf f$ is an infinitesimally small element on the boundary of $V$ such that the normal vector $\textbf f$ point outwards. Bonus points for proving this equation (Hint: use the volume of a small cube, such that the normal vectors point in $x$, $y$ and $z$ direction and assume that the vector field in question is smooth)!