# 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)!