# Charges and their Electric Fields

**Gauss's law** \(\nabla\cdot\mathbf{E}\left(\mathbf{r}\right) = \rho\left(\mathbf{r}\right)/\varepsilon_{0}\) relates the electric field to charges. Learn the basic techniques to determine the **electric field** \(\mathbf{E}\left(\mathbf{r}\right)\) and/or the **electrostatic potential** \(\phi\left(\mathbf{r}\right)\) for given **charge distributions** \(\rho\left(\mathbf{r}\right)\)!

To have a certain solution at hand is often useful construct another one out of it. This is the case for the electrostatic potential and field of the **charged ring** that can be generalized to the **homogeneously charged disk** and **cylinder**. Find out how to calculate the solutions on the axis of symmetry.

To find the electrostatic potential of a general charge distribution can be quite complicated. However, if **symmetries** are present, the problem can be boiled down to the solution of much simple(r) equation(s). Let us employ the **spherical symmetry** of the homogeneously charged sphere to determine its potential and electric field!

One of the **fundamental charge distributions** for which an **analytical expression** of the electric field can be found is that of a line charge of finite length. Nevertheless, the result we will encounter is hard to follow. Two **limiting cases** will help us understand the basic features of the result.

Sometimes it happens that a thing is more than the sum of its parts. What about two charges? Can their respective electric field behave fundamentally different in some way than just a single charge? In this problem you will learn about two main concepts in electromagnetics - the **superposition principle** and the **dipole**.

If we would make a poll about the **most fundamental question of electrostatics**, the field of a point charge is very likely the winner. You may already know the answer but are you able to **derive** it directly from **Maxwell's equations**?

A charge close to a metal induces **surface charges**. But what happens if one puts some charge inside of a **metallic shell**? What will happen to the field in- and outside of this cavity? Also find out what how the shell **screens** the inner structure of an arbitrary charge distribution.