## The Electric Field and Potential of a Homogeneously Charged Sphere

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!

## Problem Statement

A homogeneously charged sphere of radius $$R$$ is described by the charge distribution $\rho\left(\mathbf{r}\right) = \begin{cases}\rho_{0} & r<R\\0 & r\geq R\end{cases}\ .$ The charge density $$\rho_0$$ can be understood as the number of homogeneously distributed charges vs. the volume of the sphere, see right figure.

Find the electrostatic potential of the homogeneously charged sphere and its electric field. You may want to follow these steps:

• Rewrite the Laplace equation for the electrostatic potential in a suitable coordinate system.
• Simplify the equation using the spherical symmetry of the charge distribution.
• Try to integrate the equation.
• Determine the constants of integration to find the electrostatic potential.
• Finally, calculate the electric field.