## The Homogeneously Charged Disk: Electrostatic Potential and Electric Field

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.

## Problem Statement

To calculate the electrostatic potential and electric field of a disk, it is useful to follow a two-step procedure as seen below. Unfortunately, the involved integrals would become incredibly complicated if we would not restrict ourselves to the $$z$$-axis of symmetry. In the end we will see how we can use the results for the disk to find the field of a cylinder - up to some technical steps.

• Derive the electric field $$\mathbf{E}_{\,\mathrm{ring}}\left(\mathbf{r}\right)$$ of a uniformly charged ring with radius $$R$$ along the $$z$$-axis.
• Use your result for the ring charge to calculate the electric field $$\mathbf{E}_{\,\mathrm{disk}}\left(\mathbf{r}\right)$$ of a uniformly charged disk with radius $$R$$, also along the $$z$$-axis. Discuss the limiting cases $$z\ll R$$ (near field) and $$z\gg R$$ (far field).
• Write down the equation that you would use to calculate the electrostatic potential $$\phi_{\,\mathrm{ring}}\left(\mathbf{r}\right)$$ of a uniformly charged cylinder with radius $$R$$ and height $$h$$. Restrict yourself even in this general approach to the axis of symmetry.