Legendre polynomial expansion of the potential of a uniformly charged disk

TL;DR

The problem addressed is computing the electrostatic potential and field of a uniformly charged disk. The authors apply a Legendre polynomial expansion, using the axisymmetric ring potential $\phi(z)=\frac{Q}{\sqrt{z^2+R^2}}$ and extending off-axis by multiplying each $1/r^{l+1}$ term with $P_l(\cos\theta)$ to obtain $\phi_{\rm disk}$ for $r>R$ and $r<R$. The key contributions are explicit Legendre-series expressions for $\phi_{\rm disk}$ and the corresponding $E_r$ and $E_\theta$, derived by differentiating the potential. This approach offers a simpler, graduate-friendly route that avoids elliptic integrals and yields a straightforward path to the field via differentiation.

Abstract

We use a Legendre polynomial expansion to find the electrostatic potential of a uniformly charged disk. We then use the potential to find the electric field of the disk.