Applying Gauss' Law: Cylindrical Symmetry — Question 2

Question 2

A uniformly charged solid cylinder has its largest field at the boundary

An infinite nonconducting solid cylinder of radius R carries fixed uniform density $\rho>0$. No other sources are present.

1. Derive $E_r(r)$ inside and outside by calculating enclosed charge per unit length.

2. Check continuity at R and locate the maximum field magnitude over all radii.

Question 2 - Solution

A uniformly charged solid cylinder has its largest field at the boundary

(a) Method and calculation

$$\lambda_{\mathrm{enc}}(r)=\pi\rho r^2\ (r<R),\quad\lambda_{\mathrm{total}}=\pi\rho R^2.$$$$E_r(r)=\begin{cases}\rho r/(2\epsilon_0),&0\le r\le R,\\ \rho R^2/(2\epsilon_0r),&r\ge R.\end{cases}$$

(b) Interpretation and further calculation

$$E_r(R^-)=E_r(R^+)=\rho R/(2\epsilon_0).$$$$\frac{dE_r}{dr}=\frac\rho{2\epsilon_0}>0\ (r<R),\quad\frac{dE_r}{dr}=-\frac{\rho R^2}{2\epsilon_0r^2}<0\ (r>R).$$Thus the global maximum occurs at the cylinder boundary.

Physical check

There is no sheet charge at R, so the normal field is continuous even though its derivative changes.