Applying Gauss' Law: Cylindrical Symmetry — Question 8

Question 8

An oppositely charged annulus can reverse a line-source field

An infinite positive line charge $\lambda$ lies on the axis of an insulating annulus $a<r<b$ with uniform negative density $-\rho_0$, where $\rho_0>0$. There are no other sources.

1. Derive the radial field within the charged annulus and find a possible zero-field radius.

2. Give the strict condition for that zero to lie inside the material, and determine the exterior field sign in that case.

Question 8 - Solution

An oppositely charged annulus can reverse a line-source field

(a) Method and calculation

$$\lambda_{\mathrm{enc}}(r)=\lambda-\pi\rho_0(r^2-a^2),\quad E_r=\frac{\lambda-\pi\rho_0(r^2-a^2)}{2\pi\epsilon_0r}.$$$$E_r(r_0)=0\Rightarrow r_0=\sqrt{a^2+\frac{\lambda}{\pi\rho_0}}.$$

(b) Interpretation and further calculation

$$a<r_0<b\Longleftrightarrow0<\lambda<\pi\rho_0(b^2-a^2).$$$$\lambda_{\mathrm{total}}=\lambda-\pi\rho_0(b^2-a^2)<0\Rightarrow E_r(r>b)<0.$$The enclosed charge changes sign as the Gaussian radius sweeps through sufficient negative material.

Physical check

At equality the zero occurs at b and the entire exterior field vanishes; that is distinct from a zero strictly inside the annulus.