Applying Gauss' Law: Cylindrical Symmetry — Question 6

Question 6

A coaxial conductor can shield the exterior while retaining a gap field

An infinite inner conducting cylinder of radius a carries charge per length $\lambda>0$. A coaxial outer conductor occupies $b<r<c$, with $a<b<c$, and has net charge per length $\Lambda$. Everything is in electrostatic equilibrium.

1. Find the charge per length on the inner and outer surfaces of the outer conductor.

2. Derive the field in the vacuum gap and outside c, then choose $\Lambda$ so the exterior field vanishes.

Question 6 - Solution

A coaxial conductor can shield the exterior while retaining a gap field

(a) Method and calculation

$$E_{b<r<c}=0\Rightarrow\lambda+\lambda_b=0\Rightarrow\lambda_b=-\lambda.$$$$\lambda_c=\Lambda-\lambda_b=\Lambda+\lambda.$$

(b) Interpretation and further calculation

$$E_r=\frac{\lambda}{2\pi\epsilon_0r}\quad(a<r<b),\quad E_r=\frac{\lambda+\Lambda}{2\pi\epsilon_0r}\quad(r>c).$$$$\Lambda=-\lambda\Rightarrow E_{r>c}=0,\quad E_{a<r<b}\ne0.$$The Gaussian surface in the metal determines the induced inner charge before symmetry is used in the vacuum regions.

Physical check

The induced surface densities are $-\lambda/(2\pi b)$ and $(\Lambda+\lambda)/(2\pi c)$, not equal merely because total charges may cancel.