Potential of a Charged Isolated Conductor — Question 1

Question 1

A conductor interior retains the surface voltage despite having no field

Use $k=8.99\times10^9\,\mathrm{N\,m^2/C^2}$. An isolated conducting sphere of radius .150 m carries charge +6.00 nC. It is in vacuum with no other sources, and V(infinity)=0.

1. Find V(r) inside and outside, then evaluate at the center and at r=.450 m.

2. Find the exterior surface field and the field at the center; explain their relation to the potential profile.

Question 1 - Solution

A conductor interior retains the surface voltage despite having no field

(a) Method and calculation

$$V(r)=\begin{cases}kQ/R&r\le R,\\kQ/r&r\ge R.\end{cases}$$$$V(0)=359.6\,\mathrm V,\quad V(.450\,\mathrm m)=119.867\,\mathrm V.$$

(b) Interpretation and further calculation

$$E(R^+)=kQ/R^2=2397.33\,\mathrm{V/m},\quad E(0)=0.$$$$r<R:\quad dV/dr=0\Rightarrow E_r=0,$$although the constant interior voltage is nonzero.

Physical check

The sphere potential is continuous at the surface, while its normal derivative jumps because surface charge is present.