Potential of a Charged Isolated Conductor — Question 10

Question 10

A charged sphere in a uniform external field remains equipotential while its density varies

A conducting sphere of radius R carries net charge Q in a uniform external field $E_0\hat{\mathbf z}$. Its exterior potential is supplied as $V=-E_0(r-R^3/r^2)\cos\theta+kQ/r$. Here $\theta$ is measured from positive z and $E_0>0$.

1. Find the surface potential and both exterior field components at r=R.

2. Find the surface charge density and the minimum Q for it to be nonnegative everywhere. Use $\sigma=\epsilon_0 E_r(R^+)$.

Question 10 - Solution

A charged sphere in a uniform external field remains equipotential while its density varies

(a) Method and calculation

$$V(R,\theta)=kQ/R,\quad E_\theta=-E_0(1-R^3/r^3)\sin\theta\Rightarrow E_\theta(R)=0.$$$$E_r=E_0(1+2R^3/r^3)\cos\theta+kQ/r^2\Rightarrow E_r(R)=3E_0\cos\theta+kQ/R^2.$$

(b) Interpretation and further calculation

$$\sigma(\theta)=\epsilon_0[3E_0\cos\theta+kQ/R^2].$$$$\sigma_{\min}=\epsilon_0[-3E_0+kQ/R^2]\ge0\Rightarrow Q\ge3E_0R^2/k.$$

Physical check

The supplied external uniform field has no infinity-zero potential; the stated formula fixes its reference. Integrating the cosine surface term gives zero net induced charge.