Equipotential Surfaces — Question 7

Question 7

A surface potential ripple would drive charges along a conductor

A proposed electrostatic conductor surface has arclength coordinate s and $V(s)=V_0+\delta V\cos(2\pi s/L)$, with nonzero $\delta V$. Use $E_t=-dV/ds$.

1. Find the tangential field and its maximum magnitude.

2. Find the work on a positive surface charge q from s=0 to L/2 and determine whether this profile can represent equilibrium.

Question 7 - Solution

A surface potential ripple would drive charges along a conductor

(a) Method and calculation

$$E_t=\frac{2\pi\delta V}{L}\sin(2\pi s/L).$$$$|E_t|_{\max}=2\pi|\delta V|/L.$$

(b) Interpretation and further calculation

$$W_e=q[(V_0+\delta V)-(V_0-\delta V)]=2q\delta V.$$$$E_t\equiv0\ \text{in equilibrium}\Rightarrow\delta V=0.$$Mobile charge must redistribute until the entire connected conductor has one potential.

Physical check

A constant potential does not require zero normal exterior field; it requires zero tangential field at the surface.