Electric Potential — Question 9

Question 9

Zero potential at one plane does not mean zero electric force

A source gives $V(x)=-E_0x$ with $E_0>0$. A second region has spatially constant potential $V_c$. Use $dW_e=-q\,dV=F_x\,dx$.

1. Find the force at x=0 in the first region although its potential vanishes there.

2. Find the force in the constant-potential region and the work moving a probe between any two points within it.

Question 9 - Solution

Zero potential at one plane does not mean zero electric force

(a) Method and calculation

$$dV=-E_0\,dx,\quad F_x\,dx=-q\,dV=qE_0\,dx.$$$$F_x=qE_0\quad\text{including at }x=0\text{ where }V=0.$$

(b) Interpretation and further calculation

$$dV=0\Rightarrow F_x=0;\quad\nabla V_c=\mathbf0\Rightarrow\mathbf F=\mathbf0.$$$$W_e=q(V_c-V_c)=0.$$

Physical check

The slope determines the force; the selected zero of potential supplies no force information.