Potential Due to a Group of Point Charges — Question 1

Question 1

A nonzero midpoint potential can coexist with a zero electric field

Two identical positive point charges Q are fixed at x=$\pm a$. Use V(infinity)=0.

1. Find the potential and field at the origin.

2. Find the work to bring a probe q quasistatically from infinity to the origin; explain why the zero final force does not make that work zero.

Question 1 - Solution

A nonzero midpoint potential can coexist with a zero electric field

(a) Method and calculation

$$V(0)=kQ/a+kQ/a=2kQ/a.$$$$\mathbf E(0)=\frac{kQ}{a^2}\hat{\mathbf x}-\frac{kQ}{a^2}\hat{\mathbf x}=\mathbf0.$$

(b) Interpretation and further calculation

$$W_{\mathrm{ext}}=q[V(0)-V(\infty)]=2kQq/a.$$$$\mathbf E(0)=\mathbf0\quad\text{constrains }\nabla V(0),\ \text{not }V(0).$$

Physical check

Potential contributions add as scalars; opposing field vectors cancel at the midpoint.