Potential Due to a Group of Point Charges — Question 8

Question 8

A compensating center charge sets one point to zero potential without zeroing its field

Two charges +Q are at $(\pm a,0)$. An adjustable charge Q0 is at the origin. At P=(0,a), require V=0 with the infinity reference.

1. Find Q0.

2. Find the electric field at P after compensation and determine its direction.

Question 8 - Solution

A compensating center charge sets one point to zero potential without zeroing its field

(a) Method and calculation

$$V_P=\frac{2kQ}{\sqrt2a}+\frac{kQ_0}{a}=0.$$$$Q_0=-\sqrt2Q.$$

(b) Interpretation and further calculation

$$E_y=2\frac{kQ a}{(2a^2)^{3/2}}+\frac{kQ_0}{a^2}=\frac{kQ}{a^2}\left(\frac1{\sqrt2}-\sqrt2\right).$$$$\mathbf E_P=-\frac{kQ}{\sqrt2a^2}\hat{\mathbf y},\quad E_x=0.$$

Physical check

Potential cancellation weights each charge by inverse distance, while the vector field has a different distance and directional weighting.