Potential Due to a Group of Point Charges — Question 2

Question 2

Unequal opposite sources produce two axial zero-potential points

Charges +Q at x=0 and $-2Q$ at x=d have Q,d>0. Use zero potential at infinity and exclude the source points.

1. Find all finite points on the x axis where V=0.

2. Find the direction and magnitude of the field at each zero.

Question 2 - Solution

Unequal opposite sources produce two axial zero-potential points

(a) Method and calculation

$$\frac1{|x|}=\frac2{|x-d|}\Rightarrow|x-d|=2|x|.$$$$(x-d)^2=4x^2\Rightarrow x=d/3\ \text{or}\ x=-d.$$

(b) Interpretation and further calculation

$$\mathbf E(d/3)=\left(9+\frac92\right)\frac{kQ}{d^2}\hat{\mathbf x}=\frac{27kQ}{2d^2}\hat{\mathbf x}.$$$$\mathbf E(-d)=\left(-1+\frac12\right)\frac{kQ}{d^2}\hat{\mathbf x}=-\frac{kQ}{2d^2}\hat{\mathbf x}.$$

Physical check

Both roots satisfy the unsquared distance equation; neither has a vanishing field.