Potential Due to a Group of Point Charges — Question 3

Question 3

A zero-potential locus of unequal opposite charges is a sphere

Charges +Q at the origin and $-2Q$ at $(d,0,0)$ have Q,d>0. Use the infinity reference.

1. Derive the full three-dimensional V=0 surface.

2. Find its center, radius and axial intersections, and identify which source it surrounds.

Question 3 - Solution

A zero-potential locus of unequal opposite charges is a sphere

(a) Method and calculation

$$r_- =2r_+,\quad (x-d)^2+y^2+z^2=4(x^2+y^2+z^2).$$$$\left(x+\frac d3\right)^2+y^2+z^2=\left(\frac{2d}{3}\right)^2.$$

(b) Interpretation and further calculation

$$\mathbf r_c=-\frac d3\hat{\mathbf x},\quad R_s=2d/3,\quad x_{\mathrm{ends}}=-d,d/3.$$$$|0-x_c|=d/3<R_s,\quad|d-x_c|=4d/3>R_s.$$The sphere encloses the smaller positive source.

Physical check

The sphere is a voltage level, not a conducting shell; its existence does not alter either source.