Equipotential Surfaces — Question 8

Question 8

The volume between nearby spherical equipotentials grows with radius

A positive point source gives $V=kQ/r$. Two spherical equipotentials have values V and V/2, where V is positive.

1. Find their radii and the volume between them.

2. For a small voltage decrease $dV<0$, derive the positive radial spacing dr and compare equal small voltage steps at r and 2r.

Question 8 - Solution

The volume between nearby spherical equipotentials grows with radius

(a) Method and calculation

$$r_1=kQ/V,\quad r_2=2kQ/V=2r_1.$$$$\mathcal V=\frac{4\pi}{3}(r_2^3-r_1^3)=\frac{28\pi}{3}\left(\frac{kQ}{V}\right)^3.$$

(b) Interpretation and further calculation

$$dV=-\frac{kQ}{r^2}dr\Rightarrow dr=-\frac{r^2}{kQ}dV.$$$$\frac{dr|_{2r}}{dr|_r}=4\quad\text{for the same infinitesimal }|dV|.$$

Physical check

The factor four is a local small-step result; it need not hold for a large finite voltage interval.