Potential Due to an Electric Dipole — Question 10

Question 10

Opposite-angle voltage differences isolate an ideal dipole contribution

A remote source has the model potential $V(r,\theta)=kQ/r+kp\cos\theta/r^2+C$. The axis is known, but Q,p and the constant offset C are unknown. At fixed R measure $V(0)=14$ V, $V(\pi)=6$ V and $V(\pi/2)=10$ V.

1. Find $kp/R^2$ and check the angular consistency of the model.

2. An additional equatorial reading at radius 2R is 7 V. Find $kQ/R$ and C.

Question 10 - Solution

Opposite-angle voltage differences isolate an ideal dipole contribution

(a) Method and calculation

$$V(0)-V(\pi)=2kp/R^2\Rightarrow kp/R^2=4\,\mathrm V.$$$$\tfrac12[V(0)+V(\pi)]=10\,\mathrm V=V(\pi/2).$$

(b) Interpretation and further calculation

$$A+C=10,\quad A/2+C=7,\quad A\equiv kQ/R.$$$$A=6\,\mathrm V,\quad C=4\,\mathrm V.$$

Physical check

Angular differences remove both the monopole voltage and a common reference offset; a second radius separates those two remaining terms.