Potential Due to an Electric Dipole — Question 3

Question 3

Equal dipole voltage at different angles fixes a square-root radius ratio

For an ideal dipole, $V=kp\cos\theta/r^2$. Point A is at radius R and angle $30^\circ$. Point B is at angle $60^\circ$ and has the same voltage.

1. Find the radius of B relative to R.

2. Determine whether any point in the southern hemisphere $\pi/2<\theta\le\pi$ can have this same voltage at a finite positive radius.

Question 3 - Solution

Equal dipole voltage at different angles fixes a square-root radius ratio

(a) Method and calculation

$$\frac{\cos30^\circ}{R^2}=\frac{\cos60^\circ}{r_B^2}.$$$$r_B=R\sqrt{\frac{1/2}{\sqrt3/2}}=3^{-1/4}R\approx.759836R.$$

(b) Interpretation and further calculation

$$V_A=\frac{\sqrt3kp}{2R^2}>0.$$$$\cos\theta<0\Rightarrow V(r,\theta)<0\quad(r>0),$$so no southern point shares VA.

Physical check

A dipole equipotential has an angularly varying radius and occupies only the hemisphere consistent with its sign.