Potential Due to an Electric Dipole — Question 4

Question 4

Voltage measurements along two directions reconstruct a planar dipole

An ideal dipole has unknown moment $\mathbf p=p_x\hat{\mathbf x}+p_y\hat{\mathbf y}$. At radius R, measured potentials on the positive x and y axes are $V_x=3V_0$ and $V_y=4V_0$, where V0>0.

1. Find the two moment components, its magnitude and its orientation.

2. Predict the voltage at radius 2R along $(\hat{\mathbf x}+\hat{\mathbf y})/\sqrt2$.

Question 4 - Solution

Voltage measurements along two directions reconstruct a planar dipole

(a) Method and calculation

$$p_x=3V_0R^2/k,\quad p_y=4V_0R^2/k.$$$$p=5V_0R^2/k,\quad\theta_p=\tan^{-1}(4/3)=53.1301^\circ.$$

(b) Interpretation and further calculation

$$\mathbf p\cdot\hat{\mathbf r}=\frac{7V_0R^2}{k\sqrt2}.$$$$V(2R)=\frac{k}{4R^2}\frac{7V_0R^2}{k\sqrt2}=\frac7{4\sqrt2}V_0.$$

Physical check

A third independent directional measurement would be needed if a nonzero out-of-plane component were allowed.