Calculating the Field from the Potential — Question 4

Question 4

Spherical differentiation gives both dipole field components

An ideal dipole has $V(r,\theta)=kp\cos\theta/r^2$. Use $E_r=-\partial_rV$ and $E_\theta=-r^{-1}\partial_\theta V$; there is no azimuthal dependence.

1. Find the two nonzero components and the field magnitude.

2. Find the angle between E and the outward radial direction at $\theta=\pi/3$.

Question 4 - Solution

Spherical differentiation gives both dipole field components

(a) Method and calculation

$$E_r=\frac{2kp\cos\theta}{r^3},\quad E_\theta=\frac{kp\sin\theta}{r^3}.$$$$|\mathbf E|=\frac{kp}{r^3}\sqrt{1+3\cos^2\theta}.$$

(b) Interpretation and further calculation

$$\theta=\pi/3:\quad\frac{E_\theta}{E_r}=\frac{\sqrt3}{2}.$$$$\alpha=\tan^{-1}(\sqrt3/2)=40.8934^\circ\quad\text{toward increasing }\theta.$$

Physical check

A dipole field generally does not point radially; differentiating only with respect to r loses its angular component.