The Electric Field Due to an Electric Dipole — Question 7

Question 7

Equal field magnitudes around a dipole do not form a sphere

In the far field of a dipole, $|\mathbf E|=(kp/r^3)\sqrt{1+3\cos^2\theta}$. Consider the surface where the magnitude equals a specified $E_*>0$.

1. Derive its radius as a function of polar angle.

2. Find the axial-to-equatorial radius ratio and compare field direction at these two locations.

Question 7 - Solution

Equal field magnitudes around a dipole do not form a sphere

(a) Method and calculation

$$E_*r^3=kp\sqrt{1+3\cos^2\theta}.$$$$r(\theta)=\left(\frac{kp}{E_*}\right)^{1/3}(1+3\cos^2\theta)^{1/6}.$$

(b) Interpretation and further calculation

$$r_{\mathrm{axis}}/r_{\mathrm{eq}}=4^{1/6}=2^{1/3}\approx1.25992.$$$$\mathbf E_{\mathrm{axis}}\parallel\mathbf p,\quad\mathbf E_{\mathrm{eq}}\parallel-\mathbf p.$$Equal magnitude does not require equal distance or equal direction.

Physical check

The entire contour must remain far from the finite source size for the point-dipole formula to apply.