The Electric Field Due to an Electric Dipole — Question 10

Question 10

Superposed dipoles combine as vectors in the common far field

Two compact neutral sources share the same center and have moments $\mathbf p_1=p\hat{\mathbf x}$ and $\mathbf p_2=p\hat{\mathbf y}$. At distances much larger than both source sizes, use $\mathbf E=k[3(\mathbf p\cdot\hat{\mathbf r})\hat{\mathbf r}-\mathbf p]/r^3$.

1. Find the total field at $(r,0,0)$.

2. Find its magnitude and the angle below the positive x axis. Compare with the incorrect result from adding the two separate magnitudes.

Question 10 - Solution

Superposed dipoles combine as vectors in the common far field

(a) Method and calculation

$$\mathbf p_{\mathrm{total}}=p(\hat{\mathbf x}+\hat{\mathbf y}),\quad\mathbf p_{\mathrm{total}}\cdot\hat{\mathbf r}=p.$$$$\mathbf E=\frac{kp}{r^3}(2\hat{\mathbf x}-\hat{\mathbf y}).$$

(b) Interpretation and further calculation

$$|\mathbf E|=\sqrt5\frac{kp}{r^3},\quad\alpha=\tan^{-1}(1/2)\approx26.5651^\circ.$$$$|\mathbf E_1|+|\mathbf E_2|=3kp/r^3>\sqrt5kp/r^3.$$The two contributions are perpendicular at this location, so their magnitudes cannot be added directly.

Physical check

This leading-order superposition treats both compact sources as dipoles about the same center.