Potential Due to an Electric Dipole — Question 8

Question 8

A dipole pair interaction depends on both moments and their connecting axis

Two small ideal dipoles have moments p1 and p2 and center separation R much larger than their physical sizes. Use $U=-\mathbf p_2\cdot\mathbf E_1$ and $\mathbf E_1=k[3(\mathbf p_1\cdot\hat{\mathbf R})\hat{\mathbf R}-\mathbf p_1]/R^3$.

1. Find U when the moments are parallel to each other and to the connecting axis, then when parallel to each other but perpendicular to that axis.

2. For fixed orientations find the radial force $F_R=-dU/dR$ in both cases and classify attraction or repulsion.

Question 8 - Solution

A dipole pair interaction depends on both moments and their connecting axis

(a) Method and calculation

$$U_{\parallel}=-2kp_1p_2/R^3.$$$$U_{\perp}=+kp_1p_2/R^3.$$

(b) Interpretation and further calculation

$$F_{R,\parallel}=-6kp_1p_2/R^4<0\quad\text{attraction}.$$$$F_{R,\perp}=+3kp_1p_2/R^4>0\quad\text{repulsion}.$$

Physical check

Orientations are held fixed when differentiating R; allowing free rotation changes the mechanical problem.