Potential Due to an Electric Dipole — Question 7

Question 7

A dipole translation samples curvature in an external potential

A rigid dipole is constrained along x with charges +q at X+d/2 and -q at X-d/2. The external potential is $V_{\mathrm{ext}}(x)=Ax^2$, with A positive. Its moment is p=qd directed along positive x.

1. Find the exact external interaction energy as a function of its center X.

2. Find the net force from that energy and verify it by summing the forces on both charges.

Question 7 - Solution

A dipole translation samples curvature in an external potential

(a) Method and calculation

$$U=qA[(X+d/2)^2-(X-d/2)^2]=2AqdX.$$$$U(X)=2ApX,\quad F_X=-dU/dX=-2Ap.$$

(b) Interpretation and further calculation

$$E_x(x)=-2Ax,\quad F_{\mathrm{net}}=qE_x(X+d/2)-qE_x(X-d/2).$$$$F_{\mathrm{net}}=-2Aqd=-2Ap.$$

Physical check

The force is nonzero because the external field varies in space; the quadratic potential makes this finite-separation result exact.