A Dipole in an Electric Field — Question 8

Question 8

Induced dipole energy includes the cost of creating the dipole

A small polarizable object has scalar induced moment p along a positive prescribed field $E(x)$. Its model energy is $U(p,x)=p^2/(2\alpha)-pE(x)$ with $\alpha>0$. The dipole adjusts quasistatically to minimize U at each x; neglect other forces and higher multipoles.

1. Minimize over p and find the effective position-dependent energy.

2. Derive the force for $E(x)=E_0+Gx>0$, and show why inserting $p=\alpha E$ into $-pE$ alone gives an incorrect factor.

Question 8 - Solution

Induced dipole energy includes the cost of creating the dipole

(a) Method and calculation

$$\frac{\partial U}{\partial p}=p/\alpha-E=0\Rightarrow p=\alpha E,\quad\frac{\partial^2U}{\partial p^2}=1/\alpha>0.$$$$U_{\mathrm{eff}}=\frac{\alpha E^2}{2}-\alpha E^2=-\tfrac12\alpha E^2.$$

(b) Interpretation and further calculation

$$F_x=-\frac{dU_{\mathrm{eff}}}{dx}=\alpha E\frac{dE}{dx}=\alpha G(E_0+Gx).$$$$-\frac{d(-\alpha E^2)}{dx}=2\alpha E\frac{dE}{dx}=2F_x.$$The internal polarization cost supplies the missing positive term in the energy; omitting it doubles the inferred force.

Physical check

For positive polarizability the force points toward increasing $E^2$, within the stated quasistatic model.