Dielectrics: An Atomic View — Question 1

Question 1

A harmonic atomic model produces an induced dipole proportional to the field

A fixed charge +e is at x=0. Its bound negative charge -e is displaced by x and experiences restoring force -Kx in a uniform field $E\hat{\mathbf x}$. Let K>0.

1. Find the equilibrium displacement and dipole moment.

2. Include elastic and electric terms in the energy and find its minimized value; compare with inserting the final induced moment into -pE alone.

Question 1 - Solution

A harmonic atomic model produces an induced dipole proportional to the field

(a) Method and calculation

$$-Kx-eE=0\Rightarrow x_*=-eE/K.$$$$p=-ex_*=e^2E/K=\alpha E,\quad\alpha=e^2/K.$$

(b) Interpretation and further calculation

$$U(x)=\tfrac12Kx^2+eEx\Rightarrow U(x_*)=-\frac{e^2E^2}{2K}=-\tfrac12\alpha E^2.$$$$-pE=-\alpha E^2$$omits the positive elastic energy required to create the dipole.

Physical check

The negative charge moves against the field, but the induced dipole moment points with the field.