Two types of polarization with relations

Polarization is the formation of induced or aligned dipoles in a dielectric under an electric field; $P = N\alpha E$.

1. Electronic polarization

The field pushes the electron cloud of an atom one way and the nucleus the other, creating an induced dipole.

  E = 0:   ( (+) )      E -> :  (  (+))
          centres same           cloud shifts

• Take the atom as a nucleus $+Ze$ in a uniform electron cloud of radius $R$. For displacement $x$, the restoring field from the cloud is $\dfrac{Zex}{4\pi\varepsilon_0R^3}$.
• At equilibrium this equals $E$: $x = \dfrac{4\pi\varepsilon_0R^3E}{Ze}$
• Induced dipole moment: $p = Zex = 4\pi\varepsilon_0R^3E$

$$\alpha_e = \frac{p}{E} = 4\pi\varepsilon_0R^3, \qquad P_e = N\alpha_eE$$

It is independent of temperature and works up to optical frequencies.

2. Orientational (dipolar) polarization

Polar molecules (H₂O, HCl) carry permanent dipoles $p_0$ pointing randomly, so net $P = 0$. A field tries to align them; thermal agitation opposes this.

• Energy of a dipole: $U = -p_0E\cos\theta$
• Using Boltzmann statistics, for $p_0E \ll kT$: $\langle\cos\theta\rangle = \dfrac{p_0E}{3kT}$
• Average dipole moment: $p_{av} = \dfrac{p_0^2E}{3kT}$

$$\alpha_d = \frac{p_0^2}{3kT}, \qquad P_d = \frac{Np_0^2}{3kT}E$$

It decreases with rising temperature and can follow the field only up to about $10^{9}$–$10^{11}$ Hz.