Dielectrics and Gauss' Law — Question 8

Question 8

A dielectric sphere responds to its applied field and its own depolarizing field

A homogeneous dielectric sphere of radius R has relative permittivity kappa and no free charge. A distant source applies uniform field E0. Use the supplied result that uniform polarization P creates internal field $-\mathbf P/(3\epsilon_0)$ and the constitutive law $\mathbf P=\epsilon_0(\kappa-1)\mathbf E_{\mathrm{in}}$.

1. Solve self-consistently for the interior field and polarization.

2. Find its total dipole moment and bound surface density as a function of angle theta from E0.

Question 8 - Solution

A dielectric sphere responds to its applied field and its own depolarizing field

(a) Method and calculation

$$\mathbf E_{\mathrm{in}}=\mathbf E_0-\mathbf P/(3\epsilon_0)\Rightarrow\mathbf E_{\mathrm{in}}=\frac3{\kappa+2}\mathbf E_0.$$$$\mathbf P=\frac{3\epsilon_0(\kappa-1)}{\kappa+2}\mathbf E_0.$$

(b) Interpretation and further calculation

$$\mathbf p_{\mathrm{sphere}}=\frac{4\pi R^3}{3}\mathbf P=4\pi\epsilon_0R^3\frac{\kappa-1}{\kappa+2}\mathbf E_0.$$$$\sigma_b(\theta)=P\cos\theta,\quad\rho_b=0.$$

Physical check

The internal field is not E0/kappa; that reduction applies to a different boundary geometry and source constraint.