Capacitor with a Dielectric — Question 10

Question 10

An anisotropic dielectric makes capacitance depend on orientation

A uniaxial dielectric fills an ideal plate gap d of area A. Its axis makes angle theta with the plate normal. The supplied constitutive law is $D_n=\epsilon_0(\kappa_\parallel\cos^2\theta+\kappa_\perp\sin^2\theta)E_n$, and $E_n=V/d$. Let $\kappa_\parallel>\kappa_\perp>0$.

1. Find C(theta) and its extreme values.

2. At fixed voltage use the generalized electric torque $\tau_\theta=\tfrac12V^2dC/d\theta$ to determine the preferred orientation.

Question 10 - Solution

An anisotropic dielectric makes capacitance depend on orientation

(a) Method and calculation

$$Q=AD_n\Rightarrow C(\theta)=\frac{\epsilon_0A}{d}(\kappa_\parallel\cos^2\theta+\kappa_\perp\sin^2\theta).$$$$C_{\max}=\kappa_\parallel\epsilon_0A/d,\quad C_{\min}=\kappa_\perp\epsilon_0A/d.$$

(b) Interpretation and further calculation

$$\tau_\theta=\frac{\epsilon_0AV^2}{d}(\kappa_\perp-\kappa_\parallel)\sin\theta\cos\theta.$$$$0<\theta<\pi/2:\quad\tau_\theta<0,$$so the axis tends to align with the normal, maximizing capacitance.

Physical check

The fixed-voltage torque includes source work; alignment follows the larger permittivity direction.