The Electrical Properties of Solids — Question 9

Question 9

A cubic anisotropic sample can resist differently along two axes

A homogeneous solid obeys $J_x=\sigma_xE_x$, $J_y=\sigma_yE_y$, with $\sigma_x=4\sigma_0$ and $\sigma_y=\sigma_0$. Apply $E_x=E_y=E_0$.

1. Find the angle of the current density relative to x.

2. For a cube contacted on opposite x or y faces, find $R_y/R_x$.

Question 9 - Solution

A cubic anisotropic sample can resist differently along two axes

(a) Method and calculation

$$\boldsymbol J=\sigma_0E_0(4\hat x+\hat y).$$$$\boxed{\tan\theta_J=J_y/J_x=1/4,\quad\theta_J=14.04^\circ}.$$

(b) Interpretation and further calculation

$$R_i=L/(\sigma_iL^2)=1/(\sigma_iL).$$$$\boxed{R_y/R_x=\sigma_x/\sigma_y=4}.$$

Physical check

The applied field points at 45 degrees, while current does not. A scalar conductivity is insufficient for a general anisotropic crystal.