Resistance and Resistivity — Question 8

Question 8

A layered conductor has different effective resistivities along and across layers

Two materials occupy equal thicknesses in a laminated block. Their resistivities are $\rho_1=\rho$ and $\rho_2=4\rho$. Ignore interface resistance. Compare current perpendicular and parallel to layers.

1. Find the effective resistivity for perpendicular flow using equal normal current density.

2. Find it for parallel flow using equal tangential electric field, and compare the two.

Question 8 - Solution

A layered conductor has different effective resistivities along and across layers

(a) Method and calculation

$$E_{\mathrm{avg}}=\tfrac12(\rho_1J+\rho_2J).$$$$\rho_\perp=(\rho_1+\rho_2)/2=5\rho/2.$$

(b) Interpretation and further calculation

$$J_{\mathrm{avg}}=\tfrac12(E/\rho_1+E/\rho_2)=5E/(8\rho).$$$$\rho_\parallel=8\rho/5,\quad\rho_\perp/\rho_\parallel=25/16.$$

Physical check

Structural anisotropy can arise from layers of individually isotropic materials.