Resistance and Resistivity — Question 5

Question 5

Orthogonal measurements separate a material property from specimen shape

A homogeneous isotropic block has dimensions $L_x=4.0$ cm, $L_y=2.0$ cm and $L_z=1.0$ cm. Ideal full-face electrodes measure $R_x=8.0$ m$\Omega$ along x.

1. Infer resistivity and predict resistance along y.

2. Find the ratio $R_x/R_z$ and explain why these unequal resistances do not imply anisotropy.

Question 5 - Solution

Orthogonal measurements separate a material property from specimen shape

(a) Method and calculation

$$\rho=R_xL_yL_z/L_x=4.00\times10^{-5}\,\Omega\,\mathrm m.$$$$R_y=\rho L_y/(L_xL_z)=2.00\,\mathrm{m\Omega}.$$

(b) Interpretation and further calculation

$$R_z=\rho L_z/(L_xL_y)=.500\,\mathrm{m\Omega}.$$$$R_x/R_z=(L_x/L_z)^2=16.0.$$

Physical check

Resistivity is direction-independent in this model; electrode separation and face area are not.