Ohm's Law — Question 7

Question 7

A nonuniform ohmic bar remains globally linear when its profile stays fixed

A bar has constant area A and fixed conductivity profile $\sigma(x)=\sigma_0e^{x/L}$ on $0\le x\le L$. Its sidewalls are insulating and local transport is $J=\sigma E$.

1. Derive the end-to-end I-V relation.

2. Find the ratio of endpoint fields and the limiting result if the conductivity instead becomes uniform at sigma0.

Question 7 - Solution

A nonuniform ohmic bar remains globally linear when its profile stays fixed

(a) Method and calculation

$$J=I/A,\quad V=\int_0^L\frac{I}{A\sigma_0}e^{-x/L}dx.$$$$I=\frac{A\sigma_0}{L(1-e^{-1})}V.$$

(b) Interpretation and further calculation

$$E(L)/E(0)=e^{-1}.$$$$\sigma(x)=\sigma_0\Rightarrow I=(A\sigma_0/L)V.$$

Physical check

Spatial nonuniformity changes the conductance but does not alone destroy proportionality.