Resistance and Resistivity — Question 3

Question 3

A resistivity gradient gives a nonlinear potential profile

A uniform-area bar of length L has $\rho(x)=\rho_0(1+x/L)$ and steady current I toward $+x$. Define $V(0)-V(L)=V_0>0$.

1. Find total resistance and current.

2. Find $V(x)-V(L)$ and the position where half the voltage drop has occurred from x=0.

Question 3 - Solution

A resistivity gradient gives a nonlinear potential profile

(a) Method and calculation

$$R=\int_0^L\frac{\rho(x)}A dx=\frac{3\rho_0L}{2A}.$$$$I=V_0/R=2AV_0/(3\rho_0L).$$

(b) Interpretation and further calculation

$$V(x)-V(L)=\frac{I\rho_0L}{A}\left(\frac32-u-\frac{u^2}2\right),\quad u=x/L.$$$$u+u^2/2=3/4\Rightarrow x=L(\sqrt{5/2}-1)=.581139L.$$

Physical check

The field is larger in the more resistive region; potential does not generally vary linearly.