Resistance and Resistivity — Question 2

Question 2

A linearly tapered radius requires integrating local resistance

A long slender conductor has resistivity $\rho$, length L and radius $r(x)=a+(b-a)x/L$, where $a,b>0$. Treat slices as equipotential disks and neglect end corrections.

1. Derive its end-to-end resistance.

2. Compare it with a uniform wire whose radius is the arithmetic mean $(a+b)/2$.

Question 2 - Solution

A linearly tapered radius requires integrating local resistance

(a) Method and calculation

$$dR=\frac{\rho\,dx}{\pi r(x)^2},\quad R=\frac{\rho L}{\pi(b-a)}\left(\frac1a-\frac1b\right).$$$$R=\frac{\rho L}{\pi ab}\quad\text{also by continuity when }a=b.$$

(b) Interpretation and further calculation

$$R_{\mathrm{mean}}=\frac{4\rho L}{\pi(a+b)^2}.$$$$\frac{R}{R_{\mathrm{mean}}}=\frac{(a+b)^2}{4ab}=1+\frac{(a-b)^2}{4ab}\ge1.$$

Physical check

The narrow portion raises resistance more than the wide portion lowers it.