Resistance and Resistivity — Question 9

Question 9

A small neck can dominate the resistance of a nearly uniform wire

A wire of total length L and resistivity rho has area A over fraction $1-f$ of its length and area $\eta A$ over fraction f, with $0<f,\eta<1$. Transitions are idealized as short.

1. Find the total resistance and fraction of the voltage drop across the neck.

2. For $f=.0200$, find eta that makes the neck account for half the voltage drop.

Question 9 - Solution

A small neck can dominate the resistance of a nearly uniform wire

(a) Method and calculation

$$R=\frac{\rho L}A[(1-f)+f/\eta].$$$$\frac{V_{\mathrm{neck}}}V=\frac{f/\eta}{(1-f)+f/\eta}.$$

(b) Interpretation and further calculation

$$V_{\mathrm{neck}}/V=1/2\Rightarrow f/\eta=1-f.$$$$\eta=f/(1-f)=1/49=.0204082.$$

Physical check

The same current passes every segment, so voltage-drop fractions equal resistance fractions.