Resistance and Resistivity — Question 1

Question 1

Uniform drawing changes resistance quadratically with length

A wire of initial length L, area A and resistance R is drawn uniformly to length $\lambda L$, with $\lambda>1$. Its volume and resistivity remain constant.

1. Find the new area and resistance.

2. If resistance increases by 44 percent, infer the fractional increase in length and fractional decrease in diameter.

Question 1 - Solution

Uniform drawing changes resistance quadratically with length

(a) Method and calculation

$$AL=A\prime\lambda L\Rightarrow A\prime=A/\lambda.$$$$R\prime=\rho\lambda L/(A/\lambda)=\lambda^2R.$$

(b) Interpretation and further calculation

$$\lambda=\sqrt{1.44}=1.20,\quad\Delta L/L=.20.$$$$d\prime/d=\sqrt{A\prime/A}=1/\sqrt{1.20}=.912871,\quad1-d\prime/d=.087129.$$

Physical check

Both the longer path and the smaller area increase resistance; unchanged resistivity is an explicit idealization.