Resistance and Resistivity — Question 10

Question 10

Resistance uncertainty is especially sensitive to a measured diameter

A uniform round wire has $\rho=R\pi d^2/(4L)$. Independent hard bounds are $R=10.0\pm.1\,\Omega$, $d=1.00\pm.02$ mm and $L=2.00\pm.01$ m.

1. Find nominal resistivity and its exact hard-bound interval.

2. Find the first-order fractional bound and identify the dominant measurement.

Question 10 - Solution

Resistance uncertainty is especially sensitive to a measured diameter

(a) Method and calculation

$$\rho_0=10\pi(10^{-3})^2/(4\cdot2)=3.92699\times10^{-6}\,\Omega\,\mathrm m.$$$$\rho\in\left[\frac{9.9\pi(.98\times10^{-3})^2}{4(2.01)},\frac{10.1\pi(1.02\times10^{-3})^2}{4(1.99)}\right].$$

(b) Interpretation and further calculation

$$\rho\in[3.71519,4.14723]\times10^{-6}\,\Omega\,\mathrm m.$$$$|\delta\rho|/\rho\lesssim .01+2(.02)+.005=.0550.$$The diameter contributes 4 percent, the largest term.

Physical check

Hard bounds combine extreme values; the squared diameter doubles its first-order fractional contribution.