A thin-film resistor is a flat strip of nichrome of uniform thickness and uniform…

PHYSICS · Multi-part · Fermi University Physics · Electricity, Magnetism & Electromagnetism · Electric current and circuits · Electric current and circuits

Electricity, Magnetism and Electromagnetism, Ch Current, current density, conductivity and Ohm's law

Problem

A thin-film resistor is a flat strip of nichrome of uniform thickness t=0.10 mmt = 0.10\,\mathrm{mm} and uniform resistivity ρr=1.1×10−6 Ω m\rho_{\mathrm{r}} = 1.1\times10^{-6}\,\Omega\,\mathrm{m}. Current enters one end and leaves the other, a distance L=20 mmL = 20\,\mathrm{mm} apart, and over that distance the width of the strip grows linearly from w1=1.0 mmw_1 = 1.0\,\mathrm{mm} to w2=4.0 mmw_2 = 4.0\,\mathrm{mm}. Both ends are metallised, so each end face is an equipotential. The taper is gentle enough that the current density may be taken as parallel to the length of the strip and uniform over every cross-section, and the current is steady. ((a)) Starting from the local Ohm's law J=σcE\mathbf{J} = \sigma_{\mathrm{c}}\mathbf{E}, show that R=ρrLt (w2−w1)ln⁡ ⁣w2w1.R = \frac{\rho_{\mathrm{r}}L}{t\,(w_2 - w_1)}\ln\!\frac{w_2}{w_1}. ((b)) Evaluate RR.

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