A resistor is machined from carbon in the shape of a truncated cone

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 resistor is machined from carbon in the shape of a truncated cone. Its axis is straight and of length L=40 mmL = 40\,\mathrm{mm}; the two flat end faces are circular discs of radii a=2.0 mma = 2.0\,\mathrm{mm} and b=5.0 mmb = 5.0\,\mathrm{mm}, and the radius grows linearly with distance xx along the axis. The carbon has uniform resistivity ρr=3.5×10−5 Ω m\rho_{\mathrm{r}} = 3.5\times10^{-5}\,\Omega\,\mathrm{m}. Each end face is coated with metal, so each is an equipotential. The taper is gentle enough that the current density may be taken as parallel to the axis 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 the resistance between the end faces is R=ρrLπab.R = \frac{\rho_{\mathrm{r}}L}{\pi a b}. ((b)) Evaluate RR.

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