Electricity and Magnetism · Problem 4.4

THE FIELDS OF MOVING CHARGES — Edward Purcell & David Morin

Physics — E&M · spherical-resistor

Problem

Spherical resistor ** (a) The region between two concentric spherical shells is filled with a material with resistivity ρ\rho. The inner radius is r1r_1, and the outer radius r2r_2 is many times larger (essentially infinite). Show that the resistance between the shells is essentially equal to ρ/4πr1\rho/4\pi r_1. (b) Without doing any calculations, dimensional analysis suggests that the above resistance should be proportional to ρ/r1\rho/r_1, because ρ\rho has units of ohm-meters and r1r_1 has units of meters. But is this reasoning rigorous?
Intuitive explanation

This problem explores how resistance scales in a geometry where current flows radially outward from a point source, rather than through a uniform conductor. The key insight is that as current spreads spherically, the cross-sectional area grows as r2r^2, so the resistance contribution from each thin shell decreases with distance—you need to integrate these contributions carefully. Part (b) then asks whether dimensional analysis alone can pin down the exact numerical coefficient, which is a great reminder that dimensions constrain but don't determine physics.

Intuitive explanation

This problem explores how resistance scales in a geometry where current flows radially outward from a point source, rather than through a uniform conductor. The key insight is that as current spreads spherically, the cross-sectional area grows as r2r^2, so the resistance contribution from each thin shell decreases with distance—you need to integrate these contributions carefully. Part (b) then asks whether dimensional analysis alone can pin down the exact numerical coefficient, which is a great reminder that dimensions constrain but don't determine physics.

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