Question

Hollow metallic sphere has inner and outer radii a and b respectively calculate its resistance between the points a and b the resistivity of the metal is rho

Answer 1

A hollow metallic sphere has inner and outer radii a and b respectively.

let's cut an element at a distance r from the centre of metallic sphere.

then, area of elementary sphere, A = 4πr²

and thickness of elementary sphere, l = dr.

use formula, R = ρl/A

for continuous distribution, dR = ρ dr/A

or, $\int\limits^R_0{dR}=\rho\int\limits^b_a{\frac{1}{4\pi r^2}}\,dr$

or, $R=\frac{\rho}{4\pi}\left[-\frac{1}{r}\right]^b_a$

or, $R=\frac{\rho}{4\pi}\left[\frac{1}{a}-\frac{1}{b}\right]$

hence, resistance of metallic sphere between the points a and b is $\frac{\rho}{4\pi}\left[\frac{1}{a}-\frac{1}{b}\right]$