Q: There are two concentric spheres of radius a and b respectively. If the space between them is filled with medium of resistivity ρ , then the resistance of the inter gap between the two spheres will be (Assume b > a)

Sol. Consider a concentric spherical shell of radius x and thickness dx, its resistance is

$\large dR = \frac{\rho \; dx}{4 \pi x^2} $ ; (by applying the formula R = ρl/A)

Total resistance , $\large R = \int_{a}^{b} \frac{\rho \; dx}{4 \pi x^2} $

$\large R = \frac{\rho}{4 \pi} \int_{a}^{b} \frac{dx}{x^2} $

$\large R = \frac{\rho}{4 \pi} \int_{a}^{b} x^{-2} dx$

$\large R = \frac{\rho}{4 \pi} [\frac{x^{-2+1}}{-2+1}]_{a}^{b} $

$\large R = \frac{\rho}{4 \pi} [-\frac{1}{x}]_{a}^{b} $

$\large R = \frac{\rho}{4\pi} [\frac{1}{a} – \frac{1}{b}] $