Question

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

• A. $\frac{\rho }{4\pi (b+a)}$
• B. $\frac{\rho }{4\pi }(\frac{1}{b}-\frac{1}{a})$
• C. $\frac{\rho }{4\pi }(\frac{1}{a^2}-\frac{1}{b^2})$
• D. $\frac{\rho }{4\pi }(\frac{1}{a}-\frac{1}{b})$

Answer 1

The correct option is D $\frac{\rho }{4\pi }(\frac{1}{a}-\frac{1}{b})$

Consider at a distance n from center a spherical layer of thickness dx then

$dR=\frac{\rho dx}{4\pi x^2}$

$\Rightarrow {\int }_a^{b}dR={\int }_a^b\frac{edx}{4\pi x^2}$

$R_{ab}=\frac{\rho }{4\pi }[\frac{-1}{x}{]}_a^b$

$R_{ab}=\frac{\rho }{4\pi }[\frac{1}{a}-\frac{1}{b}]$