question_answer

There are two concentric spheres of radius a and b respectively. If the space between them is filed with medium of resistivity p, then the resistance of the inter gap between the two spheres will be

A)  \[\frac{\alpha }{4\pi (b+a)}\]                    

B)  \[\frac{\alpha }{4\pi }\left( \frac{1}{b}-\frac{1}{a} \right)\]

C)  \[\frac{\alpha }{4\pi }\left( \frac{1}{{{a}^{2}}}-\frac{1}{{{b}^{2}}} \right)\]            

D)  \[\frac{\alpha }{4\pi }\left( \frac{1}{a}-\frac{1}{b} \right)\]

E)  \[\frac{\alpha }{4\pi (b-a)}\]

Correct Answer: D

Solution :

                Caisider a concentric spherical shell of radius \[x\] and thickness \[dx\] as showing in figure, its resistance is\[{{d}_{R}}=\frac{\rho \,dx}{4\pi {{x}^{2}}}\] \[\therefore \]Total resistance \[R=\frac{\rho }{4\pi }\int_{a}^{b}{\frac{dx}{{{x}^{2}}}}\]                                         \[=\frac{\rho }{4\pi }\left[ \frac{1}{a}-\frac{1}{b} \right]\]