Space between two concentric conducting spheres of radii a and b (b>a) is filled with a medium of resistivity $ρ$ . The resistance between the two spheres will be:

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$\frac{ρ}{4 π} (\frac{1}{a} + \frac{1}{b})$

$\frac{ρ}{2 π} (\frac{1}{a} + \frac{1}{b})$

$\frac{ρ}{2 π} (\frac{1}{a} - \frac{1}{b})$

$\frac{ρ}{4 π} (\frac{1}{a} - \frac{1}{b})$

answer is A.

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Detailed Solution

Consider an shell of thickness dx with a radius of x

$dR = ρ \cdot \frac{dx}{4 {πx}^{2}}$

$R = \int dR = ρ \cdot \int_{a}^{b} \frac{dx}{4 {πx}^{2}}$

$R = \frac{ρ}{4 π} {[- \frac{1}{x}]}_{a}^{b}$

$R = \frac{ρ}{4 π} (\frac{1}{a} - \frac{1}{b})$

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