The region between two concentric conducting spheres with radii a and b is filled with a conducting material of resistivity $ρ$ , and maintained a potential difference V

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the radial resistance between the spheres is $\frac{ρ}{4 π} (\frac{1}{a} - \frac{1}{b})$

the radial resistance between the spheres is $\frac{ρ}{2 π} (\frac{1}{a} - \frac{1}{b})$

the expression for current density as a function of radius is $J (r) = \frac{V a b}{ρ (b - a) r^{2}}$

the expression for current density as a function of radius is $J (r) = \frac{4 V a b}{ρ (b - a) r^{2}}$

answer is A, C.

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Detailed Solution

a) Consider a shell of thickness dr and radius r. Resistance of such shell is dR
$d R = \frac{ρ d r}{4 π r^{2}} \Rightarrow R = \int_{a}^{b} d R \Rightarrow = \int_{a}^{b} \frac{ρ d r}{4 π r^{2}} = \frac{ρ}{4 π} (\frac{1}{a} - \frac{1}{b})$ Question Image
b) $I = \frac{V}{R} = \frac{V 4 π a b}{(b - a)}; J (r) = \frac{I}{4 π r^{2}} = \frac{V 4 π a b}{(b - a) 4 π r^{2}} \Rightarrow J (r) = \frac{V a b}{ρ (b - a) r^{2}}$