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A metal sphere of radius a is surrounded by a concentric metal sphere of inner radius b(b>a). The space between the sphere is filled with a material whose electrical conductivity $σ$ varies with the electric field strength E, according to the law $σ = K E$ , where K is a constant. A potential difference V is maintained between two spheres. Choose CORRECT statement(s)

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Current I between spheres is given by $I = \frac{4 π K V^{2}}{{(l o g_{e} (\frac{b}{a}))}^{2}}$

Current I between spheres is given by $I = \frac{4 π K V^{2}}{(\frac{b}{a})}$

Electric field at a distance r from centre (a < r < b) is given by $E = \frac{1}{r^{2}} \sqrt{\frac{I}{4 π K}}$ , where I is current between spheres.

Electric field at a distance r from centre (a < r < b) is given by $E = \frac{1}{r} \sqrt{\frac{I}{4 π K}}$ , where I is current between spheres.

answer is A, C.

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Detailed Solution

$I = σ E \times 4 π r^{2} = K E^{2} \times 4 π r^{2} \Rightarrow E = \frac{1}{r} \sqrt{\frac{I}{4 π K}} V = - \int_{b}^{a} E d r = \sqrt{\frac{I}{4 π K}} l o g_{e} \frac{b}{a} \Rightarrow I = \frac{4 π K V^{2}}{l o g_{e} \frac{b}{a}}$

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