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A metal sphere of radius ‘a’ is surrounded by a concentric metal sphere of radius b. The space between the spheres is filled with a material whose electrical conductivity $σ$ varies with the electric field strength E according to the relation $σ = K . E$ where ‘K’ is a constant. A potential difference V is maintained between the spheres. The electric current density near the surface of sphere of radius ‘b’ is :

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a

$\frac{4 π K V^{2}}{{[l n (\frac{b}{a})]}^{2} b^{2}}$

b

$\frac{K V^{2}}{b^{2} {[\ln (\frac{b}{a})]}^{2}}$

c

$\frac{4 K V^{2}}{b^{2} {[\ln (\frac{b}{a})]}^{2}}$

d

$\frac{K V^{2}}{4 b^{2} {[\ln (\frac{b}{a})]}^{2}}$

answer is C.

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Detailed Solution

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$j = σ E = k E^{2} \Rightarrow \frac{I}{4 π x^{2}} = K E^{2} \Rightarrow E = \sqrt{\frac{I}{4 π k}} \times \frac{1}{x} \Rightarrow \int \vec{E} . d x = V = \sqrt{\frac{I}{4 π k}} \int_{a}^{b} \frac{1}{x} d x \Rightarrow V = \sqrt{\frac{I}{4 π k}} \ln (\frac{b}{a})$

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A metal sphere of radius ‘a’ is surrounded by a concentric metal sphere of radius b. The space between the spheres is filled with a material whose electrical conductivity σ varies with the electric field strength E according to the relation σ=K.E where ‘K’ is a constant. A potential difference V is maintained between the spheres. The electric current density near the surface of sphere of radius ‘b’ is :