An infinite line charge of uniform electric charge density $λ$ lies along the axis of an electrically conducting infinite cylindrical shell of radius R. At time t=0, the space inside the cylinder is filled with a material of permittivity $ε$ and electrical conductivity $σ .$ The electrical conduction in the material follows Ohm’s law. Which one of the following graphs best describes the subsequent variation of the magnitude of current density j(t) at any point in the material ?

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answer is C.

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Detailed Solution

Electric field at a distance r from line charge $E = \frac{λ}{2 π ε r} = \frac{- d V}{d r}$ ( $λ =$ linear charge density of wire)

$d V = - \frac{λ}{2 π ε r} d r$

Current through the elemental shell $I = \frac{| d V |}{d R} = \frac{\frac{λ}{2 π ε r} d r}{\frac{1}{σ} \times \frac{d r}{2 π r l}} = \frac{λ σ l}{ε} (∵ R = ρ \frac{l}{A} ∴ d R = ρ \frac{d r}{2 π r l} = \frac{1}{σ} \frac{d r}{2 π r l})$

This current is radially outwards, $∴ \frac{d}{d t} (λ l) = \frac{- λ σ l}{ε} \Rightarrow \int_{λ_{0}}^{λ} \frac{d λ}{λ} = - (\frac{σ}{ε}) \int_{0}^{t} d t \Rightarrow λ = λ_{0} e^{- (σ / ε) t}$

$∴ J = \frac{I}{2 π r l} = \frac{λ σ l}{2 π r l \in} = \frac{λ σ}{2 π ε r}$ or $J = (\frac{λ_{0} σ}{2 π r \in}) e^{- (σ / ε) t} \Rightarrow J = J_{0} e^{- (σ / ε) t}$