An infinite line charge of uniform electric charge density lies along the axis of an electrically conducting infinite cylindrical shell of radius R. At time , the space inside the cylinder is filled with a material of permittivity and electrical conduct

Question

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? [2016]

Smart Achievers · Accepted Answer

(3)

Electric field at a distance $t = 0$ from a line charge

$E = \frac{λ}{2 π ε r} = \frac{d V}{d r}$

where $λ$ is the linear charge density of the 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 outward.

$∴$ $\frac{d}{d t} (λ l) = - \frac{λ σ l}{ε}$ $\Rightarrow \frac{d λ}{d t} = - \frac{σ}{ε} λ$

Integrating,

$\int_{λ_{0}}^{λ} \frac{d λ}{λ} = - \frac{σ}{ε} \int_{0}^{t} d t$

$\ln (\frac{λ}{λ_{0}}) = - \frac{σ}{ε} t$

$∴$ $λ = λ_{0} e^{- (σ / ε) t}$

Current density,

$J = \frac{I}{2 π r l}$ $= \frac{λ σ l}{2 π r l ε}$ $= \frac{λ σ}{2 π ε r}$

Substituting $λ = λ_{0} e^{- (σ / ε) t}$ ,

$J = (\frac{λ_{0} σ}{2 π ε r}) e^{- (σ / ε) t}$

$∴$ $J = J_{0} e^{- (σ / ε) t}$

Solution

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