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 ?

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 ?

Infinity Learn · Accepted Answer

$$j=E$$σ⇒−$$12$$$$π$$$$rldqdt=$$λ$$2$$πε$$0r$$σ put $$q=$$λ$$/$$⇒−$$1$$λdλ$$dt=$$σε$$0$$⇒∫Λ$$0$$λ dλλ$$=$$−σε$$0$$∫$$0t$$ dt⇒ln⁡λλ$$0=$$−σtε$$0$$⇒λ$$=$$λ$$0e$$−σ$$t/$$ε$$0$$ Now, $$j=E$$σ$$=$$λ$$2$$πε$$0r$$σ$$=$$σ$$2$$πε$$0r$$λ$$0e$$−σtε$$0$$ "j decreases exponentially with time and j→$$0$$ as t→∞

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