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A parallel plate capacitor with rectangular plates is being discharged. Consider a rectangular loop centered on the plates and between them. The loop measures a by $2a$, the plate measures $2a$ by $4a$. What fraction of the displacement current (i) is encircled by the loop?

$\begin {array} {1 1} (a)\;i & \quad (b)\;\large\frac{i}{4} \\ (c)\;\large\frac{i}{3} & \quad (d)\;\large\frac{i}{2} \end {array}$

Displacement current, $I = \in_o\: \large\frac{d \phi E}{dt}$$= \in_o A \large\frac{dE}{dt} = \in_o A\: \large\frac{ d\bigg( \Large\frac{q}{\in_oA’} \bigg) }{ dt} where, A’ is area of each plate of capacitor So, I = \large\frac{A}{A’}$$ \times \large\frac{ dq}{dt}$$= \large\frac{(a)(2a)dq}{(2a)(4a)dt}$
$\large\frac{i}{4}$
Ans : (b)