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

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Solution

Given, a parallel plate capacitor with rectangular plates.

Displacement current is given by,

I_{d} = ε_{0} \frac{{dϕ}_{E}}{dt} = ε_{0} A \frac{dE}{dt} = ε_{0} A \frac{d}{dt} (\frac{q}{ε_{0} A'})

where,
A' is the area of each plate of the capacitor.

∴

I_{d} = \frac{A}{A'} \frac{dq}{dt} = \frac{(a) (2 a)}{(2 a) (4 a)} \frac{dq}{dt}

= \frac{1}{4} I .

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