Question

A parallel-plate capacitor with circular plates of radius $ R $ is being charged with a current $ i $ . Find the value $ \oint B\cdot d \,s $ between the plates at radius $ r = R /5 $ from their centre

• A. $ \mu_{0}i /5 $
• B. $ \mu_{0}i/25 $
• C. $ \mu_{0}\,i $
• D. $ 5\mu_{0}\,i $

Answer 1

Answer: (B)

Surface integral of magnetic field between the circular plates of capacitor
$\oint B\cdot ds= \left|B\right|2 \pi r=\mu_{0} \,\varepsilon_{0} \frac{d}{dt} \int\int E\cdot d\,A $
$=\mu_{0}\varepsilon_{0} \frac{d\left|E\right|}{dt}\pi r^{2}=\mu_{0}\varepsilon_{0} \frac{d\left|E\right|}{dt} \pi r^{2}$
$=\frac{\mu_{0} i\,\pi r^{2}}{\pi\,R^{2}}=\frac{\mu_{0} i r^{2}}{R^{2}}$
Given, $r=\frac{R}{5}$
$\therefore \oint B\cdot dS=\frac{\mu_{0} i}{25}$