Question

A parallel-plate capacitor with circular plates of radius $R$ is being charged with a current $i$ . Find the value $\oint B\cdot ds$ 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
$\oint B\cdot ds=\left|B\right|2\pi r={\mu }_0{\epsilon }_0\frac{d}{dt}\int \int E\cdot dA$
$={\mu }_0{\epsilon }_0\frac{d|E|}{dt}\pi r^2={\mu }_0{\epsilon }_0\frac{d|E|}{dt}\pi r^2$
$=\frac{{\mu }_{0}i\pi r^2}{\pi R^2}=\frac{{\mu }_{0}i{r}^2}{R^2}$
Given, $r=\frac{R}{5}$
$\therefore \oint B\cdot dS=\frac{{\mu }_{0}i}{25}$