Question

Find magnetic field on circular loop of radius $r$, placed between circular plates of capacitor of radius $R$ having displacement current $i_d,r<R$.

• A. $\frac{{\mu }_0{i}_{d}r}{2\pi R^2}$
• B. $\frac{{\mu }_0{i}_d}{2\pi R}$
• C. $\frac{{\mu }_0{i}_d}{2\pi r}$
• D. zero

Answer 1

Answer: (A)

Consider a loop of radius $r(<R)$ between the two circular plates, placed coaxially with them. The area of the loop $=\pi r^2$.
By symmetry, magnetic field is equal in magnitude at all points on the loop. If $i_d$ is the displacement current crossing the loop and $i_d$ is the total displacement current between plates $i_d=\frac{i_d}{\pi R^2}\times \pi r^2$.
Using Ampere-Maxwell's law, we have
$B\cdot dl={\mu }_0{i}_d^{\prime }$ or $B\cdot 2\pi r={\mu }_0{i}_d\frac{\pi r^2}{\pi R^2}$ or $B=\frac{{\mu }_0{i}_{d}r}{2\pi R^2}$