Question

Two concentric and coplanar circular coils have radii a and b (>> a) as shown in the figure. Resistance of the inner coil is R. If current in the outer coil is increased from 0 to I, What will be the total charge circulating in the inner coil.

• A. $\frac{\mu_0 \pi ia^2}{2Rb}$
• B. $\frac{\mu_0 \pi ia^2}{Rb}$
• C. $\frac{\mu_0 \pi ia^2}{4Rb}$
• D. $\frac{2\mu_0 \pi ia^2}{Rb}$

Answer 1

Answer: (A)

(A)

Answer 2

The magnetic field produced by the outer coil at its center is $B = \frac{\mu_0 I}{2b}$. Due to $b \gg a$, this field is approximately uniform over the inner coil's area. The magnetic flux through the inner coil is $\Phi = B \cdot (\pi a^2) = \frac{\mu_0 \pi I a^2}{2b}$. When the current in the outer coil increases from $0$ to $I$, the change in magnetic flux is $\Delta\Phi = \frac{\mu_0 \pi I a^2}{2b}$. The total charge ($Q$) circulating in the inner coil is given by $Q = \frac{\Delta\Phi}{R}$, where $R$ is the resistance of the inner coil. Substituting $\Delta\Phi$, we get $Q = \frac{\mu_0 \pi I a^2}{2Rb}$.