N equally spaced charges each of value q , are placed on a circle of radius R . The circle rotates about its axis with an angular velocity $$\omega$$ as shown in the figure. A bigger Amperian loop B encloses the whole circle where as a smaller Amperian loop A encloses a small segment. The difference between enclosed currents, $$I_A - I_B$$, for the given Amperian loops is

For Amperian loop B:

The loop encloses the entire plane of the rotating circle.

Every charge crossing the loop surface in one direction completes a full loop and crosses back through the enclosed area $$\implies I_B = 0$$

For Amperian loop A:

The loop encloses a small segment of the circumference.

The average current due to $$N$$ discrete rotating charges passing through this segment per unit time:

$$T = \frac{2\pi}{\omega}$$

$$I_A = \frac{Q_{\text{total}}}{T} \implies I_A = \frac{Nq}{\left(\frac{2\pi}{\omega}\right)} = \frac{N}{2\pi}qw$$

Difference between enclosed currents:

$$I_A - I_B = \frac{N}{2\pi}qw - 0 = \frac{N}{2\pi}qw$$