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, whereas a smaller Amperian loop \( A \) encloses a small segment. The difference between enclosed currents, \( I_B - I_A \) for the given Amperian loops is:
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When calculating currents in rotating charge configurations, remember that the velocity of each charge is \( v = R\omega \), and the total current enclosed by an Amperian loop depends on the number of charges and the fraction of the total circle enclosed.
The current enclosed by an Amperian loop, driven by moving charges, is calculated as:
\[
I = nqv,
\]
where:
- \( n \) represents the count of charges.
- \( q \) denotes the magnitude of a single charge.
- \( v \) signifies the velocity of a charge.
Given that the charges are in rotational motion, the velocity \( v \) of each charge is related to the angular velocity \( \omega \) and the radius \( R \) of the circular path by \( v = R\omega \).
For the larger Amperian loop, designated as \( B \), the total enclosed current is the aggregate of contributions from all charges, expressed as:
\[
I_B = N \times q \times R\omega.
\]
For the smaller loop, designated as \( A \), the enclosed current is generated by only a portion of the charges. If this loop encircles a minor segment of the circular path, the fraction of the total charge encompassed is directly proportional to the ratio of the segment's length to the total circumference. Consequently, the current enclosed by loop \( A \) is:
\[
I_A = \frac{1}{2\pi} \times N \times q \times R\omega.
\]
The disparity between these currents is therefore:
\[
I_B - I_A = \frac{N}{2\pi} q\omega.
\]
Final Answer: \( \frac{N}{2\pi} q\omega \).
