Question

In a circular loop of radius \( R \), current \( I \) enters at point \( A \) and exits at point \( B \), as shown in the figure. The value of the magnetic field at the centre \( O \) of the loop is:

• A. \( \dfrac{\mu_0 I}{R} \)
• B. zero
• C. \( \dfrac{\mu_0 I}{2R} \)
• D. \( \dfrac{\mu_0 I}{4R} \)

Hint:

If current enters and exits a full circular loop at two diametrically opposite points, the current splits into two semicircles. The magnetic field at the center due to each half cancels the other if they carry equal current in opposite directions.

Answer 1

The Correct Option is B

The scenario involves a complete circular loop where current enters at point A and exits at point B. This current splits equally into two symmetrical semicircular paths, forming the upper and lower halves of the circle. Consequently, the current in the upper and lower semicircles flows in opposite directions around the loop. Each semicircular arc generates a magnetic field at the center \( O \) with identical magnitude but opposing directions. The formula for the magnetic field of a semicircular arc is given by \[B_{\text{semicircle}} = \frac{\mu_0 I}{4R}\]. Therefore, the upper semicircle produces a magnetic field \( B \) in one direction (e.g., into the page), and the lower semicircle produces a field \( B \) in the opposite direction (out of the page). Because the magnitudes are equal and the directions are opposite, the net magnetic field at the center is calculated as \[B_{\text{net}} = \frac{\mu_0 I}{4R} - \frac{\mu_0 I}{4R} = 0\]. % Final Answer Statement Answer: \( \boxed{\text{(B)} \ \text{zero}} \)