Question

A current \( i \) flows through a circular loop of radius \( R \). The ratio of the magnetic field produced at its centre to the field produced at a point at a distance \( \frac{R}{\sqrt{3}} \) from its centre on its axis is:

• A. \( 1:8 \)
• B. \( 8:1 \)
• C. \( 1:4 \)
• D. \( 4:1 \)

Hint:

The magnetic field at the center of a current-carrying loop is stronger than the field at a point on its axis, with the strength decreasing as we move away from the center.

Answer 1

The Correct Option is B

Step 1:
The magnetic field at the center of the loop is given by: \[ B_{\text{center}} = \frac{\mu_0 i}{2R} \] where \( i \) is the current and \( R \) is the radius of the loop.
Step 2:
The magnetic field at a distance \( \frac{R}{\sqrt{3}} \) on the axis of the loop is given by: \[ B_{\text{axis}} = \frac{\mu_0 i}{4R} \left( \frac{1}{\left( 1 + \left(\frac{d}{R}\right)^2 \right)^{3/2}} \right) \]
Step 3:
The ratio of the magnetic fields is: \[ \frac{B_{\text{center}}}{B_{\text{axis}}} = 8:1 \]