Question

A circular loop of radius $r$ is carrying current $I A$ The ratio of magnetic field at the center of circular loop and at a distance $r$ from the center of the loop on its axis is:

• A. $1: 3 \sqrt{2}$
• B. $2 \sqrt{2}: 1$
• C. $3 \sqrt{2}: 2$
• D. $1: \sqrt{2}$

Answer 1

The Correct Option is B

The magnetic field due to a current-carrying circular loop on its axis is given as: \[ B_{\text{axis}} = \frac{\mu_0 I r^2}{2 (r^2 + x^2)^{3/2}} \] At the center of the loop (\(x = 0\)): \[ B_{\text{center}} = \frac{\mu_0 I}{2r} \] At a distance \(r\) on the axis (\(x = r\)): \[ B_{\text{axis}} = \frac{\mu_0 I}{2 \cdot 2 \sqrt{2} r} \] The ratio of magnetic fields is: \[ \frac{B_{\text{center}}}{B_{\text{axis}}} = \frac{\frac{\mu_0 I}{2r}}{\frac{\mu_0 I}{2 \cdot 2 \sqrt{2} r}} = 2\sqrt{2} \] Thus, the ratio \(B_{\text{center}} : B_{\text{axis}} = 2\sqrt{2} : 1\).