Question

A circular loop of radius R is carrying current I. 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. \(1:\sqrt{2}\)
• B. \(1:3\sqrt{2}\)
• C. \(\sqrt{8}:1\)
• D. \(3\sqrt{2}:1\)

Hint:

Use the on-axis formula with x equal to R.

Answer 1

The Correct Option is C

Step 1: Centre
\(B_c = \frac{\mu_0I}{2R}\).

Step 2: On axis at x = R
\(B_x = \frac{\mu_0IR^2}{2(R^2+x^2)^{3/2}} = \frac{\mu_0IR^2}{2(2R^2)^{3/2}} = \frac{\mu_0I}{4\sqrt2R}\).

Step 3: Ratio
\(\frac{B_c}{B_x} = \frac{1/2}{1/(4\sqrt2)} = 2\sqrt2 = \sqrt8\). The ratio is \(\sqrt8:1\). Option (C).

Final Answer:
The ratio is root 8 : 1. \[ \boxed{\text{(C)}\ \sqrt8:1} \]