A current $I$ flows in a circular loop of radius $r$. The magnetic field at a point on its axis at distance $r$ from the center of the loop is

$\frac{μ_0I}{4\sqrt{2}r}$

The correct answer is Option (2) → $\frac{μ_0I}{4\sqrt{2}r}$

Given:

Current in circular loop: $I$

Radius of loop: $r$

Point on axis at distance $x = r$ from center

Magnetic field on axis of circular loop:

$B = \frac{\mu_0 I r^2}{2 (r^2 + x^2)^{3/2}}$

Substitute $x = r$:

$B = \frac{\mu_0 I r^2}{2 (r^2 + r^2)^{3/2}} = \frac{\mu_0 I r^2}{2 (2 r^2)^{3/2}} = \frac{\mu_0 I r^2}{2 (2^{3/2} r^3)}$

$B = \frac{\mu_0 I}{2 \cdot 2^{3/2} r} = \frac{\mu_0 I}{2 \cdot 2 \sqrt{2} r} = \frac{\mu_0 I}{4 \sqrt{2} r}$

Answer: $B = \frac{\mu_0 I}{4 \sqrt{2} r}$