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
Answer & explanation
Correct answer: option 2
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: $B = \frac{\mu_0 I}{4 \sqrt{2} r}$