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) → \( \displaystyle\frac{\mu_0 I}{4\sqrt{2r}} \)
The magnetic field on the axis of a circular current loop of radius \( R \), at a distance \( x \) from its centre, follows from the Biot-Savart law:
\( \displaystyle B = \frac{\mu_0 I R^{2}}{2\left(R^{2} + x^{2}\right)^{3/2}} \)
Here \( R = r \) and \( x = r \):
\( \displaystyle B = \frac{\mu_0 I r^{2}}{2\left(r^{2} + r^{2}\right)^{3/2}} = \frac{\mu_0 I r^{2}}{2\left(2r^{2}\right)^{3/2}} \)
\( \displaystyle \left(2r^{2}\right)^{3/2} = 2\sqrt{2}\,r^{3} \;\Rightarrow\; B = \frac{\mu_0 I r^{2}}{4\sqrt{2}\,r^{3}} = \frac{\mu_0 I}{4\sqrt{2}\,r} \)
This is the expression in Option (2).
Note: It seems that there is a mistake in the original NTA paper, which prints the square-root sign over \( r \) as well (\( \sqrt{2r} \)), and over \( 2\pi r \) in options 3 and 4, when Option 2 should read \( \mu_0 I/(4\sqrt{2}\,r) \). The “316” in Option 4 also appears garbled.