Question

Consider a circular loop of radius R on the $xy$-plane carrying a steady current anticlockwise. The magnetic field at the center of the loop is given by

• A. $\frac{\mu_0}{2R} I \hat{x}$
• B. $\frac{\mu_0}{2R} I \hat{y}$
• C. $\frac{\mu_0}{2R} I \hat{z}$
• D. $\frac{\mu_0}{R} I \hat{x}$
• E. $\frac{\mu_0}{R} I \hat{y}$

Hint:

For circular loops, the field at the center is always perpendicular to the plane of the loop. Anticlockwise current in the $xy$-plane results in a field along $+\hat{z}$, while clockwise results in $-\hat{z}$.

Answer 1

The Correct Option is C

Concept: The magnetic field produced by a circular current loop is determined using the Biot-Savart Law. For a point at the center of the loop, the field magnitude depends on the current and the radius, while the direction follows the right-hand grip rule.

Step 1: {Determine the magnitude of the magnetic field.}
For a circular loop of radius $R$ carrying a steady current $I$, the formula for the magnetic field $B$ at the center is: $$B = \frac{\mu_0 I}{2R}$$

Step 2: {Determine the direction using the right-hand rule.}
The loop is situated in the $xy$-plane with an anticlockwise current. By curling the fingers of the right hand in the anticlockwise direction, the thumb points vertically upwards along the positive $z$-axis ($+\hat{z}$).

Step 3: {Final vector representation.}
Combining the calculated magnitude and the determined direction gives the final magnetic field vector at the center: $$\vec{B} = \frac{\mu_0 I}{2R} \hat{z}$$