Question

The wire loop shown in the figure carries a steady current \( I \). Each straight section of the loop has length \( d \). A part of the loop lies in the \( xy \)-plane and the other part is tilted at \( 30^\circ \) with respect to the \( xz \)-plane. The magnitude of the magnetic dipole moment of the loop (in appropriate units) is:

 

• A. \( \sqrt{2} I d^2 \)
• B. \( 2 I d^2 \)
• C. \( \sqrt{3} I d^2 \)
• D. \( I d^2 \)

Hint:

For a current loop, the magnetic dipole moment is proportional to the current and the area enclosed by the loop. Tilt does not affect the proportionality in this case.

Answer 1

The Correct Option is D

1. The magnetic dipole moment \( \mu \) of a current loop is given by:
\[ \mu = I A \] where \( A \) is the area of the loop. The magnetic moment depends on the current and the area enclosed by the loop. For this problem, the loop consists of two straight sections: one in the \( xy \)-plane and the other tilted at \( 30^\circ \) with respect to the \( xz \)-plane.
2. The area of the loop formed in the \( xy \)-plane is a square with side length \( d \), so its area is:
\[ A_{\text{xy}} = d^2 \] 3. The area of the loop formed by the tilted section in the \( xz \)-plane is also a square with side length \( d \), but it is tilted at an angle of \( 30^\circ \). The area contribution of this tilted section is:
\[ A_{\text{tilted}} = d^2 \cos(30^\circ) = d^2 \times \frac{\sqrt{3}}{2} \] 4. Since we are only interested in the magnetic dipole moment due to the steady current in the loop, the tilt does not change the total contribution to the magnetic moment, as the loop's symmetry ensures the dipole moment remains proportional to the area of the loop. The magnetic dipole moment is:
\[ \mu = I \times d^2 \] Thus, the magnitude of the magnetic dipole moment of the loop is \( I d^2 \).