Two concentric circular coils, each of radius \( r \) and carrying a current \( I \), are placed with their planes perpendicular to each other. The resultant magnetic moment of the two coils is
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \( \sqrt{2}\,I\pi r^{2} \)
A current loop of area \( A \) has a magnetic moment \( m=IA \). It is a vector along the normal to the plane of the loop. The planes of the two coils are perpendicular, so their magnetic moments are also perpendicular. The resultant is found as for two vectors at right angles.
Magnetic moment of each coil:
\( \begin{aligned} m_1=m_2 &= IA \\ &= I\pi r^{2} \end{aligned} \)
Resultant magnetic moment:
\( \begin{aligned} m &= \sqrt{m_1^{2}+m_2^{2}} \\ &= \sqrt{(I\pi r^{2})^{2}+(I\pi r^{2})^{2}} \\ &= \sqrt{2}\,I\pi r^{2} \end{aligned} \)
From the above, \( m=\sqrt{2}\,I\pi r^{2} \). Hence, Option (1) is correct.
NCERT (2026-27):
Circular current loop as a magnetic dipole – “As earlier, we define the magnetic moment m to have a magnitude IA, m = I A.”