Question

Obtain the formula for the resultant magnetic moment of the two concentric circular coils of radius \( r \), placed perpendicular to each other on passing the same current \( i \).

Hint:

The magnetic moment of two coils placed perpendicular to each other is the vector sum of their individual moments, and in this case, it results in a factor of \( \sqrt{2} \).

Answer 1

Step 1: Magnetic Moment of a Coil.
The magnetic moment \( M \) of a single coil is given by: \[ M = i A \] where \( i \) is the current, and \( A \) is the area of the coil. For a circular coil of radius \( r \), the area \( A = \pi r^2 \). Thus, the magnetic moment of each coil is: \[ M = i \pi r^2 \]
Step 2: Resultant Magnetic Moment.
Since the two coils are placed perpendicular to each other and carry the same current \( i \), the resultant magnetic moment \( M_{\text{res}} \) is the vector sum of the individual magnetic moments. Since the angle between the two moments is 90°, the resultant magnetic moment is: \[ M_{\text{res}} = \sqrt{M_1^2 + M_2^2} = \sqrt{(i \pi r^2)^2 + (i \pi r^2)^2} = i \pi r^2 \sqrt{2} \]
Step 3: Conclusion.
The resultant magnetic moment of the two coils is \( M_{\text{res}} = i \pi r^2 \sqrt{2} \).