Question

A metal wire of length \( L \) is bent to form a circular coil of number of turns \( n \). The coil is placed in magnetic field \( B \) and current is passed through the coil. The maximum torque acting on the coil is

• A. \( \frac{BIL^2}{4 \pi n} \)
• B. \( \frac{BIL^2}{2 \pi n} \)
• C. \( \frac{B^2IL}{2 \pi n} \)
• D. \( \frac{B^2IL}{4 \pi n} \)

Hint:

The torque on a current-carrying coil in a magnetic field depends on the number of turns, magnetic field strength, current, and the area of the coil.

Answer 1

The Correct Option is A

Step 1: Torque on the coil.
The torque \( \tau \) acting on a coil in a magnetic field is given by: \[ \tau = n \cdot B \cdot I \cdot A \] where \( n \) is the number of turns, \( B \) is the magnetic field, \( I \) is the current, and \( A \) is the area of the coil. For a circular coil, the area \( A = \pi r^2 \), where \( r \) is the radius of the coil. The radius is related to the length of the wire by \( L = 2 \pi r \), so \( r = \frac{L}{2 \pi} \).
Step 2: Substituting the values.
Substitute \( A = \pi \left( \frac{L}{2 \pi} \right)^2 \) into the torque formula: \[ \tau = n \cdot B \cdot I \cdot \frac{L^2}{4 \pi^2} \] Step 3: Conclusion.
Thus, the maximum torque is \( \frac{BIL^2}{4 \pi n} \), which corresponds to option (A).