Question

A wire of length 10 m carrying current of 1 A is bent into a circular loop. If a magnetic field of \(2\pi \times 10^{-4} \, \text{T}\) is applied on the loop, then the maximum torque acting on it is:

• A. \(100 \times 10^{-4} \, \text{Nm}\)
• B. \(50 \times 10^{-4} \, \text{Nm}\)
• C. \(25 \times 10^{-4} \, \text{Nm}\)
• D. \(75 \times 10^{-4} \, \text{Nm}\)

Hint:

For a wire bent into a circular loop, use the total length to find the radius and area. Maximum torque on a loop in a magnetic field is given by \( \tau = n I A B \), where \( n \) is number of turns.

Answer 1

The Correct Option is B

Step 1: Use the formula for maximum torque on a current loop.
Maximum torque is given by: \[ \tau = n I A B \] Where: \( n = 1 \) (since it's a single loop) 
\( I = 1 \, \text{A} \) 
\( B = 2\pi \times 10^{-4} \, \text{T} \) 
\( A = \pi r^2 \) is the area of the loop 
Step 2: Calculate the radius of the loop.
The wire forms a circle, so the circumference \( C = 2\pi r = 10 \Rightarrow r = \frac{10}{2\pi} = \frac{5}{\pi} \) 
Step 3: Calculate the area.
\[ A = \pi r^2 = \pi \left(\frac{5}{\pi}\right)^2 = \frac{25}{\pi} \] Step 4: Calculate the torque.
\[ \tau = n I A B = 1 \times 1 \times \frac{25}{\pi} \times 2\pi \times 10^{-4} = 25 \times 2 \times 10^{-4} = 50 \times 10^{-4} \, \text{Nm} \] Step 5: Select the correct option.
The calculated torque is \(50 \times 10^{-4} \, \text{Nm}\), which matches option (2).