Question

The wire of length $\ell$ is bent into a circular loop of a single turn and is suspended in a magnetic field of induction $B$. When a current $I$ is passed through the loop, the maximum torque experienced by it is

• A. $\frac{1}{4\pi} BI\ell^2$
• B. $\frac{1}{4\pi} BI^2 \ell$
• C. $\frac{1}{4\pi} BI\ell$
• D. $\frac{1}{4\pi} B^2 I \ell$
• E. $\frac{1}{4\pi} B^2 I^2 \ell^2$

Hint:

Loop torque = $IAB$ and convert wire length → radius → area.

Answer 1

The Correct Option is A

Concept: Torque on current loop: \[ \tau = IAB \sin\theta \] Maximum torque: \[ \tau_{\max} = IAB \]

Step 1: Length relation Wire forms circle: \[ \ell = 2\pi r \Rightarrow r = \frac{\ell}{2\pi} \]

Step 2: Area of circle \[ A = \pi r^2 = \pi \left(\frac{\ell}{2\pi}\right)^2 \] \[ A = \frac{\ell^2}{4\pi} \]

Step 3: Substitute in torque \[ \tau = I B \cdot \frac{\ell^2}{4\pi} \] \[ \tau = \frac{1}{4\pi} BI \ell^2 \] Physical Insight:
• Larger loop → larger torque
• Torque tries to align loop with field Final Answer: \[ \frac{1}{4\pi} BI\ell^2 \]