Question

A rectangular loop of dimension \( L \) and width \( w \) moves with a constant velocity \( v \) away from an infinitely long straight wire carrying a current \( I \) in the plane of the loop as shown in the figure below. Let \( R \) be the resistance of the loop. What is the current in the loop at the instant the near-side is at a distance \( r \) from the wire? 

• A. \( \frac{\mu_0 I L w}{2\pi R (r + 2w)} \)
• B. \( \frac{\mu_0 I L w}{2\pi R (r + w)} \)
• C. \( \frac{\mu_0 I L w}{2\pi R (r + w)^2} \)
• D. \( \frac{\mu_0 I L w}{2\pi R (r + 2w)^2} \)

Hint:

The induced current in a moving loop is related to the rate of change of the magnetic flux through the loop, which depends on the distance from the current-carrying wire.

Answer 1

The Correct Option is C

Step 1: Understanding the induced EMF.
As the loop moves with a constant velocity, the magnetic flux through the loop changes due to the changing distance from the current-carrying wire. According to Faraday's law of induction, an EMF is induced in the loop. The induced EMF depends on the distance \( r \) from the wire and the velocity \( v \).
Step 2: Using the formula for induced current.
The induced current in the loop is given by \( I = \frac{\mu_0 I L w}{2\pi R (r + w)^2} \), where \( \mu_0 \) is the permeability of free space, \( I \) is the current in the wire, and \( R \) is the resistance of the loop.
Step 3: Conclusion.
Thus, the correct answer is option (C).