Question

A rectangular metallic loop is moving out of a uniform magnetic field region to a field-free region with a constant speed. When the loop is partially inside the magnetic field, the plot of the magnitude of the induced emf \( (\varepsilon) \) with time \( (t) \) is given by:

• A.
• B.
• C.
• D.

Hint:

To determine the emf induced in a moving conductor: - Use Faraday’s Law: \( \varepsilon = \left| \frac{d\Phi_B}{dt} \right| \). - If motion is constant and uniform, the change in flux is linear, leading to a linear change in emf. - For non-uniform motion, consider the velocity function to determine the rate of flux change.

Answer 1

The Correct Option is A

Step 1: Understanding Faraday's Law. According to Faraday's Law of Electromagnetic Induction, the induced emf \( \varepsilon \) in a loop is given by: \[ \varepsilon = \left| \frac{d\Phi_B}{dt} \right|, \] where \( \Phi_B \) is the magnetic flux through the loop. 

Step 2: Expressing flux in terms of motion. Since the loop is moving with constant velocity \( v \), the flux linkage \( \Phi_B \) is proportional to the area of the loop inside the magnetic field: \[ \Phi_B = B L x, \] where: - \( B \) is the magnetic field strength, - \( L \) is the width of the loop, - \( x \) is the portion of the loop still inside the field, given by \( x = vt \). 

Step 3: Computing emf. Differentiating \( \Phi_B \) with respect to time: \[ \varepsilon = B L \frac{dx}{dt} = B L v. \] Since \( v \) is constant, the emf remains constant while the loop is partially inside the field. However, as the loop starts exiting, the effective area inside the field decreases linearly, causing \( \varepsilon \) to decrease linearly to zero. 

Step 4: Identifying the correct graph. 
- Since the emf starts at zero, increases linearly while exiting, and reaches a peak before going to zero once the loop is fully out of the field, the correct choice is: \[ \boxed{1} \text{ (Linearly increasing graph)} \]