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: Understand Faraday's Law. Faraday's Law of Electromagnetic Induction states that the induced electromotive force (emf) \( \varepsilon \) in a loop is quantified by the absolute rate of change of magnetic flux \( \Phi_B \) through it: \[ \varepsilon = \left| \frac{d\Phi_B}{dt} \right|, \] where \( \Phi_B \) represents the magnetic flux within the loop.

Step 2: Relate flux to motion. Given the loop's constant velocity \( v \), the magnetic flux linkage \( \Phi_B \) is directly proportional to the area of the loop that is within the magnetic field: \[ \Phi_B = B L x, \] where: - \( B \) signifies the magnetic field strength, - \( L \) denotes the loop's width, - \( x \) is the segment of the loop still within the field, defined as \( x = vt \). 

Step 3: Calculate the emf. By differentiating \( \Phi_B \) with respect to time: \[ \varepsilon = B L \frac{dx}{dt} = B L v. \] As \( v \) is constant, the emf remains constant while the loop is partially within the field. However, upon the loop's exit, the effective area inside the field diminishes linearly, consequently causing \( \varepsilon \) to decrease linearly to zero. 

Step 4: Identify the correct graph. 
- The emf initiates at zero, increases linearly during the exit phase, attains a peak, and then returns to zero once the loop is completely outside the field. Thus, the appropriate selection is: \[ \boxed{1} \text{ (Linearly increasing graph)} \]