Motional Electromotive Force

Step 1: Understanding the Concept:
This problem involves calculating the magnetic flux through a changing area and the motional electromotive force (EMF) induced in a conductor moving through a magnetic field.

Step 2: Key Formula or Approach:
1. Magnetic Flux ( ): The magnetic flux through a surface is defined as , where is the magnetic field strength, is the area of the surface, and is the angle between the magnetic field and the normal to the surface. 2. Motional EMF ( ): When a conductor of length moves with velocity perpendicular to a uniform magnetic field , an EMF is induced across its ends, given by . Alternatively, using Faraday's Law, .

Step 3: Detailed Explanation:
Magnetic Flux Calculation:
Let the length of the conductor PQ be and its distance from the side RS be at any instant. The area of the rectangular loop PQRS is . The magnetic field is uniform and directed perpendicularly into the plane of the loop. So, the angle between and the area vector is , and . The magnetic flux linked with the loop is: Motional EMF Calculation:
Method 1: Using the motional EMF formula
The conductor PQ of length is moving with a constant velocity perpendicular to the magnetic field . The motional EMF induced across PQ is directly given by: Method 2: Using Faraday's Law of Induction
The EMF is the rate of change of magnetic flux. Since and are constant, we have: The conductor is moved towards the left, so the distance is decreasing. The rate of change of position, , is the velocity. Since decreases, . Both methods give the same result for the motional EMF.

Step 4: Final Answer:
The magnetic flux is and the motional EMF is . This matches option (C).