Faradays Laws Of Induction

The equations describe the induced electromotive force (emf) in a circuit:

The magnitude of the emf ( ) is given by the product of inductance ( ) and the rate of change of current ( ).

Substituting the values, the emf is calculated using the discrete current change over time.

Plugging in the specific time interval ( , ) and assuming the current change ( ).

The final result for the induced emf is volts.

Given Information:
Side of square loop,
Speed of loop entering the magnetic field,
Magnetic field, , is uniform and perpendicular to the plane of the loop.

Step-by-Step Explanation:

Step 1: Understanding the problem clearly:
When the loop enters the magnetic field, magnetic flux through the loop changes, inducing an emf according to Faraday’s law of electromagnetic induction.

Step 2: Determine how emf changes with time:
The induced emf is given by Faraday's law as:

Since , , and are constant, the induced emf remains constant while the loop is entering or leaving the field. However, the emf changes direction based on whether the loop is entering or leaving the magnetic field.

Step 3: Calculate time intervals clearly:
The loop enters at speed , and loop side is . Thus, the time for the loop to completely enter the field is:

After fully entering, emf becomes zero since flux remains constant. Then, as the loop exits the field, emf is induced again (but with opposite sign), taking another to exit.

Step 4: Describe the induced emf clearly over time:

• From to , emf is constant positive (clockwise as per given sign convention).
• From to , the emf is zero (flux is constant).
• From to , emf is constant negative (opposite direction).

Thus, the correct graph should show a positive constant emf for the first interval, zero emf during the second interval, and negative constant emf in the third interval.

Conclusion:
Option (C) clearly matches this pattern.

Given Information:
Length of rod,
Horizontal component of Earth's magnetic field,
Acceleration due to gravity,
Time after release,

Step-by-Step Explanation:

Step 1: Find velocity ( ) of the rod after time :

Rod is falling freely under gravity, thus initial velocity , so velocity after seconds is:

Step 2: Calculate induced emf ( ):

The induced emf in a rod moving perpendicular to a magnetic field is given by:

Substitute values into this formula:

Simplify:

Final Conclusion:
The induced emf in the rod at is .