Electromagnetic Induction

Given: - Resistance, - Change in magnetic flux, - The circuit has only resistance and inductance (no external source). - To find: Charge flown,

Key concept: According to Faraday's law and Ohm’s law, the total charge that flows due to a change in magnetic flux is given by:

Substitute values:

Final Answer: 0.05 C

We need to analyze the induced electromotive force (EMF), the induced current, the direction of the current, and the potential differences in the loop.

Step 1: Calculate the Induced EMF

As the loop is pulled through the magnetic field, an EMF is induced in the wires perpendicular to the velocity and magnetic field. The motional EMF induced in a wire of length moving with velocity perpendicular to a magnetic field is given by:

In this case, the wires and are the ones experiencing the induced EMF. Both wires have length , so the total induced EMF is:

Step 2: Calculate the Resistance of the Loop

The loop has a total length of . The resistance per unit length is . Therefore, the total resistance of the loop is:

Step 3: Calculate the Induced Current

Using Ohm's Law, the induced current in the loop is:

Step 4: Determine the Direction of the Current

Using the right-hand rule (or Lenz's Law), as the loop moves to the right, the magnetic flux through the loop decreases (since the area within the field decreases). To oppose this change in flux, the induced current will create a magnetic field pointing downwards (in the same direction as the external field). This requires a clockwise current when viewed from above.

Step 5: Analyze the Potential Differences

Consider the potential difference and . Since PQ and RS are the source of the emf then however due to the resistance, some of the potential will be lost hence is wrong.

There is definitely induction in part SR, since it is also moving through the magnetic field.

Step 6: Determine the Correct Statements

1. Current in the loop I= Bbv / (2b+2a): This statement is TRUE. Assuming the typo is instead of the number 1, and that the total length of the wire is and thus total resistance is . The formula matches what we derived:
2. Current will be in clockwise direction, looking from the top: This statement is TRUE.
3. V_P - V_S = V_Q - V_R, where V is the potential: This statement is FALSE.
4. There cannot be any induction in part SR: This statement is FALSE.

Conclusion

The correct statements are:

• Current in the loop I= Bbv / (2b+2a)
• Current will be in clockwise direction, looking from the top.

Step 1: Expression for Current as a Function of Time

The current in the coil decreases uniformly from an initial value to zero over time . This implies that the current decreases linearly with time:

$ t = 0 I(t) = I_0 t = T I(t) = 0 Q I(t) = I_0 \left( 1 - \frac{t}{T} \right) I_0 I(t) = I_0 \left( 1 - \frac{t}{T} \right) I_0 = \frac{2Q}{T} $

To analyze the given problem, we use Faraday's Law of Induction and Lenz's Law. The setup involves a circular coil placed near a current-carrying conductor, both lying on the plane of the paper. The induced current in the loop is clockwise.

Analyzing the Induced Current

1. Direction of the Induced Current:
   The induced current is clockwise. By Lenz's Law, it must oppose the change in magnetic flux.
2. Change in Magnetic Flux:
   A clockwise induced current means the magnetic flux through the loop is increasing out of the plane. To oppose this, the induced magnetic field is into the plane.
3. Current in the Wire:
   For the magnetic flux to be increasing out of the plane, the magnetic field at the loop’s location must be increasing and directed out of the plane. According to the right-hand rule, this happens if the current in the wire is downward.
4. Time Dependence:
   Since the flux is changing, the current in the wire must be increasing with time — i.e., time-dependent.

Conclusion

The current in the wire is time-dependent and downward.

Final Answer: (A): time-dependent and downward