Question

Draw the plots showing the variation of magnetic flux φ linked with the loop with time t and variation of induced emf E with time t. Mark the relevant values of E, φ and t on the graphs.

Answer 1

Magnetic Flux and Induced emf vs Time

When a square loop moves at a constant velocity through a uniform magnetic field, the magnetic flux \( \Phi \) and induced emf \( \mathcal{E} \) change over time during its entry and exit from the field. Three time intervals are defined:

• From \( t = 0 \) to \( t_1 \): Loop enters the magnetic field.
• From \( t_1 \) to \( t_2 \): Loop is fully within the magnetic field.
• From \( t_2 \) to \( t_3 \): Loop exits the magnetic field.

Magnetic Flux \( \Phi \) vs Time \( t \)

• From \( 0 \leq t < t_1 \): Magnetic flux increases linearly as the loop enters the field, directly proportional to the increasing area within the field.
• From \( t_1 \leq t \leq t_2 \): Magnetic flux remains constant because the loop is entirely within the field.
• From \( t_2 < t \leq t_3 \): Magnetic flux decreases linearly as the loop exits the field, directly proportional to the decreasing area within the field.
• For \( t > t_3 \): Magnetic flux becomes zero once the loop is completely outside the field.

Induced emf \( \mathcal{E} \) vs Time \( t \)

• From \( 0 \leq t < t_1 \): A constant, non-zero induced emf is generated due to the changing magnetic flux.
• From \( t_1 \leq t \leq t_2 \): The induced emf is zero because the magnetic flux is constant.
• From \( t_2 < t \leq t_3 \): A constant induced emf, opposite in sign to the previous interval, is generated as the flux decreases.
• For \( t > t_3 \): The induced emf is zero because there is no change in magnetic flux.

Summary:

The magnetic flux \( \Phi \) and induced emf \( \mathcal{E} \) follow a predictable progression as the loop traverses the magnetic field:

• Magnetic Flux: \[ \Phi = \begin{cases} \text{Increases linearly,} & 0 \leq t < t_1 \\ \text{Constant,} & t_1 \leq t \leq t_2 \\ \text{Decreases linearly,} & t_2 < t \leq t_3 \\ \text{Zero,} & t > t_3 \end{cases} \]
• Induced emf: \[ \mathcal{E} = \begin{cases} \text{Constant and non-zero,} & 0 \leq t < t_1 \\ 0, & t_1 \leq t \leq t_2 \\ \text{Constant and opposite,} & t_2 < t \leq t_3 \\ 0, & t > t_3 \end{cases} \]