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

Consider a square loop moving with a constant velocity through a uniform magnetic field region. The magnetic flux \( \Phi \) and induced emf \( \mathcal{E} \) vary over time as the loop enters and exits the field. Let’s define three distinct intervals:

• From \( t = 0 \) to \( t_1 \): The loop enters the magnetic field.
• From \( t_1 \) to \( t_2 \): The loop is entirely inside the field.
• From \( t_2 \) to \( t_3 \): The loop exits the magnetic field.

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

• From \( 0 \leq t < t_1 \): The magnetic flux increases linearly as the loop enters the magnetic field. As the area of the loop inside the magnetic field increases, so does the flux.
• From \( t_1 \leq t \leq t_2 \): The magnetic flux remains constant as the loop is fully inside the magnetic field.
• From \( t_2 < t \leq t_3 \): The magnetic flux decreases linearly as the loop exits the magnetic field. As the area of the loop inside the magnetic field decreases, the flux decreases.
• For \( t > t_3 \): The magnetic flux becomes zero as the loop is completely outside the magnetic field.

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

• From \( 0 \leq t < t_1 \): The induced emf is constant and non-zero. The change in magnetic flux induces a constant emf.
• From \( t_1 \leq t \leq t_2 \): The induced emf is zero, as there is no change in flux while the loop is fully inside the field.
• From \( t_2 < t \leq t_3 \): The induced emf is constant but opposite in sign, as the flux decreases while the loop exits the magnetic field.
• For \( t > t_3 \): The induced emf becomes zero, as there is no change in flux once the loop is outside the magnetic field.

Summary:

The magnetic flux \( \Phi \) and induced emf \( \mathcal{E} \) change in a predictable pattern as the loop moves through 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: \[ \mathcal{E} = \begin{cases} \text{Constant and non-zero,} & 0 \leq t < t_1 \\ 0, & t_1 \leq t \leq t_2 \\ \text{Constant and opposite,} & t_2 < t \leq t_3 \\ 0, & t > t_3 \end{cases} \]