Question

when acircular loop is moved outof uniform magnetic field sketch the graph for variation of fluxinduced emf and power decipated as function of time

Answer 1

The magnetic flux, induced EMF, and power dissipated will vary as the circular loop moves out of the uniform magnetic field. Here's a summary of how they change with time:

1. Magnetic Flux ($\Phi_B$):

   • Before exiting: Constant and maximum.
   • During exiting: Decreases non-linearly from its maximum value to zero.
   • After exiting: Zero.

2. Induced EMF ($\mathcal{E}$):

   • Before exiting: Zero.
   • During exiting: Increases from zero to a maximum, then decreases back to zero.
   • After exiting: Zero.

3. Power Dissipated (P):

   • Before exiting: Zero.
   • During exiting: Increases from zero to a maximum, then decreases back to zero.
   • After exiting: Zero.

The graphs for these variations are pulse-like shapes for EMF and Power, and a decreasing curve for Magnetic Flux.

Answer 2

When a circular loop moves out of a uniform magnetic field, the magnetic flux through the loop changes, inducing an electromotive force (EMF) and causing power to be dissipated if the loop has resistance.

Let's assume the following:

1. The magnetic field $\vec{B}$ is uniform and perpendicular to the plane of the loop.
2. The loop moves at a constant velocity $\vec{v}$ perpendicular to the field boundary.
3. The loop starts exiting the field at $t=0$ and is completely out at time $T$. Due to the circular shape, the area of the loop inside the field does not decrease linearly with time.

1. Magnetic Flux ($\Phi_B$)

• Before exiting ($t < 0$): The loop is fully inside the uniform magnetic field. The magnetic flux is constant and maximum: $\Phi_B = B \cdot (\text{Area of loop}) = B \pi R^2$.
• During exiting ($0 \le t \le T$): As the loop moves out, the area enclosed by the loop that is still within the magnetic field decreases. For a circular loop, this decrease is not linear. The rate of change of the area initially increases, then decreases. Therefore, the magnetic flux decreases smoothly from its maximum value to zero.
• After exiting ($t > T$): The loop is completely out of the magnetic field. The magnetic flux is zero: $\Phi_B = 0$.

The graph of $\Phi_B$ vs. time will show a constant maximum value, followed by a smooth, non-linear decrease to zero, and then remain at zero. The curve in the decreasing region will be concave up initially and then concave down.

2. Induced EMF ($\mathcal{E}$)

• According to Faraday's Law of Induction, $\mathcal{E} = -\frac{d\Phi_B}{dt}$.
• Before exiting ($t < 0$): $\Phi_B$ is constant, so $\frac{d\Phi_B}{dt} = 0$, and thus $\mathcal{E} = 0$.
• During exiting ($0 \le t \le T$): $\Phi_B$ is decreasing, so $\frac{d\Phi_B}{dt}$ is negative. This means $\mathcal{E}$ will be positive (or negative, depending on the chosen direction, but its magnitude will be non-zero).
  • When the loop just begins to exit, the rate of change of the area inside the field is small, so $|\mathcal{E}|$ is small.
  • As more of the loop exits, the chord length (the part of the loop boundary at the field edge) increases, reaching a maximum when the center of the loop is at the field boundary. At this point, the rate of change of area is maximum, leading to a maximum induced EMF.
  • As the loop continues to exit, the chord length decreases again, and the rate of change of area decreases, causing $|\mathcal{E}|$ to decrease back to zero.
  • Therefore, the magnitude of the induced EMF starts from zero, increases to a maximum, and then decreases back to zero.
• After exiting ($t > T$): $\Phi_B$ is constant (zero), so $\frac{d\Phi_B}{dt} = 0$, and thus $\mathcal{E} = 0$.

The graph of $\mathcal{E}$ vs. time will show zero EMF, followed by a pulse-like shape (symmetric if the field is uniform and the exit is symmetric), starting from zero, peaking, and returning to zero, then remaining at zero.

3. Power Dissipated (P)

• The power dissipated in the loop is given by $P = \frac{\mathcal{E}^2}{R}$, where $R$ is the resistance of the loop.
• Before exiting ($t < 0$): $\mathcal{E} = 0$, so $P = 0$.
• During exiting ($0 \le t \le T$): Since $P$ is proportional to $\mathcal{E}^2$, and $\mathcal{E}$ follows a pulse-like shape, $P$ will also follow a similar shape, but always positive and peaking at the same time as $\mathcal{E}$ (or rather, its magnitude). The peaks will be sharper due to squaring.
• After exiting ($t > T$): $\mathcal{E} = 0$, so $P = 0$.

The graph of P vs. time will show zero power dissipated, followed by a positive pulse-like shape (symmetric and peaking at the same time as EMF), and then returning to zero.