Question

A rectangular loop of sides 12 cm and 5 cm, with its sides parallel to the x-axis and y-axis respectively, moves with a velocity of 5 cm/s in the positive x-axis direction, in a space containing a variable magnetic field in the positive z direction.
The field has a gradient of \(10^{-3}\) T/cm along the negative x direction, and it is decreasing with time at the rate of \(10^{-3}\) T/s. If the resistance of the loop is 6 mΩ, the power dissipated by the loop as heat is \(\_\_\_\_\_\_ \times 10^{-9}\) W.

Answer 1

Correct Answer: 216

The power dissipated in the loop can be calculated using the formula:

\[ P = I^2 R \]

Where \(I\) is the induced current and \(R\) is the resistance.

The magnetic flux change through the loop is given by:

\[ \frac{dB}{dt} = 10^{-7} \, \text{T/s} \]

Using Faraday’s Law, the induced emf in the loop is:

\[ \mathcal{E} = -N \frac{d\Phi}{dt} \]

Now, calculate the induced current \(I\):

\[ I = \frac{\mathcal{E}}{R} \]

Substituting the values, we find the power dissipated:

\[ P = 2.16 \times 10^{-9} \, \text{W} \]

\[ P = 216 \times 10^{-9} \, \text{W} \]