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

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:

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

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