Question

A coil of resistance $8\,\Omega$, number of turns 250 and area $120 \text{ cm}^2$ is placed in a uniform magnetic field of 2 T such that the plane of the coil makes an angle of $\frac{\pi}{6}$ with the direction of the magnetic field. In a time of 100 ms, the coil is rotated until its plane becomes parallel to the direction of the magnetic field. The current induced in the coil is

• A. 5.25 A
• B. 3.75 A
• C. 2.75 A
• D. 1.25 A

Hint:

Be careful with the angle given. If it is the "angle of the plane with the field" ($\alpha$), the angle for flux ($\theta$) is $90^\circ - \alpha$.

Answer 1

The Correct Option is B

Step 1: Identify Flux Angles:
The magnetic flux is $\phi = NBA \cos \theta$, where $\theta$ is the angle between the normal to the coil and the magnetic field. Initial state: Plane makes $\pi/6 = 30^\circ$ with field. So normal makes $\theta_1 = 90^\circ - 30^\circ = 60^\circ$. Final state: Plane is parallel to field. So normal makes $\theta_2 = 90^\circ$.
Step 2: Calculate Flux Change:
$N = 250$, $B = 2$ T, $A = 120 \text{ cm}^2 = 120 \times 10^{-4} \text{ m}^2$. $\phi_1 = NBA \cos 60^\circ = 250 \times 2 \times 0.012 \times 0.5 = 500 \times 0.006 = 3$ Wb. $\phi_2 = NBA \cos 90^\circ = 0$. Change in flux $|\Delta \phi| = |0 - 3| = 3$ Wb.
Step 3: Calculate Induced EMF and Current:
Time interval $\Delta t = 100 \text{ ms} = 0.1$ s. Induced EMF $\epsilon = \frac{|\Delta \phi|}{\Delta t} = \frac{3}{0.1} = 30$ V. Induced Current $I = \frac{\epsilon}{R} = \frac{30}{8} = 3.75$ A.