A circular coil of area `8m^2` and number of turns 20 is placed in a magnetic field of 2T with its plane perpendicular to it. It is rotated with an angular velocity of 20rev/s about its natural axis. The emf induced is

Options

A. 400V
B. `800 pi V`
C. 0
D. `400 pi V`

Answer

To solve the problem, we need to determine the induced electromotive force (emf) in a circular coil that is rotating in a magnetic field. Here’s a step-by-step breakdown of the solution:

### Step 1: Understand the Given Information
- Area of the coil, \( A = 8 \, m^2 \)
- Number of turns in the coil, \( N = 20 \)
- Magnetic field strength, \( B = 2 \, T \)
- Angular velocity, \( \omega = 20 \, \text{rev/s} \)

### Step 2: Convert Angular Velocity to Radians per Second
Since the standard unit for angular velocity in physics is radians per second, we need to convert revolutions per second to radians per second:
\[ \omega = 20 \, \text{rev/s} \times 2\pi \, \text{rad/rev} = 40\pi \, \text{rad/s} \]

### Step 3: Determine the Magnetic Flux
The magnetic flux \( \Phi \) through the coil is given by the formula:
\[ \Phi = B \cdot A \cdot \cos(\theta) \]
Where \( \theta \) is the angle between the magnetic field and the normal to the coil's surface. Since the coil is rotating with its plane perpendicular to the magnetic field, \( \theta = 0^\circ \) and \( \cos(0) = 1 \):
\[ \Phi = B \cdot A = 2 \, T \cdot 8 \, m^2 = 16 \, Wb \]

### Step 4: Analyze the Change in Magnetic Flux
When the coil is rotating about its natural axis, the angle \( \theta \) does not change, and thus the magnetic flux through the coil remains constant. Since the flux is constant, there is no change in magnetic flux over time.

### Step 5: Calculate the Induced EMF
According to Faraday's law of electromagnetic induction, the induced emf \( \mathcal{E} \) is given by the rate of change of magnetic flux:
\[ \mathcal{E} = -N \frac{d\Phi}{dt} \]
Since \( \frac{d\Phi}{dt} = 0 \) (as the flux is constant), we have:
\[ \mathcal{E} = -N \cdot 0 = 0 \]

### Final Answer
The induced emf in the coil is:
\[ \mathcal{E} = 0 \, V \]

### Conclusion
Thus, the induced emf is zero because the magnetic flux through the coil does not change while it is rotating in the magnetic field.

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