A rectangular coil `20 cm xx 10 cm` having 500 turns rotates in a magnetic field of `5 xx 10^(-3) T` with a frequency of `1200 rev min^(-1)` about an axis perpendicular to the field.

Options

A. The maximum value of the induced emf `2pi//5` volt.
B. The instantaneous emf when the plane of the coil is perpendicular to the field is zero
C. The instantaneous emf when the plane of the coil makes an angle of `60^(@)` with the field is `pi//5` volt.
D. The instantaneous emf when the plane of the coil makes an angle of `30^(@)` with the field is `pi//10` volt

Explanation

To solve the problem step by step, we will calculate the maximum induced EMF in the rectangular coil rotating in a magnetic field.

### Step 1: Identify the given parameters
- Dimensions of the coil: Length \( l = 20 \, \text{cm} = 0.2 \, \text{m} \) and Width \( w = 10 \, \text{cm} = 0.1 \, \text{m} \)
- Number of turns \( N = 500 \)
- Magnetic field \( B = 5 \times 10^{-3} \, \text{T} \)
- Frequency \( f = 1200 \, \text{rev/min} \)

### Step 2: Convert frequency to revolutions per second
\[ f = \frac{1200 \, \text{rev/min}}{60 \, \text{s/min}} = 20 \, \text{rev/s} \]

### Step 3: Calculate the area of the coil
\[ A = l \times w = 0.2 \, \text{m} \times 0.1 \, \text{m} = 0.02 \, \text{m}^2 \]

### Step 4: Calculate the angular frequency
\[ \omega = 2\pi f = 2\pi \times 20 = 40\pi \, \text{rad/s} \]

### Step 5: Calculate the maximum induced EMF
The formula for the maximum induced EMF (\( \mathcal{E}_{\text{max}} \)) in a rotating coil is given by:
\[ \mathcal{E}_{\text{max}} = N \cdot B \cdot A \cdot \omega \]
Substituting the known values:
\[ \mathcal{E}_{\text{max}} = 500 \cdot (5 \times 10^{-3}) \cdot (0.02) \cdot (40\pi) \]

### Step 6: Calculate the values step by step
1. Calculate \( N \cdot B \cdot A \):
\[ N \cdot B \cdot A = 500 \cdot 5 \times 10^{-3} \cdot 0.02 = 500 \cdot 0.0001 = 0.05 \]
2. Now multiply by \( \omega \):
\[ \mathcal{E}_{\text{max}} = 0.05 \cdot (40\pi) = 2\pi \approx 6.2832 \, \text{V} \]

### Final Result
The maximum induced EMF in the coil is approximately \( 6.28 \, \text{V} \).

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