A rectangular coil of `300` turns has an average area of average area of `25 cmxx10 cm` the cooil rotates with a speed of `50 cps ` in a uniform magnetic field of strength `4xx10^(-2)T` about an axis perpendicular of the field. The peak value of the induced e.m.f. is (in volt)`

Options

A. `3000pi`
B. `300pi`
C. `30pi`
D. `3pi`

Explanation

To find the peak value of the induced electromotive force (e.m.f.) in the given rectangular coil, we can use the formula for induced e.m.f. due to electromagnetic induction:

\[ E = N \cdot B \cdot A \cdot \omega \]

Where:
- \(E\) = induced e.m.f. (in volts)
- \(N\) = number of turns in the coil
- \(B\) = magnetic field strength (in tesla)
- \(A\) = area of the coil (in square meters)
- \(\omega\) = angular velocity (in radians per second)

### Step 1: Convert the area from cm² to m²
The area of the coil is given as \(25 \, \text{cm} \times 10 \, \text{cm}\).

\[ A = 25 \, \text{cm} \times 10 \, \text{cm} = 250 \, \text{cm}^2 \]

To convert cm² to m², we use the conversion factor \(1 \, \text{cm}^2 = 10^{-4} \, \text{m}^2\):

\[ A = 250 \, \text{cm}^2 \times 10^{-4} \, \text{m}^2/\text{cm}^2 = 0.025 \, \text{m}^2 \]

### Step 2: Calculate the angular velocity \(\omega\)
The coil rotates at a speed of \(50 \, \text{cps}\) (cycles per second). To convert this to angular velocity in radians per second, we use the relation:

\[ \omega = 2\pi \cdot f \]

Where \(f\) is the frequency in hertz (cycles per second):

\[ \omega = 2\pi \cdot 50 \, \text{cps} = 100\pi \, \text{rad/s} \]

### Step 3: Substitute the values into the e.m.f. formula
Now we can substitute the values into the e.m.f. formula:

- \(N = 300\) (number of turns)
- \(B = 4 \times 10^{-2} \, \text{T}\) (magnetic field strength)
- \(A = 0.025 \, \text{m}^2\) (area)
- \(\omega = 100\pi \, \text{rad/s}\)

\[ E = N \cdot B \cdot A \cdot \omega \]

Substituting the values:

\[ E = 300 \cdot (4 \times 10^{-2}) \cdot 0.025 \cdot (100\pi) \]

### Step 4: Calculate the e.m.f.
Calculating step by step:

1. Calculate \(300 \cdot 4 \times 10^{-2} = 12\)
2. Calculate \(0.025 \cdot 100\pi \approx 7.85\) (using \(\pi \approx 3.14\))
3. Now multiply:

\[ E = 12 \cdot 7.85 \approx 94.2 \, \text{V} \]

### Final Result
The peak value of the induced e.m.f. is approximately:

\[ E \approx 94.2 \, \text{V} \]