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

Answer

To solve the problem step by step, we will follow the principles of electromagnetic induction.

### Step 1: Identify the given data
- Number of turns (N) = 300
- Area of the coil (A) = 25 cm x 10 cm = 250 cm² = 250 x 10^(-4) m²
- Frequency (f) = 50 cycles per second (cps)
- Magnetic field strength (B) = 4 x 10^(-2) T

### Step 2: Convert the area into square meters
To convert the area from cm² to m²:
\[ A = 25 \, \text{cm} \times 10 \, \text{cm} = 250 \, \text{cm}^2 = 250 \times 10^{-4} \, \text{m}^2 \]

### Step 3: Calculate the angular frequency (ω)
The angular frequency (ω) is related to the frequency (f) by the formula:
\[ \omega = 2\pi f \]
Substituting the value of f:
\[ \omega = 2\pi \times 50 = 100\pi \, \text{rad/s} \]

### Step 4: Use the formula for peak induced EMF
The peak value of the induced EMF (ε) can be calculated using the formula:
\[ \epsilon = N \cdot B \cdot A \cdot \omega \]
Substituting the known values:
\[ \epsilon = 300 \cdot (4 \times 10^{-2}) \cdot (250 \times 10^{-4}) \cdot (100\pi) \]

### Step 5: Simplify the expression
Calculating the expression step-by-step:
1. Calculate \( B \cdot A \):
\[ B \cdot A = (4 \times 10^{-2}) \cdot (250 \times 10^{-4}) = 1 \times 10^{-5} \, \text{T m}^2 \]
2. Now substitute this back into the EMF formula:
\[ \epsilon = 300 \cdot (1 \times 10^{-5}) \cdot (100\pi) \]
3. Calculate:
\[ \epsilon = 300 \cdot 100 \cdot \pi \cdot 10^{-5} = 30000\pi \times 10^{-5} = 300\pi \times 10^{-2} \]

### Step 6: Calculate the numerical value
Using the approximate value of π (3.14):
\[ \epsilon \approx 300 \cdot 3.14 \times 10^{-2} \approx 942 \times 10^{-2} \approx 9.42 \, \text{V} \]

### Final Answer
The peak value of the induced EMF is approximately **9.42 V**.