A circular coil of mean radius of `7 cm` and having `4000` turns Is rotate at the rate of `1800` revolution per minute in the earth 's magnetic field (B=0.5 gauss), the maximum e.m.f. induced in coil will be

Options

A. `1.158 V`
B. `0.57 V`
C. `0.29 V`
D. `5.8 V`

Explanation

To solve the problem of finding the maximum e.m.f. induced in a circular coil rotating in a magnetic field, we can follow these steps:

### Step 1: Understand the given data
- Mean radius of the coil, \( R = 7 \, \text{cm} = 0.07 \, \text{m} \)
- Number of turns, \( N = 4000 \)
- Rate of rotation, \( f = 1800 \, \text{revolutions per minute} \)
- Magnetic field strength, \( B = 0.5 \, \text{gauss} = 0.5 \times 10^{-4} \, \text{T} \)

### Step 2: Convert the rate of rotation to radians per second
To convert revolutions per minute to radians per second, we use the conversion factor:
\[ \omega = 2\pi \times \frac{f}{60} \]
Substituting \( f = 1800 \):
\[ \omega = 2\pi \times \frac{1800}{60} = 2\pi \times 30 = 60\pi \, \text{rad/s} \]

### Step 3: Calculate the area of the coil
The area \( A \) of the circular coil is given by:
\[ A = \pi R^2 \]
Substituting \( R = 0.07 \, \text{m} \):
\[ A = \pi (0.07)^2 = \pi \times 0.0049 \approx 0.0154 \, \text{m}^2 \]

### Step 4: Use the formula for maximum induced e.m.f.
The maximum induced e.m.f. \( \mathcal{E}_{\text{max}} \) in a rotating coil is given by:
\[ \mathcal{E}_{\text{max}} = N B A \omega \]
Substituting the values:
\[ \mathcal{E}_{\text{max}} = 4000 \times (0.5 \times 10^{-4}) \times 0.0154 \times 60\pi \]

### Step 5: Calculate the maximum e.m.f.
Calculating step by step:
1. Calculate \( N \times B = 4000 \times 0.5 \times 10^{-4} = 0.2 \)
2. Calculate \( 0.2 \times 0.0154 \approx 0.00308 \)
3. Calculate \( 0.00308 \times 60\pi \approx 0.00308 \times 188.4 \approx 0.58 \, \text{V} \)

### Final Result
Thus, the maximum e.m.f. induced in the coil is approximately:
\[ \mathcal{E}_{\text{max}} \approx 0.58 \, \text{V} \]