A rectangular coil of wire has dimensions `0.2 m xx 0.1` m. The coil has 2000 turns. The coil rotates in a magnetic field about an axis parallel to its length and perpendicular to the magnetic field of `0.02 "Wb m"^(-2)` The speed of rotation of the coil is 4200 rpm.
What is the maximum value of induced emf in the coil?

Options

A. 3000 V
B. 3520 V
C. 3200 V
D. 3430 V

Explanation

To find the maximum value of induced emf in the coil, we can use the formula:

\[ E_0 = N \cdot B \cdot A \cdot \omega \]

where:
- \(E_0\) = maximum induced emf
- \(N\) = number of turns in the coil
- \(B\) = magnetic field strength (in Weber per square meter)
- \(A\) = area of the coil (in square meters)
- \(\omega\) = angular velocity (in radians per second)

### Step 1: Calculate the area of the coil
The area \(A\) of the rectangular coil can be calculated using the formula:

\[ A = \text{length} \times \text{width} \]

Given dimensions:
- Length = 0.2 m
- Width = 0.1 m

Calculating the area:

\[ A = 0.2 \, \text{m} \times 0.1 \, \text{m} = 0.02 \, \text{m}^2 \]

### Step 2: Convert the magnetic field strength
The magnetic field strength \(B\) is given as:

\[ B = 0.02 \, \text{Wb/m}^2 \]

### Step 3: Calculate the angular velocity \(\omega\)
The speed of rotation is given in revolutions per minute (rpm). We need to convert this to radians per second.

The conversion factor is:

\[ \omega = \text{rpm} \times \frac{2\pi \, \text{radians}}{1 \, \text{revolution}} \times \frac{1 \, \text{minute}}{60 \, \text{seconds}} \]

Given speed of rotation:

\[ \text{rpm} = 4200 \]

Calculating \(\omega\):

\[ \omega = 4200 \times \frac{2\pi}{60} = 4200 \times \frac{\pi}{30} = 140\pi \, \text{radians/second} \]

### Step 4: Substitute values into the induced emf formula
Now we can substitute the values into the induced emf formula:

\[ E_0 = N \cdot B \cdot A \cdot \omega \]

Given:
- \(N = 2000\)
- \(B = 0.02 \, \text{Wb/m}^2\)
- \(A = 0.02 \, \text{m}^2\)
- \(\omega = 140\pi \, \text{radians/second}\)

Substituting these values:

\[ E_0 = 2000 \cdot 0.02 \cdot 0.02 \cdot (140\pi) \]

Calculating:

\[ E_0 = 2000 \cdot 0.02 \cdot 0.02 \cdot 140 \cdot \frac{22}{7} \]

\[ E_0 = 2000 \cdot 0.0004 \cdot 140 \cdot \frac{22}{7} \]

\[ E_0 = 2000 \cdot 0.0004 \cdot 140 \cdot 3.14 \approx 352 \, \text{Volts} \]

### Final Answer
The maximum value of induced emf in the coil is approximately:

\[ E_0 \approx 352 \, \text{Volts} \]

---