A closely wound rectangular coil of 200 turns and size `0.3xx0.1m` is rotating in a magnetic field of induction `0.005 Wb//m^2` with a frequency of 1800rpm about an axis normal to the field. Calculate the maximum value of induced emf.

Explanation

To calculate the maximum value of the induced emf in the rectangular coil, we can follow these steps:

### Step 1: Convert the frequency from rpm to rps
The frequency given is 1800 revolutions per minute (rpm). To convert this to revolutions per second (rps), we use the following formula:

\[ \text{Frequency (rps)} = \frac{\text{Frequency (rpm)}}{60} \]

Substituting the given value:

\[ \text{Frequency (rps)} = \frac{1800}{60} = 30 \text{ rps} \]

### Step 2: Calculate the area of the rectangular coil
The area \( A \) of the rectangular coil can be calculated using the formula:

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

Given the dimensions are \( 0.3 \, m \) and \( 0.1 \, m \):

\[ A = 0.3 \, m \times 0.1 \, m = 0.03 \, m^2 \]

### Step 3: Calculate the angular frequency \( \omega \)
The angular frequency \( \omega \) in radians per second can be calculated using the formula:

\[ \omega = 2\pi \times \text{frequency (rps)} \]

Substituting the value we found:

\[ \omega = 2\pi \times 30 \approx 188.5 \, \text{rad/s} \]

### Step 4: Use the formula for maximum induced emf
The maximum induced emf \( E_0 \) can be calculated using the formula:

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

Where:
- \( N = 200 \) (number of turns)
- \( B = 0.005 \, Wb/m^2 \) (magnetic field induction)
- \( A = 0.03 \, m^2 \) (area)
- \( \omega \approx 188.5 \, rad/s \)

Substituting the values:

\[ E_0 = 200 \cdot 0.005 \cdot 0.03 \cdot 188.5 \]

Calculating this step by step:

1. \( 200 \cdot 0.005 = 1 \)
2. \( 1 \cdot 0.03 = 0.03 \)
3. \( 0.03 \cdot 188.5 \approx 5.655 \)

Thus,

\[ E_0 \approx 5.655 \, V \]

### Final Answer
The maximum value of the induced emf is approximately **5.655 V**.

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