A rectangular coil has 60 turns and its length and width is 20 cm and 10 cm recpectively . The coil rotates at a speed of 1800 rotation per minute in a uniform magnetic field of 0.5 T about its one of the diameter. The maximum induced emf will be :-

Options

A. 98 V
B. 110 V
C. 113 V
D. 118 V

Answer

To solve the problem of finding the maximum induced EMF in a rectangular coil rotating in a magnetic field, we can follow these steps:

### Step 1: Identify the given values
- Number of turns (N) = 60
- Length of the coil (l) = 20 cm = 0.2 m
- Width of the coil (w) = 10 cm = 0.1 m
- Magnetic field (B) = 0.5 T
- Speed of rotation = 1800 rotations per minute

### Step 2: Convert the dimensions of the coil to meters and calculate the area
The area (A) of the rectangular coil can be calculated using the formula:
\[ A = l \times w \]
\[ A = 0.2 \, \text{m} \times 0.1 \, \text{m} = 0.02 \, \text{m}^2 \]

### Step 3: Convert the speed of rotation to radians per second
To find the angular frequency (ω), we need to convert the rotations per minute to radians per second:
\[ \text{Rotations per second} = \frac{1800 \, \text{rotations}}{60 \, \text{seconds}} = 30 \, \text{rotations/second} \]
Since one rotation is \(2\pi\) radians:
\[ \omega = 30 \times 2\pi \, \text{radians/second} = 60\pi \, \text{radians/second} \]

### Step 4: Use the formula for maximum induced EMF
The maximum induced EMF (E_max) in a rotating coil can be calculated using the formula:
\[ E_{\text{max}} = N \cdot B \cdot A \cdot \omega \]
Substituting the values we have:
\[ E_{\text{max}} = 60 \cdot 0.5 \cdot 0.02 \cdot 60\pi \]

### Step 5: Calculate E_max
Now, let's calculate it step by step:
1. Calculate \(N \cdot B \cdot A\):
\[ N \cdot B \cdot A = 60 \cdot 0.5 \cdot 0.02 = 0.6 \]
2. Now calculate \(E_{\text{max}}\):
\[ E_{\text{max}} = 0.6 \cdot 60\pi \]
\[ E_{\text{max}} = 0.6 \cdot 60 \cdot 3.14 \]
\[ E_{\text{max}} \approx 113.04 \, \text{V} \]

### Final Answer
The maximum induced EMF will be approximately 113 V.

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