A boy peddles a stationary bicycle the pedals of the bicycle are attached to a 200 turn coil of area `0.10m^(2)`. The coil rotates at half a revolution per second and it is placed in a uniform magnetic field of 0.02 T perpendicular to the axis of rotation of the coil. The maximum voltage generated in the coil is

Options

A. `1.26V`
B. `2.16V`
C. `3.24V`
D. `4.12V`

Explanation

To find the maximum voltage generated in the coil, we will use the formula for electromotive force (emf) generated in a rotating coil in a magnetic field:

\[ E_0 = n \cdot B \cdot A \cdot \omega \cdot \sin \theta \]

Where:
- \(E_0\) is the maximum emf (voltage),
- \(n\) is the number of turns in the coil,
- \(B\) is the magnetic field strength,
- \(A\) is the area of the coil,
- \(\omega\) is the angular velocity,
- \(\theta\) is the angle between the magnetic field and the normal to the coil's surface.

Since the coil is rotating perpendicular to the magnetic field, the maximum value of \(\sin \theta\) is 1. Therefore, we can simplify the equation to:

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

### Step 1: Identify the given values
From the problem statement, we have:
- Number of turns, \(n = 200\)
- Area of the coil, \(A = 0.10 \, m^2\)
- Magnetic field strength, \(B = 0.02 \, T\)
- The coil rotates at half a revolution per second, which means the frequency \(f = 0.5 \, Hz\).

### Step 2: Calculate the angular velocity (\(\omega\))
The angular velocity \(\omega\) can be calculated using the formula:

\[ \omega = 2 \pi f \]

Substituting the frequency:

\[ \omega = 2 \pi (0.5) = \pi \, \text{rad/s} \]

### Step 3: Substitute the values into the emf formula
Now we can substitute the values into the simplified emf formula:

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

Substituting the known values:

\[ E_0 = 200 \cdot 0.02 \cdot 0.10 \cdot \pi \]

### Step 4: Calculate the maximum voltage
Now, we calculate:

\[ E_0 = 200 \cdot 0.02 \cdot 0.10 \cdot 3.14 \]

Calculating step by step:

1. \(200 \cdot 0.02 = 4\)
2. \(4 \cdot 0.10 = 0.4\)
3. \(0.4 \cdot 3.14 \approx 1.256\)

Thus, the maximum voltage generated in the coil is approximately:

\[ E_0 \approx 1.26 \, V \]

### Final Answer
The maximum voltage generated in the coil is \(1.26 \, V\).

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