A circular coil of radius 2cm and 200 numbers of turns is placed perpendicular to uniform magnetic field of 2 T. The coil is turned through angle `180^@` in time interval of 0.5 s. What will be the average emf induced in the coil ?

Answer

To find the average electromotive force (emf) induced in a circular coil when it is turned through an angle of 180 degrees in a uniform magnetic field, we can follow these steps:

### Step 1: Identify the given values
- Radius of the coil, \( r = 2 \, \text{cm} = 0.02 \, \text{m} \)
- Number of turns, \( N = 200 \)
- Magnetic field strength, \( B = 2 \, \text{T} \)
- Angle turned, \( \Delta \theta = 180^\circ = \pi \, \text{radians} \)
- Time interval, \( \Delta t = 0.5 \, \text{s} \)

### Step 2: Calculate the area of the coil
The area \( A \) of the circular coil can be calculated using the formula:
\[ A = \pi r^2 \]
Substituting the radius:
\[ A = \pi (0.02)^2 = \pi (0.0004) = 1.25664 \times 10^{-3} \, \text{m}^2 \]

### Step 3: Calculate the initial and final magnetic flux
The magnetic flux \( \Phi \) through the coil is given by:
\[ \Phi = N \cdot B \cdot A \cdot \cos(\theta) \]
- Initial angle \( \theta_1 = 0^\circ \) (perpendicular to the magnetic field)
- Final angle \( \theta_2 = 180^\circ \) (opposite direction)

Calculating the initial flux \( \Phi_1 \):
\[ \Phi_1 = N \cdot B \cdot A \cdot \cos(0) = 200 \cdot 2 \cdot 1.25664 \times 10^{-3} \cdot 1 = 0.502656 \, \text{Wb} \]

Calculating the final flux \( \Phi_2 \):
\[ \Phi_2 = N \cdot B \cdot A \cdot \cos(180^\circ) = 200 \cdot 2 \cdot 1.25664 \times 10^{-3} \cdot (-1) = -0.502656 \, \text{Wb} \]

### Step 4: Calculate the change in magnetic flux
The change in magnetic flux \( \Delta \Phi \) is:
\[ \Delta \Phi = \Phi_2 - \Phi_1 = -0.502656 - 0.502656 = -1.005312 \, \text{Wb} \]

### Step 5: Calculate the average induced emf
The average induced emf \( \mathcal{E} \) can be calculated using Faraday's law:
\[ \mathcal{E} = -\frac{\Delta \Phi}{\Delta t} \]
Substituting the values:
\[ \mathcal{E} = -\frac{-1.005312}{0.5} = 2.010624 \, \text{V} \]

### Final Answer
The average emf induced in the coil is approximately:
\[ \mathcal{E} \approx 2.01 \, \text{V} \]

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