A coil of area `0.04 m^(2)` having 1000 turns is suspended perpendicular to a magnetic field of `5.0 xx 10^(-5)` Wb `m^(2)`. It is rotated through `90^(@)` in 0.2 second. Calculate the average emf induced in it.

Answer

To calculate the average emf induced in a coil when it is rotated in a magnetic field, we can follow these steps:

### Step 1: Understand the Given Data
- Area of the coil, \( A = 0.04 \, \text{m}^2 \)
- Number of turns, \( N = 1000 \)
- Magnetic field strength, \( B = 5.0 \times 10^{-5} \, \text{Wb/m}^2 \)
- Angle of rotation, \( \Delta \theta = 90^\circ \)
- Time taken for rotation, \( \Delta t = 0.2 \, \text{s} \)

### Step 2: Calculate the Initial Magnetic Flux
The magnetic flux \( \Phi \) through the coil is given by:
\[ \Phi = B \cdot A \cdot \cos(\theta) \]
Initially, when the coil is perpendicular to the magnetic field, \( \theta = 0^\circ \):
\[ \Phi_{\text{initial}} = B \cdot A \cdot \cos(0) = B \cdot A = (5.0 \times 10^{-5}) \cdot (0.04) \]
Calculating this gives:
\[ \Phi_{\text{initial}} = 2.0 \times 10^{-6} \, \text{Wb} \]

### Step 3: Calculate the Final Magnetic Flux
After rotating the coil by \( 90^\circ \), the angle \( \theta = 90^\circ \):
\[ \Phi_{\text{final}} = B \cdot A \cdot \cos(90) = B \cdot A \cdot 0 = 0 \]

### Step 4: Calculate the Change in Magnetic Flux
The change in magnetic flux \( \Delta \Phi \) is given by:
\[ \Delta \Phi = \Phi_{\text{final}} - \Phi_{\text{initial}} = 0 - (2.0 \times 10^{-6}) = -2.0 \times 10^{-6} \, \text{Wb} \]

### Step 5: Calculate the Average Induced EMF
According to Faraday's law of electromagnetic induction, the average induced emf \( \mathcal{E} \) is given by:
\[ \mathcal{E} = -N \frac{\Delta \Phi}{\Delta t} \]
Substituting the values:
\[ \mathcal{E} = -1000 \cdot \frac{-2.0 \times 10^{-6}}{0.2} \]
Calculating this gives:
\[ \mathcal{E} = 1000 \cdot \frac{2.0 \times 10^{-6}}{0.2} = 1000 \cdot 10^{-5} = 10^{-2} \, \text{V} = 0.01 \, \text{V} = 10 \, \text{mV} \]

### Final Answer
The average emf induced in the coil is \( 10 \, \text{mV} \).

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