A coil has `2000` turns and area of `70cm^(2)`. The magnetic field perpendicular to the plane of the coil is `0.3 Wb//m^(2)` and takes `0.1` sec to rotate through `180^(0)`. The value of the induced e.m.f. will be

Options

A. `8.4 V`
B. `84 V`
C. `42 V`
D. `4.2 V`

Explanation

To find the induced electromotive force (e.m.f.) in the coil, we can use Faraday's law of electromagnetic induction, which states that the induced e.m.f. (ε) in a closed loop is equal to the negative rate of change of magnetic flux through the loop.

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Number of turns in the coil (N) = 2000
- Area of the coil (A) = 70 cm² = 70 × 10⁻⁴ m² = 7 × 10⁻³ m²
- Magnetic field strength (B) = 0.3 Wb/m²
- Time taken to rotate (Δt) = 0.1 s
- The coil rotates through an angle of 180°.

2. **Calculate the Initial and Final Magnetic Flux:**
- The magnetic flux (Φ) through the coil is given by:
\[ Φ = B \cdot A \cdot \cos(θ) \]
- Initially (θ = 0°):
\[ Φ_{initial} = B \cdot A \cdot \cos(0) = 0.3 \cdot (7 \times 10^{-3}) \cdot 1 = 0.0021 \text{ Wb} \]
- Finally (θ = 180°):
\[ Φ_{final} = B \cdot A \cdot \cos(180) = 0.3 \cdot (7 \times 10^{-3}) \cdot (-1) = -0.0021 \text{ Wb} \]

3. **Calculate the Change in Magnetic Flux (ΔΦ):**
\[ ΔΦ = Φ_{final} - Φ_{initial} = -0.0021 - 0.0021 = -0.0042 \text{ Wb} \]

4. **Calculate the Induced e.m.f. (ε):**
- According to Faraday's law:
\[ ε = -\frac{ΔΦ}{Δt} \]
- Plugging in the values:
\[ ε = -\frac{-0.0042}{0.1} = 0.042 \text{ V} \]

5. **Multiply by the Number of Turns (N):**
- The total induced e.m.f. in the coil is:
\[ ε_{total} = N \cdot ε = 2000 \cdot 0.042 = 84 \text{ V} \]

### Final Answer:
The value of the induced e.m.f. will be **84 V**.