What is the emf induced in a coil of 200 turns and cross sectional area `0.2 m^2`, when a magnetic field perpendicular to the plane of the coil change from `0.1 Wbm^(-2)` to `0.5 wbm^(-2)` at a uniform rate over a period of 0.05 sec?

Options

A. 300 V
B. 320 V
C. 310 V
D. 290 V

Explanation

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

### Step-by-Step Solution:

1. **Identify the given values:**
- Number of turns (N) = 200
- Cross-sectional area (A) = 0.2 m²
- Initial magnetic field (B₁) = 0.1 Wb/m²
- Final magnetic field (B₂) = 0.5 Wb/m²
- Time interval (t) = 0.05 s

2. **Calculate the change in magnetic field (ΔB):**
\[ \Delta B = B_2 - B_1 = 0.5 \, \text{Wb/m}^2 - 0.1 \, \text{Wb/m}^2 = 0.4 \, \text{Wb/m}^2 \]

3. **Calculate the change in magnetic flux (ΔΦ):**
The magnetic flux (Φ) through the coil is given by:
\[ \Phi = B \cdot A \]
Therefore, the change in magnetic flux (ΔΦ) is:
\[ \Delta \Phi = N \cdot A \cdot \Delta B = 200 \cdot 0.2 \cdot 0.4 \]

4. **Calculate ΔΦ:**
\[ \Delta \Phi = 200 \cdot 0.2 \cdot 0.4 = 16 \, \text{Wb} \]

5. **Calculate the induced emf (E):**
Using Faraday's law:
\[ E = -\frac{\Delta \Phi}{t} \]
Substitute the values:
\[ E = -\frac{16 \, \text{Wb}}{0.05 \, \text{s}} = -320 \, \text{V} \]

6. **Final Result:**
The induced emf in the coil is:
\[ E = 320 \, \text{V} \quad (\text{magnitude only, as emf is typically expressed as a positive value}) \]