The radius of a circular coil having 50 turns is 2 cm. Its plane is normal to the magnetic field. The magnetic field changes from 2T to 4T in 3.14 sec. The induced emf in coil will be :-

Options

A. 0.4V
B. 0.04V
C. 4mV
D. 0.12V

Answer

To solve the problem, we need to calculate the induced electromotive force (emf) in a circular coil when the magnetic field changes. Here are the steps to find the solution:

### Step 1: Gather Given Information
- Number of turns (N) = 50
- Radius of the coil (r) = 2 cm = 0.02 m (conversion from cm to m)
- Initial magnetic field (B_initial) = 2 T
- Final magnetic field (B_final) = 4 T
- Time interval (Δt) = 3.14 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:
\[ A = \pi (0.02)^2 = \pi (0.0004) \]
\[ A = 0.0004\pi \, \text{m}^2 \]

### Step 3: Calculate the Initial and Final Magnetic Flux
Magnetic flux (Φ) is given by the formula:
\[ \Phi = B \cdot A \cdot N \]
- **Initial Flux (Φ_initial)**:
\[ \Phi_{\text{initial}} = B_{\text{initial}} \cdot A \cdot N = 2 \cdot (0.0004\pi) \cdot 50 \]
\[ \Phi_{\text{initial}} = 2 \cdot 0.0004\pi \cdot 50 = 0.04\pi \, \text{Wb} \]

- **Final Flux (Φ_final)**:
\[ \Phi_{\text{final}} = B_{\text{final}} \cdot A \cdot N = 4 \cdot (0.0004\pi) \cdot 50 \]
\[ \Phi_{\text{final}} = 4 \cdot 0.0004\pi \cdot 50 = 0.08\pi \, \text{Wb} \]

### Step 4: Calculate the Change in Magnetic Flux
The change in magnetic flux (ΔΦ) is given by:
\[ \Delta \Phi = \Phi_{\text{final}} - \Phi_{\text{initial}} \]
\[ \Delta \Phi = 0.08\pi - 0.04\pi = 0.04\pi \, \text{Wb} \]

### Step 5: Calculate the Induced EMF
The induced emf (ε) can be calculated using Faraday's law of electromagnetic induction:
\[ \epsilon = -\frac{\Delta \Phi}{\Delta t} \]
Substituting the values:
\[ \epsilon = -\frac{0.04\pi}{3.14} \]

Using the approximation \( \pi \approx 3.14 \):
\[ \epsilon = -\frac{0.04 \cdot 3.14}{3.14} = -0.04 \, \text{V} \]

### Step 6: Final Answer
The magnitude of the induced emf is:
\[ \epsilon = 0.04 \, \text{V} \]
The negative sign indicates the direction of the induced emf according to Lenz's law.

### Summary
The induced emf in the coil is **0.04 V**.