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 of finding the induced emf in a circular coil, we will follow these steps:

### Step 1: Understand the parameters given
- Number of turns (N) = 50
- Radius of the coil (r) = 2 cm = 0.02 m
- Initial magnetic field (B1) = 2 T
- Final magnetic field (B2) = 4 T
- Time interval (Δt) = 3.14 s

### Step 2: Calculate the area of the circular coil
The area (A) of a circular coil is given by the formula:
\[ A = \pi r^2 \]
Substituting the radius:
\[ A = \pi (0.02)^2 = \pi (0.0004) = 0.00125664 \, \text{m}^2 \]

### Step 3: Calculate the initial and final magnetic flux
The magnetic flux (Φ) is given by:
\[ \Phi = B \cdot A \cdot \cos(\theta) \]
Since the plane of the coil is normal to the magnetic field, \(\theta = 0^\circ\) and \(\cos(0) = 1\).

- Initial magnetic flux (Φ1):
\[ \Phi_1 = B_1 \cdot A = 2 \cdot 0.00125664 = 0.00251328 \, \text{Wb} \]

- Final magnetic flux (Φ2):
\[ \Phi_2 = B_2 \cdot A = 4 \cdot 0.00125664 = 0.00502656 \, \text{Wb} \]

### Step 4: Calculate the change in magnetic flux (ΔΦ)
\[ \Delta \Phi = \Phi_2 - \Phi_1 = 0.00502656 - 0.00251328 = 0.00251328 \, \text{Wb} \]

### Step 5: Use Faraday's law to calculate the induced emf (ε)
According to Faraday's law:
\[ \epsilon = -N \frac{\Delta \Phi}{\Delta t} \]
Substituting the values:
\[ \epsilon = -50 \cdot \frac{0.00251328}{3.14} \]
Calculating:
\[ \epsilon = -50 \cdot 0.0008001 \approx -0.04 \, \text{V} \]
The negative sign indicates the direction of the induced emf (Lenz's law), but we are interested in the magnitude:
\[ \epsilon \approx 0.04 \, \text{V} \]

### Final Answer
The induced emf in the coil is approximately **0.04 V**.

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