The radius of a circular coil having 100 turns is 5 cm .its plane is normal to the magnetic field. The magnetic field changes from 2T to 10 T in 3.14 sec. The induced emf in coil will be

Options

A. 2 V
B. 4 V
C. 6V
D. 8V

Explanation

To solve the problem step by step, we need to calculate the induced electromotive force (emf) in the circular coil using Faraday's law of electromagnetic induction.

### Step 1: Identify the given values
- Number of turns (N) = 100
- Radius of the coil (r) = 5 cm = 0.05 m
- Initial magnetic field (B1) = 2 T
- Final magnetic field (B2) = 10 T
- Time interval (Δt) = 3.14 s

### Step 2: Calculate the change in magnetic field (ΔB)
\[ \Delta B = B2 - B1 = 10 \, \text{T} - 2 \, \text{T} = 8 \, \text{T} \]

### Step 3: Calculate the area (A) of the circular coil
The area of a circle is given by the formula:
\[ A = \pi r^2 \]
Substituting the radius:
\[ A = \pi (0.05)^2 = \pi (0.0025) \approx 0.007854 \, \text{m}^2 \]

### Step 4: Calculate the change in magnetic flux (ΔΦ)
The change in magnetic flux is given by:
\[ \Delta \Phi = N \cdot A \cdot \Delta B \]
Substituting the values:
\[ \Delta \Phi = 100 \cdot 0.007854 \cdot 8 \]
Calculating this:
\[ \Delta \Phi \approx 100 \cdot 0.007854 \cdot 8 \approx 6.2832 \, \text{Wb} \]

### Step 5: Calculate the induced emf (ε)
According to Faraday's law, the induced emf is given by:
\[ \epsilon = -\frac{\Delta \Phi}{\Delta t} \]
Substituting the values:
\[ \epsilon = -\frac{6.2832}{3.14} \approx -2 \, \text{V} \]
The negative sign indicates the direction of the induced emf (Lenz's law), but we are interested in the magnitude, which is:
\[ \epsilon \approx 2 \, \text{V} \]

### Final Answer
The induced emf in the coil is approximately **2 V**.

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