The radius of a circular coil having 100 turns is 2 cm. its plane is normal to the magnetic field.
The magnetic field changes from 4T to 8T in 3.14 sec. The induced emf in coil will be

Options

A. 0.08 V
B. 0.04 V
C. 0.16V
D. 0.12 V

Explanation

To find the induced emf in the circular coil, we can follow these steps:

### Step 1: Understand the parameters given
- Number of turns (N) = 100
- Radius of the coil (r) = 2 cm = 0.02 m (convert cm to m)
- Initial magnetic field (B_initial) = 4 T
- Final magnetic field (B_final) = 8 T
- Time interval (Δt) = 3.14 s

### Step 2: Calculate the change in magnetic field (ΔB)
\[ \Delta B = B_{\text{final}} - B_{\text{initial}} = 8\,T - 4\,T = 4\,T \]

### Step 3: Calculate the area (A) of the coil
The area \( A \) of a circular coil is given by the formula:
\[ A = \pi r^2 \]
Substituting the radius:
\[ A = \pi (0.02\,m)^2 = \pi (0.0004\,m^2) = 0.00125664\,m^2 \quad (\text{using } \pi \approx 3.14) \]

### Step 4: Calculate the induced emf (ε)
According to Faraday's law of electromagnetic induction, the induced emf (ε) is given by:
\[ \epsilon = -N \frac{\Delta \Phi}{\Delta t} \]
Where \( \Delta \Phi \) is the change in magnetic flux, which can be calculated as:
\[ \Delta \Phi = A \Delta B = A (B_{\text{final}} - B_{\text{initial}}) = A \Delta B \]
Substituting the values:
\[ \Delta \Phi = 0.00125664\,m^2 \times 4\,T = 0.00502656\,Wb \]

Now substituting \( \Delta \Phi \) back into the emf equation:
\[ \epsilon = -N \frac{\Delta \Phi}{\Delta t} = -100 \frac{0.00502656\,Wb}{3.14\,s} \]
Calculating this gives:
\[ \epsilon = -100 \times 0.001604 = -0.1604\,V \]
The negative sign indicates the direction of the induced emf (Lenz's law), but we are interested in the magnitude:
\[ \epsilon \approx 0.16\,V \]

### Final Answer
The induced emf in the coil is approximately **0.16 V**.

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