A coil having an area `2m^(2)` is placed in a magnetic field which changes from `1 Wb//m^(2)` to `4Wb//m^(2)` in an interval of `2` second. The average e.m.f. induced in the coil will be

Options

A. `4V`
B. `3V`
C. `1.5V`
D. `2V`

Explanation

To solve the problem of finding the average e.m.f. induced in the coil, we will follow these steps:

### Step 1: Identify the given values
- Area of the coil, \( A = 2 \, m^2 \)
- Initial magnetic field, \( B_i = 1 \, Wb/m^2 \)
- Final magnetic field, \( B_f = 4 \, Wb/m^2 \)
- Time interval, \( \Delta t = 2 \, s \)

### Step 2: Calculate the initial magnetic flux
The magnetic flux \( \Phi \) through the coil is given by the formula:
\[ \Phi = B \cdot A \]
Calculating the initial flux:
\[ \Phi_i = B_i \cdot A = 1 \, Wb/m^2 \cdot 2 \, m^2 = 2 \, Wb \]

### Step 3: Calculate the final magnetic flux
Now, calculate the final flux using the final magnetic field:
\[ \Phi_f = B_f \cdot A = 4 \, Wb/m^2 \cdot 2 \, m^2 = 8 \, Wb \]

### Step 4: Calculate the change in magnetic flux
The change in magnetic flux \( \Delta \Phi \) is given by:
\[ \Delta \Phi = \Phi_f - \Phi_i = 8 \, Wb - 2 \, Wb = 6 \, Wb \]

### Step 5: Calculate the average induced e.m.f.
The average induced e.m.f. \( \mathcal{E} \) can be calculated using Faraday's law of electromagnetic induction:
\[ \mathcal{E} = -\frac{\Delta \Phi}{\Delta t} \]
Substituting the values we found:
\[ \mathcal{E} = -\frac{6 \, Wb}{2 \, s} = -3 \, V \]
Since we are interested in the magnitude of the induced e.m.f., we take the absolute value:
\[ \mathcal{E} = 3 \, V \]

### Final Answer
The average e.m.f. induced in the coil is \( 3 \, V \).

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