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

Explanation

To solve the problem, we need to calculate the average electromotive force (e.m.f.) induced in the coil using Faraday's law of electromagnetic induction. The formula for average e.m.f. is given by:

\[ \text{Average e.m.f.} = -\frac{d\Phi}{dt} \]

Where:
- \(\Phi\) is the magnetic flux through the coil.
- \(d\Phi\) is the change in magnetic flux.
- \(dt\) is the time interval over which the change occurs.

### Step 1: Calculate the change in magnetic field (B)
The magnetic field changes from \(B_1 = 1 \, \text{Wb/m}^2\) to \(B_2 = 4 \, \text{Wb/m}^2\).

\[ \Delta B = B_2 - B_1 = 4 \, \text{Wb/m}^2 - 1 \, \text{Wb/m}^2 = 3 \, \text{Wb/m}^2 \]

### Step 2: Calculate the change in magnetic flux (Φ)
The magnetic flux (\(\Phi\)) through the coil is given by the product of the magnetic field (B) and the area (A) of the coil:

\[ \Phi = B \cdot A \]

The area of the coil is given as \(A = 2 \, \text{m}^2\).

The change in magnetic flux (\(d\Phi\)) can be calculated as:

\[ d\Phi = A \cdot \Delta B = 2 \, \text{m}^2 \cdot 3 \, \text{Wb/m}^2 = 6 \, \text{Wb} \]

### Step 3: Calculate the time interval (dt)
The time interval (\(dt\)) is given as \(2 \, \text{s}\).

### Step 4: Calculate the average e.m.f.
Now, we can substitute the values into the formula for average e.m.f.:

\[ \text{Average e.m.f.} = -\frac{d\Phi}{dt} = -\frac{6 \, \text{Wb}}{2 \, \text{s}} = -3 \, \text{V} \]

Since we are interested in the magnitude of the induced e.m.f., we take the absolute value:

\[ \text{Average e.m.f.} = 3 \, \text{V} \]

### Conclusion
The average e.m.f. induced in the coil is \(3 \, \text{V}\).

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