A coil having an area of `2m^(2)` is placed in a magnetic field which changes from `2 Wb//m^(2)` to `5Wb//m^(2)` in 3 seconds. The e.m.f. Induced in the coil is

Options

A. 4V
B. 3V
C. 2V
D. 1V

Explanation

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

### Step 1: Understand the formula for induced e.m.f.
The induced e.m.f. (ε) in a coil is given by Faraday's law of electromagnetic induction, which states:
\[ \varepsilon = -\frac{d\Phi}{dt} \]
where \( \Phi \) is the magnetic flux through the coil and \( dt \) is the time interval.

### Step 2: Calculate the change in magnetic flux (dΦ)
The magnetic flux \( \Phi \) through the coil is given by:
\[ \Phi = B \cdot A \]
where \( B \) is the magnetic field strength and \( A \) is the area of the coil.

Given:
- Area \( A = 2 \, m^2 \)
- Initial magnetic field \( B_1 = 2 \, Wb/m^2 \)
- Final magnetic field \( B_2 = 5 \, Wb/m^2 \)

The change in magnetic field \( dB \) is:
\[ dB = B_2 - B_1 = 5 \, Wb/m^2 - 2 \, Wb/m^2 = 3 \, Wb/m^2 \]

Now, we can calculate the change in magnetic flux \( d\Phi \):
\[ d\Phi = A \cdot dB = 2 \, m^2 \cdot 3 \, Wb/m^2 = 6 \, Wb \]

### Step 3: Calculate the time interval (dt)
The time interval over which the change occurs is given as:
\[ dt = 3 \, seconds \]

### Step 4: Substitute into the formula for e.m.f.
Now we can substitute \( d\Phi \) and \( dt \) into the formula for induced e.m.f.:
\[ \varepsilon = -\frac{d\Phi}{dt} = -\frac{6 \, Wb}{3 \, s} = -2 \, V \]

The negative sign indicates the direction of the induced e.m.f. according to Lenz's law, but we are interested in the magnitude:
\[ \varepsilon = 2 \, V \]

### Final Answer
The induced e.m.f. in the coil is \( 2 \, V \).

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