A coil having an area `A_(0)` is placed in a magnetic field which changes from `B_(0)` to `4B_(0)` in time interval t. The emf induced in the coil will be

Options

A. `3A_(0)B_(0)//t`
B. `4A_(0)B_(0)//t`
C. `3B_(0)//A_(0)t`
D. `4B_(0)//A_(0)t`

Explanation

To find the induced EMF in the coil when the magnetic field changes from \( B_0 \) to \( 4B_0 \) over a time interval \( t \), we can follow these steps:

### Step 1: Calculate the Initial Magnetic Flux
The initial magnetic flux (\( \Phi_i \)) through the coil can be calculated using the formula:
\[ \Phi_i = B_0 \cdot A_0 \]
where \( B_0 \) is the initial magnetic field and \( A_0 \) is the area of the coil.

### Step 2: Calculate the Final Magnetic Flux
The final magnetic flux (\( \Phi_f \)) when the magnetic field changes to \( 4B_0 \) is given by:
\[ \Phi_f = 4B_0 \cdot A_0 \]

### Step 3: Determine the Change in Magnetic Flux
The change in magnetic flux (\( \Delta \Phi \)) is calculated as:
\[ \Delta \Phi = \Phi_f - \Phi_i = (4B_0 \cdot A_0) - (B_0 \cdot A_0) = 3B_0 \cdot A_0 \]

### Step 4: Apply Faraday's Law of Electromagnetic Induction
According to Faraday's law, the induced EMF (\( \mathcal{E} \)) is given by the negative rate of change of magnetic flux:
\[ \mathcal{E} = -\frac{\Delta \Phi}{\Delta t} \]
Substituting the change in flux and the time interval \( t \):
\[ \mathcal{E} = -\frac{3B_0 \cdot A_0}{t} \]

### Step 5: Determine the Magnitude of the Induced EMF
The magnitude of the induced EMF is:
\[ |\mathcal{E}| = \frac{3B_0 \cdot A_0}{t} \]

### Final Answer
Thus, the induced EMF in the coil is:
\[ \mathcal{E} = \frac{3B_0 A_0}{t} \]

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