A coil has `2000` turns and area of `70cm^(2)`. The magnetic field perpendicular to the plane of the coil is `0.3 Wb//m^(2)` and takes `0.1` sec to rotate through `180^(0)`. The value of the induced e.m.f. will be

Options

A. 8.4 V
B. 84 V
C. 42 V
D. 4.2 V

Answer

To find the induced electromotive force (e.m.f.) in the coil, we can follow these steps:

### Step 1: Calculate the area of the coil in square meters
The area of the coil is given as \(70 \, \text{cm}^2\). We need to convert this to square meters.

\[ \text{Area} = 70 \, \text{cm}^2 = 70 \times 10^{-4} \, \text{m}^2 = 0.007 \, \text{m}^2 \]

### Step 2: Calculate the initial magnetic flux
The magnetic flux (\(\Phi\)) through one turn of the coil is given by the formula:

\[ \Phi = B \times A \]

Where:
- \(B = 0.3 \, \text{Wb/m}^2\) (magnetic field)
- \(A = 0.007 \, \text{m}^2\) (area of the coil)

Calculating the initial flux for one turn:

\[ \Phi = 0.3 \, \text{Wb/m}^2 \times 0.007 \, \text{m}^2 = 0.0021 \, \text{Wb} \]

### Step 3: Calculate the total magnetic flux for all turns
Since the coil has \(2000\) turns, the total magnetic flux (\(\Phi_{\text{total}}\)) is:

\[ \Phi_{\text{total}} = N \times \Phi = 2000 \times 0.0021 \, \text{Wb} = 4.2 \, \text{Wb} \]

### Step 4: Calculate the change in magnetic flux
When the coil rotates through \(180^\circ\), the magnetic field direction reverses, leading to a change in flux. The change in flux (\(\Delta \Phi\)) is:

\[ \Delta \Phi = \Phi_{\text{final}} - \Phi_{\text{initial}} = -\Phi_{\text{total}} - \Phi_{\text{total}} = -2 \times \Phi_{\text{total}} = -2 \times 4.2 \, \text{Wb} = -8.4 \, \text{Wb} \]

### Step 5: Calculate the induced e.m.f.
Using Faraday's law of electromagnetic induction, the induced e.m.f. (\(E\)) is given by:

\[ E = -\frac{\Delta \Phi}{\Delta t} \]

Where \(\Delta t = 0.1 \, \text{s}\). Substituting the values:

\[ E = -\frac{-8.4 \, \text{Wb}}{0.1 \, \text{s}} = \frac{8.4 \, \text{Wb}}{0.1 \, \text{s}} = 84 \, \text{V} \]

### Final Answer
The value of the induced e.m.f. is \(84 \, \text{V}\).

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