A rectangular coil of 100 turns and size `0.1mxx0.05m` is placed perpendicular to a magnetic field of 0.1 T. If the field drops to 0.05 T in 0.05 s, the magnitude of the emf induced in the coil is

Options

A. `sqrt2`
B. `sqrt3`
C. `sqrt(0.6)`
D. 0.5

Explanation

To find the magnitude of the induced emf in the rectangular coil, we can follow these steps:

### Step 1: Identify the given parameters
- Number of turns (N) = 100
- Size of the coil = 0.1 m x 0.05 m
- Initial magnetic field (B_initial) = 0.1 T
- Final magnetic field (B_final) = 0.05 T
- Time interval (Δt) = 0.05 s

### Step 2: Calculate the area of the coil
The area (A) of the rectangular coil can be calculated using the formula:
\[ A = \text{length} \times \text{width} \]
\[ A = 0.1 \, \text{m} \times 0.05 \, \text{m} = 0.005 \, \text{m}^2 \]

### Step 3: Calculate the change in magnetic flux
The magnetic flux (Φ) through the coil is given by:
\[ \Phi = B \times A \]
The change in magnetic flux (ΔΦ) can be calculated as:
\[ \Delta \Phi = \Phi_{\text{final}} - \Phi_{\text{initial}} \]
Where:
\[ \Phi_{\text{final}} = B_{\text{final}} \times A \]
\[ \Phi_{\text{initial}} = B_{\text{initial}} \times A \]

Calculating the initial and final flux:
\[ \Phi_{\text{initial}} = 0.1 \, \text{T} \times 0.005 \, \text{m}^2 = 0.0005 \, \text{Wb} \]
\[ \Phi_{\text{final}} = 0.05 \, \text{T} \times 0.005 \, \text{m}^2 = 0.00025 \, \text{Wb} \]

Now, calculate the change in flux:
\[ \Delta \Phi = 0.00025 \, \text{Wb} - 0.0005 \, \text{Wb} = -0.00025 \, \text{Wb} \]

### Step 4: Calculate the induced emf (ε)
The induced emf (ε) can be calculated using Faraday's law of electromagnetic induction:
\[ \varepsilon = -\frac{\Delta \Phi}{\Delta t} \]

Substituting the values:
\[ \varepsilon = -\frac{-0.00025 \, \text{Wb}}{0.05 \, \text{s}} \]
\[ \varepsilon = \frac{0.00025 \, \text{Wb}}{0.05 \, \text{s}} = 0.005 \, \text{V} \]

### Step 5: Multiply by the number of turns
Since the coil has 100 turns, the total induced emf is:
\[ \varepsilon_{\text{total}} = N \times \varepsilon \]
\[ \varepsilon_{\text{total}} = 100 \times 0.005 \, \text{V} = 0.5 \, \text{V} \]

### Final Answer
The magnitude of the induced emf in the coil is **0.5 V**.

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