A horizontal ring of radius `r = (1)/(2) m` is kept in a vertical constant magentic field `1 T`. The ring is collapsed from maximum area to zero area in `1 s`. Then the emf induced in the ring is

Options

A. `1 V`
B. `(pi//4) V`
C. `(pi//2) V`
D. `pi V`

Explanation

To solve the problem of finding the induced emf in a collapsing ring in a magnetic field, we can follow these steps:

### Step 1: Identify the given values
- Radius of the ring, \( r = \frac{1}{2} \, \text{m} \)
- Magnetic field, \( B = 1 \, \text{T} \)
- Time taken for the collapse, \( t = 1 \, \text{s} \)

### Step 2: Calculate the initial area of the ring
The area \( A \) of the ring can be calculated using the formula for the area of a circle:
\[ A = \pi r^2 \]
Substituting the value of \( r \):
\[ A = \pi \left(\frac{1}{2}\right)^2 = \pi \left(\frac{1}{4}\right) = \frac{\pi}{4} \, \text{m}^2 \]

### Step 3: Determine the change in area
The ring collapses from its maximum area to zero area. Therefore, the change in area \( \Delta A \) is:
\[ \Delta A = A_{\text{final}} - A_{\text{initial}} = 0 - \frac{\pi}{4} = -\frac{\pi}{4} \, \text{m}^2 \]

### Step 4: Calculate the induced emf using Faraday's Law
According to Faraday's Law of electromagnetic induction, the induced emf \( \mathcal{E} \) is given by:
\[ \mathcal{E} = -B \frac{\Delta A}{\Delta t} \]
Substituting the values:
\[ \mathcal{E} = -1 \cdot \left(-\frac{\pi}{4}\right) \cdot \frac{1}{1} = \frac{\pi}{4} \, \text{V} \]

### Step 5: State the final answer
The induced emf in the ring is:
\[ \mathcal{E} = \frac{\pi}{4} \, \text{V} \]

### Conclusion
Thus, the correct answer is option 2: \( \frac{\pi}{4} \, \text{V} \).

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