A conducing circular loop is placed in a uniform magnetic field of indution `B` tesla with its plane normal to the field. Now, radius of the loop starts shrinking at the rate `(dr//dt)`. Then the induced e.m.f. at the instant when the radius is `r` is:

Options

A. `pirB(dr//dt)`
B. `2pirB(dr//dt)`
C. `pir^(2)(dB//dt)`
D. `(pir^(2)//2)B(dr//dt)`

Explanation

To solve the problem of finding the induced electromotive force (e.m.f.) in a conducting circular loop placed in a uniform magnetic field, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Problem**: We have a conducting circular loop with radius \( r \) placed in a uniform magnetic field \( B \) tesla. The radius of the loop is shrinking at a rate of \( \frac{dr}{dt} \).

2. **Identify the Relevant Formula**: According to Faraday's law of electromagnetic induction, the induced e.m.f. (\( \mathcal{E} \)) in a closed loop is given by:
\[ \mathcal{E} = -\frac{d\Phi}{dt} \]
where \( \Phi \) is the magnetic flux through the loop.

3. **Calculate the Magnetic Flux**: The magnetic flux \( \Phi \) through the loop is given by:
\[ \Phi = B \cdot A \]
where \( A \) is the area of the loop. For a circular loop, the area \( A \) is:
\[ A = \pi r^2 \]
Therefore, the magnetic flux becomes:
\[ \Phi = B \cdot \pi r^2 \]

4. **Differentiate the Magnetic Flux**: To find the induced e.m.f., we need to differentiate the magnetic flux with respect to time:
\[ \frac{d\Phi}{dt} = \frac{d}{dt}(B \cdot \pi r^2) \]
Since \( B \) is constant, we can take it out of the differentiation:
\[ \frac{d\Phi}{dt} = B \cdot \pi \frac{d}{dt}(r^2) \]

5. **Apply the Chain Rule**: Using the chain rule, we differentiate \( r^2 \):
\[ \frac{d}{dt}(r^2) = 2r \frac{dr}{dt} \]
Thus, we have:
\[ \frac{d\Phi}{dt} = B \cdot \pi \cdot 2r \frac{dr}{dt} \]

6. **Substitute Back into the Induced e.m.f. Formula**: Now we substitute this expression back into the formula for induced e.m.f.:
\[ \mathcal{E} = -\frac{d\Phi}{dt} = -B \cdot \pi \cdot 2r \frac{dr}{dt} \]

7. **Final Expression for Induced e.m.f.**: The induced e.m.f. at the instant when the radius is \( r \) is:
\[ \mathcal{E} = -2\pi B r \frac{dr}{dt} \]

### Final Answer:
The induced e.m.f. at the instant when the radius is \( r \) is:
\[ \mathcal{E} = -2\pi B r \frac{dr}{dt} \]