A conducting circualr loop is placed in a unifrom magentic field , B= 0.025 T with its plane perpendicular to the loop .The radius of the loop is made to shrink at a constant rate of 1 `mms^(-1)` .The induced emf when the radius is `2cm` ,is

Options

A. `2pimuV`
B. `pimuV`
C. `(pi)/(2)muV`
D. `2muV`

Explanation

To solve the problem, we will follow these steps:

### Step 1: Identify the given values
- Magnetic field strength, \( B = 0.025 \, \text{T} \)
- Radius of the loop, \( r = 2 \, \text{cm} = 0.02 \, \text{m} \)
- Rate of change of radius, \( \frac{dr}{dt} = -1 \, \text{mm/s} = -1 \times 10^{-3} \, \text{m/s} \) (negative because the radius is shrinking)

### Step 2: Calculate the area of the loop
The area \( A \) of a circular loop is given by the formula:
\[ A = \pi r^2 \]
Substituting the value of \( r \):
\[ A = \pi (0.02)^2 = \pi (0.0004) = 0.0004\pi \, \text{m}^2 \]

### Step 3: Calculate the magnetic flux
The magnetic flux \( \Phi \) through the loop is given by:
\[ \Phi = B \cdot A \]
Substituting the values:
\[ \Phi = 0.025 \cdot (0.0004\pi) = 0.00001\pi \, \text{Wb} \]

### Step 4: Determine the induced EMF
The induced EMF \( \mathcal{E} \) is given by Faraday's law of electromagnetic induction:
\[ \mathcal{E} = -\frac{d\Phi}{dt} \]
Using the chain rule:
\[ \mathcal{E} = -B \cdot \frac{dA}{dt} \]
Now, since \( A = \pi r^2 \), we can find \( \frac{dA}{dt} \):
\[ \frac{dA}{dt} = \frac{d}{dt}(\pi r^2) = 2\pi r \frac{dr}{dt} \]
Substituting \( r = 0.02 \, \text{m} \) and \( \frac{dr}{dt} = -1 \times 10^{-3} \, \text{m/s} \):
\[ \frac{dA}{dt} = 2\pi (0.02)(-1 \times 10^{-3}) = -0.00004\pi \, \text{m}^2/\text{s} \]

### Step 5: Substitute into the EMF formula
Now substituting \( \frac{dA}{dt} \) into the EMF equation:
\[ \mathcal{E} = -B \cdot \left(-0.00004\pi\right) = 0.025 \cdot 0.00004\pi \]
Calculating this:
\[ \mathcal{E} = 0.000001\pi \, \text{V} = \pi \times 10^{-6} \, \text{V} \]
Converting to microvolts:
\[ \mathcal{E} = \pi \, \mu\text{V} \approx 3.14 \, \mu\text{V} \]

### Final Answer
The induced EMF when the radius is \( 2 \, \text{cm} \) is approximately \( 3.14 \, \mu\text{V} \).

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