If the radius of a coil is changing at the rate of `10^(-2)` unit in a normal magnetic field of `10^(-3)` units, the induced emf is `1muV`. What the final radius of the coil?

Options

A. 1.6 cm
B. 16 cm
C. 12 cm
D. 1.2 cm

Answer

To find the final radius of the coil, we will use the formula for induced electromotive force (emf) in terms of the magnetic flux. Here are the steps to solve the problem:

### Step 1: Understand the relationship between induced emf and magnetic flux
The induced emf (ε) is given by the rate of change of magnetic flux (Φ) through the coil:
\[ \epsilon = -\frac{d\Phi}{dt} \]
where Φ is the magnetic flux given by:
\[ \Phi = B \cdot A \]
In this case, since the coil is circular, the area \( A \) is given by:
\[ A = \pi r^2 \]
Thus, the magnetic flux becomes:
\[ \Phi = B \cdot \pi r^2 \]

### Step 2: Differentiate the magnetic flux with respect to time
To find the induced emf, we need to differentiate the magnetic flux with respect to time:
\[ \frac{d\Phi}{dt} = B \cdot \frac{d}{dt}(\pi r^2) = B \cdot \pi \cdot 2r \cdot \frac{dr}{dt} \]
Here, \( \frac{dr}{dt} \) is the rate of change of the radius, which is given as \( 10^{-2} \) units.

### Step 3: Substitute the values into the equation
We know:
- \( B = 10^{-3} \) units
- \( \frac{dr}{dt} = 10^{-2} \) units
- Induced emf \( \epsilon = 1 \mu V = 1 \times 10^{-6} V \)

Substituting these values into the equation for induced emf:
\[ 1 \times 10^{-6} = B \cdot \pi \cdot 2r \cdot \frac{dr}{dt} \]
\[ 1 \times 10^{-6} = (10^{-3}) \cdot \pi \cdot 2r \cdot (10^{-2}) \]

### Step 4: Simplify the equation
Rearranging the equation gives:
\[ 1 \times 10^{-6} = 2\pi \cdot 10^{-5} \cdot r \]
Now, we can isolate \( r \):
\[ r = \frac{1 \times 10^{-6}}{2\pi \cdot 10^{-5}} \]

### Step 5: Calculate the radius
Calculating the value of \( r \):
\[ r = \frac{1 \times 10^{-6}}{2\pi \cdot 10^{-5}} = \frac{1}{2\pi} \times 10^{-1} \]
\[ r \approx \frac{1}{6.2832} \times 10^{-1} \approx 0.15915 \text{ units} \]

### Step 6: Convert to centimeters
Since the question asks for the radius in centimeters:
\[ r \approx 0.15915 \text{ units} \approx 1.5915 \text{ cm} \approx 1.6 \text{ cm} \]

### Final Answer
Thus, the final radius of the coil is approximately:
\[ \boxed{1.6 \text{ cm}} \]