A copper disc of radius `0.1 m` rotates about its centre with `10` revolutuion per second in a uniform magnetic field of `0.1` tesla. The emf induced across the radius of the disc is

Options

A. `(pi)/(10)V`
B. `(2pi)/(10)V`
C. `10pimV`
D. `20pimV`

Explanation

To solve the problem of finding the induced emf across the radius of a rotating copper disc in a magnetic field, we can follow these steps:

### Step 1: Identify the given values
- Radius of the disc (r) = 0.1 m
- Frequency of rotation (f) = 10 revolutions per second
- Magnetic field strength (B) = 0.1 T

### Step 2: Convert frequency to angular frequency
Angular frequency (ω) can be calculated using the formula:
\[ \omega = 2\pi f \]
Substituting the given frequency:
\[ \omega = 2\pi \times 10 = 20\pi \, \text{rad/s} \]

### Step 3: Use the formula for induced emf
The induced emf (E) in a rotating disc in a magnetic field can be calculated using the formula:
\[ E = \frac{1}{2} B \omega r^2 \]
Substituting the values of B, ω, and r:
\[ E = \frac{1}{2} \times 0.1 \times (20\pi) \times (0.1)^2 \]

### Step 4: Simplify the expression
Calculating the components:
- \( (0.1)^2 = 0.01 \)
- Thus, the expression becomes:
\[ E = \frac{1}{2} \times 0.1 \times 20\pi \times 0.01 \]
- Simplifying further:
\[ E = \frac{1}{2} \times 0.1 \times 20 \times 0.01 \times \pi \]
\[ E = \frac{1}{2} \times 0.1 \times 0.2 \times \pi \]
\[ E = 0.01\pi \, \text{V} \]

### Step 5: Calculate the numerical value
Using the approximate value of \(\pi \approx 3.14\):
\[ E \approx 0.01 \times 3.14 = 0.0314 \, \text{V} \]

### Final Answer
The induced emf across the radius of the disc is approximately **0.0314 V**.

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