A copper disc of radius 0.1 m is roated about its centre with 20 revolutions per second in a uniform magnetic field of 0.1 T with its plane perpendicular the field. The emf induced across the radius of disc is

Options

A. `pi/20` volt
B. `pi/10` volt
C. `20 pi` millivolt
D. `100 pi` millivolt

Answer

To find 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, \( R = 0.1 \, \text{m} \)
- Revolutions per second, \( f = 20 \, \text{rev/s} \)
- Magnetic field strength, \( B = 0.1 \, \text{T} \)

### Step 2: Convert revolutions per second to angular velocity
The angular velocity \( \omega \) in radians per second can be calculated using the formula:
\[ \omega = 2\pi f \]
Substituting the value of \( f \):
\[ \omega = 2\pi \times 20 = 40\pi \, \text{rad/s} \]

### Step 3: Set up the formula for induced EMF
The induced EMF (\( \varepsilon \)) across the radius of the disc can be calculated using the formula:
\[ \varepsilon = \frac{1}{2} B \omega R^2 \]
Where:
- \( B \) is the magnetic field,
- \( \omega \) is the angular velocity,
- \( R \) is the radius of the disc.

### Step 4: Substitute the values into the formula
Now we substitute the values we have:
\[ \varepsilon = \frac{1}{2} \times 0.1 \, \text{T} \times 40\pi \, \text{rad/s} \times (0.1 \, \text{m})^2 \]
Calculating \( (0.1 \, \text{m})^2 \):
\[ (0.1)^2 = 0.01 \, \text{m}^2 \]
Now substituting this back into the equation:
\[ \varepsilon = \frac{1}{2} \times 0.1 \times 40\pi \times 0.01 \]

### Step 5: Calculate the induced EMF
Calculating the values step-by-step:
\[ \varepsilon = \frac{1}{2} \times 0.1 \times 40\pi \times 0.01 = 0.0005 \times 40\pi \]
\[ \varepsilon = 0.02\pi \, \text{V} \]
Now, calculating \( 0.02\pi \):
\[ \varepsilon \approx 0.0628 \, \text{V} \]
Converting to millivolts:
\[ \varepsilon \approx 62.8 \, \text{mV} \]

### Final Answer
The induced EMF across the radius of the disc is approximately **62.8 mV**.

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