A wheel with 20 metallic spokes each of length 8.0 m long is rotated with a speed of 120 revolution per minute in a plane normal to the horizontal component of earth magnetic field H at a place. If `H=0.4xx10^(-4)` T at the place, then induced emf between the axle the rim of the wheel is

Options

A. `2.3xx10^(-4)V`
B. `3.1xx10^(-4)V`
C. `2.9xx10^(-4)V`
D. `1.61xx10^(-4)V`

Explanation

To find the induced electromotive force (emf) between the axle and the rim of the wheel, we can use Faraday's law of electromagnetic induction. The formula for induced emf (ε) in a rotating system can be expressed as:

\[ \epsilon = B \cdot v \cdot L \]

Where:
- \( B \) is the magnetic field strength (in tesla),
- \( v \) is the linear velocity of the spokes (in meters per second),
- \( L \) is the length of the spokes (in meters).

### Step-by-Step Solution:

**Step 1: Calculate the angular velocity (ω) of the wheel.**

The wheel rotates at 120 revolutions per minute (rpm). To convert this to radians per second, we use the conversion factor \( 2\pi \) radians per revolution and \( 60 \) seconds per minute:

\[ \omega = 120 \, \text{rpm} \times \frac{2\pi \, \text{radians}}{1 \, \text{revolution}} \times \frac{1 \, \text{minute}}{60 \, \text{seconds}} = \frac{120 \times 2\pi}{60} = 4\pi \, \text{radians/second} \]

**Step 2: Calculate the linear velocity (v) of the spokes.**

The linear velocity \( v \) of the end of the spokes can be calculated using the formula:

\[ v = \omega \cdot r \]

Where \( r \) is the length of the spokes. Given that the length of each spoke is \( 8.0 \, \text{m} \):

\[ v = 4\pi \, \text{radians/second} \times 8.0 \, \text{m} = 32\pi \, \text{m/s} \]

**Step 3: Calculate the induced emf (ε).**

Now, we can substitute the values into the induced emf formula. We have:
- \( B = 0.4 \times 10^{-4} \, \text{T} \)
- \( L = 8.0 \, \text{m} \)
- \( v = 32\pi \, \text{m/s} \)

Substituting these values:

\[ \epsilon = B \cdot v \cdot L = (0.4 \times 10^{-4} \, \text{T}) \cdot (32\pi \, \text{m/s}) \cdot (8.0 \, \text{m}) \]

Calculating this gives:

\[ \epsilon = 0.4 \times 10^{-4} \cdot 32\pi \cdot 8.0 \]

Calculating \( 32 \cdot 8 = 256 \):

\[ \epsilon = 0.4 \times 10^{-4} \cdot 256\pi \]

Now, using \( \pi \approx 3.14 \):

\[ \epsilon \approx 0.4 \times 10^{-4} \cdot 256 \cdot 3.14 \approx 0.4 \times 10^{-4} \cdot 804.16 \approx 0.032 \, \text{V} \]

Thus, the induced emf between the axle and the rim of the wheel is approximately:

\[ \epsilon \approx 0.032 \, \text{V} \text{ or } 32 \, \text{mV} \]