A wheel with ten metallic spokes each 0.50 m long is rotated with a speed of 120 rev/min in a plane normal to the earth's magnetic field at the place. If the magnitude of the field is 0.4 gauss, the induced e.m.f. between the axle and the rim of the wheel is equal to

Options

A. `1.256xx10^(-3)V`
B. `6.28xx10^(-4)V`
C. `1.256xx10^(-4)V`
D. `6.28xx10^(-5)V`

Explanation

To solve the problem, we need to calculate the induced electromotive force (e.m.f.) between the axle and the rim of the wheel using the given parameters. Here’s a step-by-step solution:

### Step 1: Convert the speed of rotation from revolutions per minute (rev/min) to radians per second (rad/s).
The formula to convert revolutions per minute to radians per second is:
\[ \omega = 2\pi \times \left(\frac{f}{60}\right) \]
where \( f \) is the frequency in revolutions per minute.

Given:
\[ f = 120 \text{ rev/min} \]
Substituting the value:
\[ \omega = 2\pi \times \left(\frac{120}{60}\right) = 2\pi \times 2 = 4\pi \text{ rad/s} \]

### Step 2: Convert the magnetic field from Gauss to Tesla.
1 Gauss = \( 10^{-4} \) Tesla. Thus:
\[ B = 0.4 \text{ Gauss} = 0.4 \times 10^{-4} \text{ Tesla} = 4 \times 10^{-5} \text{ Tesla} \]

### Step 3: Calculate the induced e.m.f. using the formula.
The formula for induced e.m.f. (E) in a rotating wheel is given by:
\[ E = \frac{1}{2} B \omega L^2 \]
where:
- \( B \) is the magnetic field in Tesla,
- \( \omega \) is the angular velocity in rad/s,
- \( L \) is the length of one spoke.

Given:
- \( L = 0.5 \text{ m} \)

Substituting the values:
\[ E = \frac{1}{2} \times (4 \times 10^{-5}) \times (4\pi) \times (0.5)^2 \]
Calculating:
\[ E = \frac{1}{2} \times (4 \times 10^{-5}) \times (4\pi) \times (0.25) \]
\[ E = 2 \times 10^{-5} \times 4\pi \]
\[ E = 8\pi \times 10^{-5} \text{ volts} \]
Using \( \pi \approx 3.14 \):
\[ E \approx 8 \times 3.14 \times 10^{-5} \approx 25.12 \times 10^{-5} \text{ volts} \approx 2.512 \times 10^{-4} \text{ volts} \]

### Step 4: Final Result
The induced e.m.f. between the axle and the rim of the wheel is approximately:
\[ E \approx 2.51 \times 10^{-4} \text{ volts} \text{ or } 0.251 \text{ mV} \]