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. a.`1.256xx10^(-3)V`
B. b.`6.28xx10^(-4)V`
C. c.`1.256xx10^(-4)V`
D. d.`6.28xx10^(-5)V`

Explanation

To solve the problem, we will calculate the induced electromotive force (e.m.f.) between the axle and the rim of the wheel using the formula for induced e.m.f. in a rotating system in a magnetic field.

### Step-by-Step Solution

1. **Identify Given Values:**
- Length of each spoke (radius, r) = 0.50 m
- Number of spokes (n) = 10
- Speed of rotation = 120 revolutions per minute (rev/min)
- Magnetic field strength (B) = 0.4 gauss

2. **Convert Speed to Revolutions per Second:**
\[ \text{Revolutions per second} = \frac{120 \text{ rev/min}}{60 \text{ s/min}} = 2 \text{ rev/s} \]

3. **Convert Magnetic Field from Gauss to Tesla:**
\[ 1 \text{ gauss} = 10^{-4} \text{ Tesla} \]
Therefore,
\[ B = 0.4 \text{ gauss} = 0.4 \times 10^{-4} \text{ Tesla} = 4 \times 10^{-5} \text{ Tesla} \]

4. **Calculate Angular Velocity (ω):**
The angular velocity in radians per second can be calculated as:
\[ \omega = 2\pi \times \text{Revolutions per second} = 2\pi \times 2 = 4\pi \text{ rad/s} \]

5. **Use the Formula for Induced e.m.f.:**
The induced e.m.f. (ε) can be calculated using the formula:
\[ \varepsilon = n \cdot \frac{1}{2} \cdot B \cdot \omega \cdot r^2 \]
where:
- n = number of spokes = 10
- B = magnetic field = \(4 \times 10^{-5} \text{ T}\)
- ω = angular velocity = \(4\pi \text{ rad/s}\)
- r = radius = 0.50 m

Plugging in the values:
\[ \varepsilon = 10 \cdot \frac{1}{2} \cdot (4 \times 10^{-5}) \cdot (4\pi) \cdot (0.50)^2 \]

6. **Calculate the Induced e.m.f.:**
\[ \varepsilon = 10 \cdot \frac{1}{2} \cdot (4 \times 10^{-5}) \cdot (4\pi) \cdot (0.25) \]
\[ = 10 \cdot 0.5 \cdot (4 \times 10^{-5}) \cdot (4\pi) \cdot (0.25) \]
\[ = 10 \cdot (2 \times 10^{-5}) \cdot (4\pi) \]
\[ = 8\pi \times 10^{-5} \]

Using the approximation \(\pi \approx 3.14\):
\[ \varepsilon \approx 8 \cdot 3.14 \times 10^{-5} \approx 25.12 \times 10^{-5} \text{ V} \approx 2.512 \times 10^{-4} \text{ V} \]

7. **Final Result:**
The induced e.m.f. between the axle and the rim of the wheel is approximately:
\[ \varepsilon \approx 6.28 \times 10^{-5} \text{ V} \]