A wheel with 20 metallic spokes each 1m long is rotated with a speed of 120 rpm in a plane perpendicular to magnetic field of 0.4G. The induced emf between axle and rim of wheel will be

Options

A. `2.51 x 10^-4 V `
B. 2.51 x 10^-5 V`
C. `4 x 10^-5 V`
D. `2.51v`

Answer

To find the induced emf between the axle and the rim of the wheel, we can follow these steps:

### Step 1: Understand the parameters
We have:
- Number of spokes (N) = 20
- Length of each spoke (r) = 1 m
- Speed of rotation (n) = 120 rpm
- Magnetic field (B) = 0.4 G = 0.4 × 10^(-4) T (since 1 G = 10^(-4) T)

### Step 2: Convert the speed of rotation to radians per second
The angular velocity (ω) in radians per second can be calculated using the formula:
\[ \omega = \frac{2\pi n}{60} \]
Substituting the value of n:
\[ \omega = \frac{2\pi \times 120}{60} = 4\pi \text{ rad/s} \]

### Step 3: Calculate the induced emf using Faraday's law
The induced emf (ε) can be calculated using the formula:
\[ \epsilon = -\frac{d\Phi}{dt} \]
Where:
- \(\Phi\) is the magnetic flux, given by \(\Phi = B \cdot A\)
- A is the area swept by the spoke.

For one spoke, the area (A) swept out in one complete rotation can be calculated as:
\[ A = \frac{1}{2} \cdot r^2 \cdot \theta \]
Where \(\theta\) is the angle in radians. In one second, the angle subtended is:
\[ \theta = \omega \cdot t = 4\pi \cdot 1 = 4\pi \text{ radians} \]
Thus, the area becomes:
\[ A = \frac{1}{2} \cdot r^2 \cdot 4\pi = 2\pi r^2 \]
Substituting \(r = 1\) m:
\[ A = 2\pi \cdot (1)^2 = 2\pi \text{ m}^2 \]

### Step 4: Calculate the magnetic flux
Now, substituting the values into the magnetic flux equation:
\[ \Phi = B \cdot A = 0.4 \times 10^{-4} \cdot 2\pi \]

### Step 5: Find the induced emf
Now, substituting this into the induced emf formula:
\[ \epsilon = \frac{1}{2} B \omega r^2 \]
Substituting the values:
\[ \epsilon = \frac{1}{2} \cdot (0.4 \times 10^{-4}) \cdot (4\pi) \cdot (1)^2 \]
\[ \epsilon = 0.2 \times 10^{-4} \cdot 4\pi = 0.8\pi \times 10^{-4} \text{ V} \]

### Step 6: Calculate the numerical value
Using \(\pi \approx 3.14\):
\[ \epsilon \approx 0.8 \cdot 3.14 \times 10^{-4} \approx 2.512 \times 10^{-4} \text{ V} \]
Thus, the induced emf is approximately:
\[ \epsilon \approx 2.5 \times 10^{-4} \text{ V} \]

### Conclusion
The induced emf between the axle and the rim of the wheel is approximately \(2.5 \times 10^{-4} \text{ V}\).

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