A wheel has three spokes and is an uniform magnetic field perpendicular to its plane, with the axis of rotation of the wheel parallel to the magnetic field. When the wheel rotates with a uniform angualr velocity `omega`, the emf induced between the centre and rim of the wheel is `'e'`. If another wheel having same radius but with six spokes is kept in the same field and rotated with a uniform angular velocity `'omega//2'`, the emf induced between the centre and the rim will be

Options

A. `e`
B. `e//2`
C. `2e`
D. `e//4`

Explanation

To solve the problem, we need to determine the induced electromotive force (emf) in a wheel with six spokes when it is rotated at half the angular velocity of a wheel with three spokes.

### Step-by-Step Solution:

1. **Understanding the Induced EMF**:
The induced emf (E) in a rotating wheel in a magnetic field is given by the formula:
\[ E = \frac{1}{2} B \omega R^2 \]
where:
- \( B \) is the magnetic field strength,
- \( \omega \) is the angular velocity,
- \( R \) is the radius of the wheel.

2. **Induced EMF for the First Wheel**:
For the first wheel with three spokes rotating at angular velocity \( \omega \):
\[ E_1 = \frac{1}{2} B \omega R^2 \]

3. **Induced EMF for the Second Wheel**:
The second wheel has six spokes and is rotating at angular velocity \( \frac{\omega}{2} \):
\[ E_2 = \frac{1}{2} B \left(\frac{\omega}{2}\right) R^2 \]

4. **Calculating \( E_2 \)**:
Substituting the value of angular velocity into the emf equation for the second wheel:
\[ E_2 = \frac{1}{2} B \left(\frac{\omega}{2}\right) R^2 = \frac{1}{2} B \frac{\omega^2}{4} R^2 = \frac{1}{4} B \omega R^2 \]

5. **Relating \( E_2 \) to \( E_1 \)**:
Now, we can relate \( E_2 \) to \( E_1 \):
\[ E_2 = \frac{1}{4} B \omega R^2 = \frac{1}{2} \left(\frac{1}{2} B \omega R^2\right) = \frac{1}{2} E_1 \]
Since \( E_1 = \frac{1}{2} B \omega R^2 \), we find:
\[ E_2 = \frac{1}{2} E_1 \]

6. **Final Result**:
Since we know \( E_1 = E \), we conclude:
\[ E_2 = \frac{E}{2} \]

### Conclusion:
The induced emf between the center and the rim of the second wheel with six spokes, when rotated at angular velocity \( \frac{\omega}{2} \), is \( \frac{E}{2} \).