Two identical cycle wheels (geometrically have different number of spokes connected from center to rim. One if having `20` spokes and the other having only `10` (the rim and the spokes are resistance less). One resistance of value `R` is connected between centre and rim. The current in `R` will be

Options

A. (a) double in the first wheel than in the second wheel
B. (b) four times in the first wheel than in the second wheel
C. ( C) will be double in the second wheel than that of the first wheel
D. (d) will be equal in both these wheels

Explanation

To solve the problem, we need to analyze the situation involving two identical cycle wheels with different numbers of spokes, and how the induced electromotive force (emf) affects the current through a resistance \( R \) connected between the center and the rim.

### Step-by-Step Solution:

1. **Understanding the Setup**:
- We have two identical cycle wheels, one with 20 spokes and the other with 10 spokes.
- Both wheels are rotated in a uniform magnetic field with the same angular speed \( \omega \).
- A resistance \( R \) is connected between the center and the rim of each wheel.

2. **Induced EMF Calculation**:
- The induced emf \( E \) in a rotating wheel can be calculated using the formula:
\[ E = \frac{1}{2} B \omega r \]
where \( B \) is the magnetic field strength, \( \omega \) is the angular speed, and \( r \) is the radius of the wheel.

3. **Distribution of EMF Across Spokes**:
- Each spoke acts like a conductor that contributes to the total induced emf.
- Since the spokes are resistance-less, the induced emf across each spoke will be the same.
- For both wheels, the total induced emf across the rim will be the same since they are geometrically identical and rotating in the same conditions.

4. **Equivalent EMF in Parallel Configuration**:
- The spokes in each wheel can be considered as being in parallel.
- For parallel configurations of identical cells (or spokes), the equivalent emf remains the same as that of a single spoke.
- Therefore, the total emf across the rim for both wheels is \( E \).

5. **Current Calculation**:
- The current \( I \) flowing through the resistance \( R \) connected between the center and the rim can be calculated using Ohm's law:
\[ I = \frac{E}{R} \]
- Since the induced emf \( E \) is the same for both wheels, the current \( I \) will also be the same.

6. **Conclusion**:
- The current in the resistance \( R \) connected to both wheels will be equal.
- Therefore, the answer to the question is that the current in \( R \) will be the same for both the wheels.

### Final Answer:
The current in resistance \( R \) will be equal in both cases.