Two identical coherent sources are placed on a diameter of a circle of radius R at a separation `d(ltltR)(d=5lambda)` symmetrically about the centre of the circle. The sources emit identical wavelength `lambda` each. What will be the number of points on the circle with constructive interference?

Options

A. 20
B. 22
C. 24
D. 26

Answer

To solve the problem of finding the number of points on a circle with constructive interference from two coherent sources, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Setup**:
- We have two coherent sources \( S_1 \) and \( S_2 \) separated by a distance \( d = 5\lambda \), where \( \lambda \) is the wavelength of the emitted waves. The sources are placed symmetrically about the center of a circle of radius \( R \).

2. **Path Difference for Constructive Interference**:
- For constructive interference, the path difference \( \Delta x \) between the waves from the two sources at any point on the circle must satisfy the condition:
\[ \Delta x = n\lambda \]
where \( n \) is an integer (0, 1, 2, ...).

3. **Calculating Path Differences**:
- The maximum path difference occurs when the angle subtended by the line connecting the two sources at the point on the circle is at its maximum. The maximum path difference can be calculated as:
\[ \Delta x_{\text{max}} = d = 5\lambda \]

4. **Finding Possible Values of \( n \)**:
- The possible values of \( n \) for constructive interference can be determined by the equation:
\[ n\lambda \leq 5\lambda \]
- This gives us:
\[ n \leq 5 \]
- Therefore, \( n \) can take the values \( 0, 1, 2, 3, 4, 5 \) which gives us a total of 6 values.

5. **Counting the Points on the Circle**:
- For each value of \( n \), there are two points on the circle where constructive interference occurs (one on each side of the line connecting the two sources).
- Thus, for each \( n \) from 0 to 5, we have:
\[ \text{Total Points} = 2 \times (\text{number of } n \text{ values}) = 2 \times 6 = 12 \]

6. **Conclusion**:
- The total number of points on the circle where constructive interference occurs is 12.

### Final Answer:
The number of points on the circle with constructive interference is **12**.