Two identical coherent sources are placed on a diameter of a circle of radius R at separation x `(lt lt R)` symmetrical about the center of the circle. The sources emit identical wavelength `lambda` each. The number of points on the circle of maximum intensity is `(x = 5 lambda)`

Options

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

Explanation

To solve the problem, we need to determine the number of points on the circumference of a circle where the intensity of the interference pattern produced by two coherent sources is at a maximum. The sources are separated by a distance \( x = 5\lambda \), where \( \lambda \) is the wavelength of the emitted waves.

### Step-by-Step Solution:

1. **Understanding the Setup**:
- We have two coherent sources \( S_1 \) and \( S_2 \) placed on the diameter of a circle of radius \( R \).
- The separation between the sources is \( x = 5\lambda \).

2. **Identifying Maximum Intensity Conditions**:
- Maximum intensity occurs when the path difference between the waves from the two sources at any point on the circle is an integer multiple of the wavelength \( \lambda \).
- Mathematically, this can be expressed as:
\[ \Delta x = S_1P - S_2P = n\lambda \]
where \( n \) is any integer (0, ±1, ±2, ±3, ...).

3. **Calculating Maximum Path Difference**:
- The maximum path difference occurs when the distance between the two sources is equal to the separation \( x \).
- Given that \( x = 5\lambda \), the maximum path difference \( \Delta x \) can take values from \( -5\lambda \) to \( 5\lambda \).

4. **Finding Possible Values of \( n \)**:
- The integer values of \( n \) that satisfy the condition \( -5\lambda \leq n\lambda \leq 5\lambda \) are:
\[ n = -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5 \]
- This gives us a total of 11 possible values for \( n \).

5. **Counting the Maxima**:
- Each integer value of \( n \) corresponds to a point of maximum intensity on the circle.
- Therefore, the total number of points on the circle where the intensity is maximum is 11.

### Final Answer:
The number of points on the circle of maximum intensity is **11**.