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 will analyze the situation step by step.

### Step 1: Understand the Configuration
We have two coherent sources \( S_1 \) and \( S_2 \) placed on a diameter of a circle of radius \( R \) with a separation \( x \) (where \( x \ll R \)). The sources emit waves of identical wavelength \( \lambda \).

**Hint:** Visualize the setup by sketching the circle and marking the positions of the two sources.

### Step 2: Determine the Path Difference
The path difference \( \Delta x \) between the waves from the two sources at any point on the circle is given by the formula:
\[ \Delta x = S_1P - S_2P \]
where \( P \) is any point on the circumference of the circle. For constructive interference (maxima), the path difference must be an integer multiple of the wavelength:
\[ \Delta x = n\lambda \]
where \( n \) is an integer (0, 1, 2, ...).

**Hint:** Remember that maxima occur when the path difference is a multiple of the wavelength.

### Step 3: Calculate Maximum Points
Given that the separation \( x = 5\lambda \), we can find the number of maxima on the circle. The maximum path difference between the two sources occurs at points on the circle directly opposite each other, which is equal to the separation \( x \).

The maximum path difference that can occur on the circle is \( x \). Since \( x = 5\lambda \), we can have:
\[ \Delta x = 0, \lambda, 2\lambda, 3\lambda, 4\lambda, 5\lambda \]
This means that there are 6 values of \( n \) (from 0 to 5) for which maxima can occur.

**Hint:** Count the number of integer values of \( n \) from 0 to the maximum path difference divided by \( \lambda \).

### Step 4: Symmetry of the Circle
Due to the symmetry of the arrangement, for each maximum on one side of the center, there is an equivalent maximum on the other side. Therefore, for each \( n \) from 1 to 5, there are two maxima (one on each side of the center).

**Hint:** Consider how symmetry affects the number of maxima on both sides of the center.

### Step 5: Total Count of Maxima
Including the central maximum (where \( n = 0 \)), we have:
- 1 maximum at \( n = 0 \)
- 5 maxima from \( n = 1 \) to \( n = 5 \), each contributing 2 maxima due to symmetry.

Thus, the total number of maxima is:
\[ 1 + 5 \times 2 = 1 + 10 = 11 \]

**Hint:** Make sure to include the central maximum when counting.

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