Two coherent sources `S_1` and `S_2` area separated by a distance four times the wavelength `lambda` of the source. The sources lie along y axis whereas a detector moves along +x axis. Leaving the origin and far off points the number of points where maxima are observed is

Options

A. 2
B. 3
C. 4
D. 5

Answer

To solve the problem of determining the number of points where maxima are observed when two coherent sources \( S_1 \) and \( S_2 \) are separated by a distance of \( 4\lambda \), we can follow these steps:

### Step 1: Understand the Geometry
We have two coherent sources, \( S_1 \) and \( S_2 \), separated by a distance \( d = 4\lambda \) along the y-axis. A detector moves along the x-axis. The goal is to find the positions along the x-axis where constructive interference (maxima) occurs.

### Step 2: Set Up the Problem
Let the distance from the origin (where the sources are located) to the detector be \( x \). The distance from \( S_1 \) to the detector is \( S_1D \) and from \( S_2 \) to the detector is \( S_2D \).

### Step 3: Calculate Path Difference
The path difference \( \Delta \) between the waves arriving at the detector from \( S_1 \) and \( S_2 \) is given by:
\[ \Delta = S_2D - S_1D \]
Using the Pythagorean theorem, we can express \( S_1D \) and \( S_2D \) in terms of \( x \):
\[ S_1D = \sqrt{x^2 + (2\lambda)^2} \quad \text{(since } S_1 \text{ is at } (0, 0) \text{ and } S_2 \text{ is at } (0, 4\lambda)\text{)} \]
\[ S_2D = \sqrt{x^2 + (2\lambda)^2} \]

### Step 4: Condition for Maxima
For constructive interference (maxima), the path difference must be an integer multiple of the wavelength:
\[ \Delta = n\lambda \quad (n = 0, 1, 2, \ldots) \]

### Step 5: Substitute and Simplify
Using the path difference, we have:
\[ \Delta = S_2D - S_1D = 4\lambda - 0 = 4\lambda \]
Setting this equal to \( n\lambda \), we can solve for \( n \):
\[ 4\lambda = n\lambda \implies n = 4 \]

### Step 6: Determine the Number of Maxima
The values of \( n \) can range from \( 0 \) to \( 4 \) (inclusive), giving us \( n = 0, 1, 2, 3, 4 \). Thus, there are a total of 5 maxima points.

### Conclusion
The number of points where maxima are observed is **5**.