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. (a) `2`
B. (b) `3`
C. (c) `4`
D. (d) `5`

Explanation

To solve the problem step by step, we need to determine the number of points where maxima are observed when two coherent sources are separated by a distance of \( 4\lambda \) and a detector moves along the x-axis.

### Step 1: Understand the setup
- We have two coherent sources \( S_1 \) and \( S_2 \) on the y-axis, separated by a distance \( d = 4\lambda \).
- A detector moves along the positive x-axis.

### Step 2: Path difference for maxima
- The path difference \( \Delta x \) between the two sources at any point \( P \) on the x-axis is given by:
\[ \Delta x = S_1P - S_2P \]
- For constructive interference (maxima), the path difference must be an integer multiple of the wavelength:
\[ \Delta x = n\lambda \quad (n = 0, 1, 2, \ldots) \]

### Step 3: Use the geometry of the setup
- Let \( S_1 \) be at \( (0, 0) \) and \( S_2 \) be at \( (0, 4\lambda) \).
- A point \( P \) on the x-axis can be represented as \( (x, 0) \).
- The distances from \( P \) to \( S_1 \) and \( S_2 \) can be calculated using the Pythagorean theorem:
\[ S_1P = \sqrt{x^2 + 0^2} = x \]
\[ S_2P = \sqrt{x^2 + (4\lambda)^2} = \sqrt{x^2 + 16\lambda^2} \]

### Step 4: Calculate the path difference
- The path difference is:
\[ \Delta x = S_2P - S_1P = \sqrt{x^2 + 16\lambda^2} - x \]

### Step 5: Set up the equation for maxima
- For maxima, we set the path difference equal to \( n\lambda \):
\[ \sqrt{x^2 + 16\lambda^2} - x = n\lambda \]
- Rearranging gives:
\[ \sqrt{x^2 + 16\lambda^2} = n\lambda + x \]
- Squaring both sides:
\[ x^2 + 16\lambda^2 = (n\lambda + x)^2 \]
- Expanding the right side:
\[ x^2 + 16\lambda^2 = n^2\lambda^2 + 2n\lambda x + x^2 \]
- Simplifying:
\[ 16\lambda^2 = n^2\lambda^2 + 2n\lambda x \]
- Rearranging gives:
\[ 2n\lambda x = 16\lambda^2 - n^2\lambda^2 \]
\[ x = \frac{16\lambda - n^2\lambda}{2n} = \frac{(16 - n^2)\lambda}{2n} \]

### Step 6: Determine valid values of \( n \)
- For \( x \) to be positive, \( 16 - n^2 > 0 \) must hold, which gives:
\[ n^2 < 16 \implies n < 4 \]
- Thus, \( n \) can take values \( 0, 1, 2, 3 \).

### Step 7: Count the maxima
- The values of \( n \) that yield maxima are \( n = 0, 1, 2, 3 \), which gives us a total of 4 maxima.

### Final Answer
The number of points where maxima are observed, excluding the origin and far-off points, is **3**.

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