Question

An interference is observed due to two coherent sources, A and B, separated by a distance \(4\lambda\) along the Y-axis, where \(\lambda\) is the wavelength of the source. Source B is at the origin. A detector D is moved along the positive X-axis. The number of points on the X-axis, excluding \(x=0\) and \(x\rightarrow \infty\), at which maxima will be observed, is

• A. \(3\)
• B. \(4\)
• C. \(5\)
• D. \(6\)

Hint:

Find the path difference between the two sources at a point on the X-axis and count the integer multiples of the wavelength it can take.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept
A maximum occurs where the path difference is a whole number of wavelengths: \(\Delta = n\lambda\).

Step 2: Path difference
Source B is at the origin, A is at \((0, 4\lambda)\) and the detector is at \((x, 0)\).
\(\Delta = \sqrt{x^{2}+16\lambda^{2}} - x\).

Step 3: Range of the path difference
At \(x = 0\), \(\Delta = 4\lambda\). As \(x \rightarrow \infty\), \(\Delta \rightarrow 0\). It decreases steadily in between.

Step 4: Counting the maxima
Both end points are excluded, so \(0 \lt \Delta \lt 4\lambda\). Hence \(n = 1, 2, 3\), and each value occurs at exactly one point.
There are 3 maxima.

Final Answer:
Exactly 3 points on the positive X-axis show maxima. \[ \boxed{\text{(A) }3} \]