Question

The phase difference between the two superimposing waves that give rise to a bright spot in a Young's double-slit experiment is (n is an integer) :

• A. \( 2n\pi \)
• B. \( 2n\pi + \frac{\pi}{4} \)
• C. \( 2n\pi + \frac{\pi}{2} \)
• D. \( 2n\pi + \pi \)

Hint:

Bright spot (Constructive Interference): Phase difference = \( 2n\pi \), Path difference = \( n\lambda \).
Dark spot (Destructive Interference): Phase difference = \( (2n \pm 1)\pi \), Path difference = \( (n \pm 0.5)\lambda \).

Answer 1

The Correct Option is A

Concept:
• In a Young's Double-Slit Experiment (YDSE), the pattern on the screen is formed by the superposition of two coherent light waves originating from two slits.

• The resultant intensity \( I \) at any point on the screen depends heavily on the phase difference \( \Delta \phi \) between the two arriving waves.

• The intensity formula is \( I = I_1 + I_2 + 2\sqrt{I_1 I_2} \cos(\Delta \phi) \).

• A "bright spot" (or maxima) is produced by purely constructive interference, which happens when the intensity is maximized.

Step 1: Determine the mathematical condition for maximum intensity
Looking at the intensity equation \( I = I_1 + I_2 + 2\sqrt{I_1 I_2} \cos(\Delta \phi) \), the only variable factor is the cosine term.
To maximize the overall intensity \( I \), the cosine function must reach its maximum possible value.
The maximum value of the cosine function is \( +1 \).
Therefore, the mathematical condition for a bright spot is:
\[ \cos(\Delta \phi) = 1 \]

Step 2: Solve for the phase difference
We need to find the angles (phase differences) for which the cosine equals 1.
From basic trigonometry, cosine equals 1 at \( 0, 2\pi, 4\pi, 6\pi, \dots \).
In other words, the phase difference must be an even integer multiple of \( \pi \).
This sequence can be concisely represented algebraically by using an integer variable \( n \) (where \( n = 0, \pm 1, \pm 2, \dots \)):
\[ \Delta \phi = 2n\pi \]
Conversely, a dark spot (destructive interference) requires minimum intensity, meaning \( \cos(\Delta \phi) = -1 \), which occurs at odd multiples of \( \pi \) or \( (2n+1)\pi \).
Since the question asks specifically for a bright spot, the condition is definitively \( 2n\pi \).

Step 3: Conclusion
The phase difference giving rise to a bright spot is universally \( 2n\pi \).
This corresponds directly to option (A).