Read the information given below to answer
In Young’s double slit experiment as shown in figure, interference of light waves were observed on the screen. Thomas Young made two pinholes \( S_1 \) and \( S_2 \) (very close to each other) on an opaque screen in front of parent source \( S \)
\( S_1 \) and \( S_2 \) behaved as coherent sources, and produced interference pattern on the screen which has alternate bright fringe and dark fringes. This experiment proved Huygens’ wave theory of Light.
Path difference between the waves meeting at point \( P \) is given by
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \( \displaystyle\frac{xd}{D} \)
In the figure, \( P \) is at height \( x \) above the central point \( O \), the slits are \( d \) apart and the screen is at distance \( D \). For \( D \gg d \) the two rays to \( P \) are nearly parallel, making a small angle \( \theta \) with the central line, and the path difference is
\( S_2P - S_1P = d\sin\theta \)
For a small angle, \( \sin\theta \approx \tan\theta = x/D \), so
\( \displaystyle S_2P - S_1P = d\,\frac{x}{D} = \frac{xd}{D} \)
Shortcut: Units check: a path difference is a length. \( xd/D \) has the unit of length and grows with the slit separation \( d \), as it must; \( xD/d \) and \( 2xD/d \) would grow as the slits come closer, which is impossible.