Derive an expression for path difference in Youngs doubleslit experiment and obtain the conditions for constructive and destructive interference

Answers (1)
S safeer

d<<D

For constructive interference

S_{2}P-S_{1}P=n\lambda

\left [ D^{2} +\left ( x+\frac{d}{2} \right )^{2}\right ]-\left [ D^{2} +\left ( x-\frac{d}{2} \right )^{2}\right ]

=S_{2}P^{2}-S_{1}P^{2}

\Rightarrow S_{2}P^{2}-S_{1}P^{2}=xd

(S_{2}P-S_{1}P)(S_{2}P+S_{1}P)=xd

S_{2}P-S_{1}P=\frac{xd}{D}\; \; \; \; \; \; \; \; (d<<D)

\therefore \frac{xd}{D}=n\lambda

or x_{n}=\frac{n\lambda D}{d}\; \; \; \; (n=o,\pm 1,\pm 2)

for destructive interference

\frac{xd}{D}=\left ( n+\frac{1}{2} \right )\lambda                 (n=o,\pm 1,\pm 2)

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