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#### CHAPTER 10 WAVE OPTICS

Young’s double slit experiment

Thomas Young made two pinholes S1 and S2 (very close to each other) on an opaque screen. These were illuminated by another pinhole that was in turn, lit by a bright source. Light waves spread out from S and fall on both S1 and S2. S1 and S2 then behave like two coherent sources because light waves coming out from S1 and S2 are derived from the same original source and any abrupt phase change in S will manifest in exactly similar phase changes in the light coming out from S1 and S2. Thus, the two sources S1 and S2 will be locked in phase; i.e., they will be coherent like the two vibrating needles. Spherical waves originating from S1 and S2 will produce interference fringes on the screen GG’.

For an arbitrary point P on the line GG’ to have a maximum,

S2P – S1P = nλ; n = 0, 1, 2 ...

From the geometry of the figure,

which gives

$\left({\mathrm{S}}_{2}\mathrm{P}-{\mathrm{S}}_{2}\mathrm{P}\right)\left({\mathrm{S}}_{2}\mathrm{P}+{\mathrm{S}}_{2}\mathrm{P}\right)=2\mathrm{x}\mathrm{d}$

$⇒{\mathrm{S}}_{2}\mathrm{P}-{\mathrm{S}}_{2}\mathrm{P}=\frac{2\mathrm{x}\mathrm{d}}{{\mathrm{S}}_{2}\mathrm{P}+{\mathrm{S}}_{2}\mathrm{P}}$

If x << D and d << D, then,

Hence,

${\mathrm{S}}_{2}\mathrm{P}-{\mathrm{S}}_{2}\mathrm{P}=\frac{\mathrm{x}\mathrm{d}}{\mathrm{D}}$

For constructive interference or bright fringes,

For destructive interference or dark fringes,