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  4. The source that illuminates the double - slit in 'double - slit interference experiment' emits two distinct monochromatic waves of wavelength 500 nm and 600 nm, each of them producing its own pattern on the screen. At the central point of the pattern when path difference is zero, maxima of both the patterns coincide and the resulting interference pattern is most distinct at the region of zero path difference. But as one moves out of this central region, the two fringe systems are gradually out of step such that maximum due to on wavelength coincides with the minimum due to the other and the combined fringe system becomes completely indistinct. This may happen when path difference in nm is:

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Q. The source that illuminates the double - slit in 'double - slit interference experiment' emits two distinct monochromatic waves of wavelength $500\, nm$ and $600\, nm$, each of them producing its own pattern on the screen. At the central point of the pattern when path difference is zero, maxima of both the patterns coincide and the resulting interference pattern is most distinct at the region of zero path difference. But as one moves out of this central region, the two fringe systems are gradually out of step such that maximum due to on wavelength coincides with the minimum due to the other and the combined fringe system becomes completely indistinct. This may happen when path difference in nm is:

JEE MainJEE Main 2013Wave Optics

Solution:

This happens when path difference is integral multiple of an
$\Delta x=n\,\lambda_{1}\,\,\, (I)$
1) $\Delta x_{2}\left(m+\frac{1}{2}\right) \lambda_{2}$
2)$n \lambda_{1}=\left(m+\frac{1}{2}\right) \lambda_{2}$
3) $500 n=\left(m+\frac{1}{2}\right) 600$
4) $5 n=6 m+3\,\,\, (II)$
if $m=2, n=3$
satifies (II)
putting $n=3$ in $1$ all we get
$\Delta x_{2}= n \lambda_{1}=3 \times 500$
$=1500\,nm$