Question

A 6200 line/cm diffraction grating is 3.14 cm wide. If light with wavelengths near 624 nm falls on the grating, how close can two wavelengths be if they are to be resolved in any order

Answer 1

Answer:

$1.6026299569\times 10^{-11}\ m$

Explanation:

Grating constant

$d=\dfrac{1}{6200}=0.000161\ cm=0.000161\times 10^{-2}\ m$

Number of slits

$N=3.14\times 6200=19468$

Order

$m=\dfrac{d}{\lambda}\\\Rightarrow m=\dfrac{0.000161\times 10^{-2}}{624\times 10^{-9}}\\\Rightarrow m\approx 2$

At m = 1

$\Delta\lambda=\dfrac{\lambda}{mN}\\\Rightarrow \Delta\lambda=\dfrac{624\times 10^{-9}}{1\times 19468}\\\Rightarrow \Delta\lambda=3.2052599137\times 10^{-11}\ m$

At m = 2

$\Delta\lambda=\dfrac{\lambda}{mN}\\\Rightarrow \Delta\lambda=\dfrac{624\times 10^{-9}}{2\times 19468}\\\Rightarrow \Delta\lambda=1.6026299569\times 10^{-11}\ m$

The wavelengths can be close by $1.6026299569\times 10^{-11}\ m$