A miniature spectrometer used for chemical analysis has a di

A miniature spectrometer used for chemical analysis has a diffraction grating with800 slits/mm set 25.0 mm in front of the detector \"screen.\" The detector can barely distinguish two bright lines that are 30 m apart in the first-order spectrum. What is the resolution of the spectrometer at a wavelength of 600 nm ? That is, if two distinct wavelengths can barely be distinguished, one of them being 600 nm , what is the wavelength difference between the two?

Solution

grating separation d = 1/800 mm= 1.25 e-6 m

wave length l= 600 nm

dSin() = l

– arcsin(600e-9/1.25e-6) = 28.68⁰

distance of the maxima from the central line = 25*Tan(28.68) = 13.68 mm

resolution of the detector = 30e-6 m

if l is the separation of the wave lengths then the line for (l+l) has to form at

13.68 + 0.03 = 13.71 mm from the central line.

angle of diffraction

_x = arctan(13.7/1.25) =28.74⁰

(l+l) = 1.25e-6*Sin(28.74) = 601 nm

resolution of the spectrometer = 1nm