The hydrogen spectrum includes a red line at $656$ nm and a blue$-$violet line at $434$ nm$.$ What are the angular separations between these two spectral lines for all visible orders obtained with a diffraction grating that has $4444$ grooves$/$cm$?($In this problem assume that the light is incident normally on the gratings.$)$

Step $1$

Many diffraction gratings yield several spectral orders within the

$90$

viewing range to either side. We estimate the angle between red and blue lines to be between about

$10$

and

$30.$

Step $2$

The angular separation is the difference between the angles corresponding to the red and blue wavelengths for each visible spectral order according to the diffraction grating equation

d sin $=$ m$,$

where d is the slit distance, $$ is the diffraction angle, and m is the order of the spectral line.

Step $3$

The grating spacing is the reciprocal of grooves$/$m$.$ We have

d $=$

Your response differs significantly from the correct answer. Rework your solution from the beginning and check each step carefully. $102$ m

$=106$ m$.$