A light with a wavelength of #540nm# passes trough a diffraction grating that has #5000# slits and is #16 mm# wide. What is the angle #theta# that will the highest order visible diffraction maxima be?

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Answer 1 · 2018-02-25T09:10:30

This is what I get

We know that when light of wavelength #lambda# is normally incident on grating, the diffracted light has maxima at angles #θ_m# given by

#d sin\ θ_m = |m| λ# ......(1)
where #d# is of slit separation, #m# is an integer and order of the maxima

From (1) we have

# |m| =(d sin\ θ_m)/λ# ........(2)

For the highest order visible diffraction maxima we must have #sin\ theta# term maximum#=1#. From (2) we get after inserting given values

# |m| =((16xx10^-3)/5000)/(540xx10^-9)#
#=> |m| =(16xx10^3)/(5xx540)#
#=> |m| =5.9#

Maximum order #=5#
From (1) we get

#θ_m =sin^-1(( |m| λ)/d)#

Inserting calculated and given values we get

#θ_5 =sin^-1(( 5xx5xx540xx10^-9)/((16xx10^-3)/5000))#
#=>θ_5 =sin^-1(( 25xx540xx10^-3)/16)#
#=> θ_5 =57.5^@#

.-.-.-.-.-.-.-.-.-.-

If a plane wave is incident at an angle of #θ_i#, above equation becomes

#d ( sin\ θ_i − sin\ θ_m ) = |m| λ#

Imposing given condition: #sin\ theta_m# must be maximum and #sin\ theta_i# minimum. We get back to (2)