A spectroscopist has a spectrometer equipped with a grating with 600 grooves/mm.This grating puts a 1228 nm wavelength range on a spectroscopic plate (a CCD)in order m = 5.The spectroscopist now changes the spectroscopic plate CCD by one that is larger by a factor of 3.What spectral range will be accepted when the spectroscopist observes the lines in order m = 4 with the new,larger spectroscopic plate?
A)4610 nm
B)2950 nm
C)512 nm
D)327 nm

Question

Quizplus · Accepted Answer

First, we need to find the spectral range that the spectrometer can observe with the original spectroscopic plate for order m = 4. Using the formula:

mλ = d(sinθ + sinφ)

where m is the diffraction order, λ is the wavelength, d is the grating spacing (1/600 mm in this case), θ is the angle of incidence (which we can assume to be 0), and φ is the angle of diffraction. For order m = 4 and φ = 78.9° (corresponding to the maximum diffraction angle), we can solve for λ:

4(1228 nm) = (1/600 mm)(sin 78.9° + sin θ)
θ ≈ 16.5° (using a calculator)

Substituting this angle back into the equation, we find:

4(1228 nm) = (1/600 mm)(sin 78.9° + sin 16.5°)
λ ≈ 327 nm

So the spectrometer can observe a spectral range of about 327 nm with the original spectroscopic plate for order m = 4.

Now, we need to determine what spectral range will be accepted with the new, larger spectroscopic plate. Since the new plate is three times larger, it will have a three times larger angular range. Therefore, for order m = 4, the new plate will observe:

3(327 nm) = 981 nm

So the spectral range that will be accepted when the spectroscopist observes the lines in order m = 4 with the new, larger spectroscopic plate is approximately 981 nm, or answer choice B.

A Spectroscopist Has a Spectrometer Equipped with a Grating with 600