In a diffraction grating, for fixed λ and d, if m increases, what happens to θ?

In a diffraction grating, for fixed λ and d, if m increases, what happens to θ?

• A. θ decreases
• B. θ stays the same
• C. θ increases
• D. There is no diffraction

Answer: (C)

The situation hinges on how the grating equation links order, wavelength, and angle. For a diffraction grating, the condition for a bright fringe is mλ = d sin θ, where m is the order, λ is the wavelength, d is the spacing between grating lines, and θ is the angle of the bright fringe from the normal. With the wavelength and the grating spacing fixed, sin θ must increase as the order m increases. Since sin θ grows with θ in the range where the angle is physical (0 to 90 degrees), θ must increase as m increases. There is a real limit, though: sin θ cannot exceed 1, so only orders up to m such that mλ ≤ d appear; beyond that, no diffraction maximum exists. Within the allowed orders, higher m corresponds to larger angles.

The situation hinges on how the grating equation links order, wavelength, and angle. For a diffraction grating, the condition for a bright fringe is mλ = d sin θ, where m is the order, λ is the wavelength, d is the spacing between grating lines, and θ is the angle of the bright fringe from the normal.

With the wavelength and the grating spacing fixed, sin θ must increase as the order m increases. Since sin θ grows with θ in the range where the angle is physical (0 to 90 degrees), θ must increase as m increases. There is a real limit, though: sin θ cannot exceed 1, so only orders up to m such that mλ ≤ d appear; beyond that, no diffraction maximum exists. Within the allowed orders, higher m corresponds to larger angles.