A diffraction grating with N = 500 slits produces principal maxima. A second grating has N = 2000 slits with the same slit spacing d. Compared to the first grating, the second grating's principal maxima are:
AAt four times the angular separation from each other
BAt the same angular positions but four times sharper
CFour times as bright but at slightly different angles
DFour times as bright and equally sharp
The grating equation d sin θ = mλ determines WHERE maxima appear — it depends on slit spacing d, not the number of slits N. More slits (same d) do not shift positions. What changes is the sharpness: with more coherent sources, any slight angular deviation causes destructive interference to accumulate rapidly, making the peaks dramatically narrower. This is the key distinction from double-slit: the equation is the same, but N controls resolution, not position.
Question 2 Multiple Choice
A physicist needs to resolve two spectral lines near λ = 500 nm separated by Δλ = 0.05 nm. Using diffraction order m = 1, what minimum number of illuminated slits N is required?
A100 slits — the grating equation provides enough angular separation at this order
B500 slits — resolving power scales with the slit spacing, not N
C10,000 slits — resolving power R = mN must equal λ/Δλ = 10,000
DN does not matter — only slit spacing d determines resolution
Resolving power R = mN must satisfy R ≥ λ/Δλ = 500/0.05 = 10,000. With m = 1, we need N ≥ 10,000. The common misconception is thinking that angular separation (set by d) is what determines resolution — but two wavelengths can have distinct angular positions and still be unresolvable if the peaks are too broad. It is the sharpness of the peaks, controlled by N, that determines whether nearby maxima can be distinguished.
Question 3 True / False
A diffraction grating with more slits per millimeter produces its principal maxima at larger angles than a coarser grating, for the same wavelength and diffraction order.
TTrue
FFalse
Answer: True
True. More slits per millimeter means smaller slit spacing d. From the grating equation d sin θ = mλ, smaller d requires larger sin θ — and thus larger θ — for the same m and λ. This is how finely ruled gratings achieve greater angular dispersion: the maxima are spread over a wider angular range, making spectral lines easier to separate spatially.
Question 4 True / False
A diffraction grating and a glass prism both separate white light into its component colors through the same underlying physical mechanism.
TTrue
FFalse
Answer: False
False. A prism separates colors through dispersion: its refractive index varies with wavelength, so different colors bend by different amounts upon entering and exiting the glass. A grating separates colors through interference: the constructive interference condition d sin θ = mλ makes different wavelengths satisfy the condition at different angles. The physics is entirely different — and gratings are preferred for precision spectroscopy because their angular dispersion is linear in λ and scales predictably with N, unlike the nonlinear dispersion of prisms.
Question 5 Short Answer
Why does increasing the number of slits in a diffraction grating improve its ability to resolve closely spaced spectral lines, even though the angular positions of the principal maxima are unchanged?
Think about your answer, then reveal below.
Model answer: More slits make the principal maxima sharper (narrower in angle). With more coherent sources all contributing, any slight deviation from the exact constructive-interference angle causes many slits to partially cancel each other, dropping the combined amplitude steeply toward zero. Sharper peaks mean two nearby wavelengths have less overlapping intensity — they can be distinguished as separate maxima. The resolving power R = mN captures this directly: more slits means narrower peaks means finer wavelength discrimination.
The grating equation sets the position of each maximum; the number of slits sets its width. The first minimum adjacent to a principal maximum is displaced by λ/(Nd cos θ) in angle — narrower for larger N. Two wavelengths are just resolvable when one's maximum coincides with the other's first minimum (the Rayleigh criterion). The more slits illuminated, the narrower each peak, and the closer together two wavelengths can be and still meet this criterion. This is why scientific spectrographs are built to illuminate as many grating lines as possible.