1. Diffraction angles
θ(400 nm) = 28.69°; θ(700 nm) = 57.14°.
| Criterion | Points |
|---|---|
| Correct application of the grating equation | 10 |
| Correct both angles | 10 |
Worked Solutions
2. Angular dispersion
Average dispersion ≈ (57.14-28.69)°/(700-400) nm = 0.0948 °/nm, consistent in order of magnitude with the exact derivative m/(d cosθ_m) evaluated near 550 nm.
| Criterion | Points |
|---|---|
| Correct average-dispersion arithmetic | 8 |
| Correct qualitative consistency check | 7 |
3. Resolving power requirement
R = λ/Δλ = 700/0.01 = 70000.
| Criterion | Points |
|---|---|
| Correct formula application | 8 |
| Correct final value | 7 |
4. Illuminated groove count
N_required = R/m = 70000. A 10 cm grating at 1200 grooves/mm has N_available = 120000 illuminated grooves -- more than enough to meet the requirement.
| Criterion | Points |
|---|---|
| Correct N_required arithmetic | 12 |
| Correct comparison and conclusion | 8 |
Grading Notes
Award full marks for correct method, units, and a numeric result consistent with the arithmetic shown (allow reasonable rounding). Verify that the student re-derives the number for this problem's specific inputs rather than quoting a lecture example without adaptation. Watch specifically for unit-conversion errors (arcsec/radian, nm/m, mm/m) and for comparative-magnitude or comparative-resolution statements given in the wrong direction (e.g., "finer" versus "coarser" resolution, or "more" versus "less" light-gathering power) -- these are the most common sources of order-of-magnitude mistakes in this course.