Convert line density into slit spacing and calculate angular orders.
Method and worked examples
A transmission grating has many closely spaced slits. Diffracted waves from adjacent slits reinforce at dsinθ=mλ. Convert line density to spacing using d=n1 when n is in lines per metre. The zero-order maximum lies straight ahead. The positive and negative orders occur symmetrically for normal incidence; they are not separate wavelengths.
Because ∣sinθ∣ cannot exceed one, an order exists only if ∣m∣λ≤d. Count zero plus both sides when asked for the total number of maxima. With white light, the zero order is white, while nonzero orders separate colours: longer wavelengths appear at larger angles for a fixed order. Gratings give narrow maxima that help distinguish closely spaced spectral lines, though different orders can overlap.
Example 1
A grating has 500 lines per mm. Find slit spacing.
n=500×1000=5.00×105 lines m−1. d=n1=2.00×10−6m.
Example 2
For d=2.00μm and λ=500 nm, find the second-order angle.
sinθ=dmλ=2×2000500=0.500, so θ=30.0∘.
Watch out:Spacing is the reciprocal of line density after converting the density to consistent units.
4. Diffraction gratings and spectra | Year 12 Physics | Ascent Arcade