Derivation of dsinθ = nλ Use of the spectrometer will not be tested.

The idea

The grating equation d sinθ = nλ links the spacing d between adjacent slits, the diffraction angle θ and the wavelength λ for the nth order maximum. Derive it by considering two adjacent slits: the extra distance travelled by light from the second slit is d sinθ. For constructive interference this path difference must equal a whole number of wavelengths, nλ, giving d sinθ = nλ. The equation shows that larger λ or smaller d spreads the pattern more, and that n counts the order. Although you must be able to derive and use the equation, practical work with a spectrometer is not examined, so focus on the geometry, the meaning of each symbol and calculations involving d, θ, n and λ.

What earns the marks

• State that the path difference between light from two adjacent slits is d sinθ, where d is the slit spacing and θ is the angle from the straight-through direction.
• Explain that constructive interference occurs when the path difference equals a whole number of wavelengths, so d sinθ = nλ.
• Identify n as the order of the maximum, with n = 0 for the central maximum and n = 1, 2, 3 … for successive maxima.
• Use the equation to calculate d, θ, n or λ, converting angles to radians where required and using sinθ correctly.
• Recognise that the derivation applies to any pair of adjacent slits and that all such pairs reinforce at the same angle.
• Explain how the equation predicts that decreasing d or increasing λ increases the diffraction angle for a given order.

Common mistakes and how to put them right

• Using the angle from the grating surface instead of the angle from the normal.

  Put it right by measuring θ between the diffracted ray and the straight-through direction.

• Treating d as the number of lines per metre rather than the slit spacing.

  Put it right by using d = 1 / N, where N is the number of lines per metre.

• Forgetting that n must be an integer for a maximum.

  Put it right by checking that the calculated order is a whole number and rejecting non-integer values.

Exam tips

• Sketch two adjacent slits and mark the path difference d sinθ before writing the equation; this makes the derivation clear and earns method credit.
• When calculating, keep d and λ in the same unit, usually metres, and give the angle to an appropriate number of significant figures.
• If asked to derive the equation, state the condition for constructive interference explicitly rather than just quoting d sinθ = nλ.