Double-slit fringe separation λ = ax/D
● VERBATIM — examined word-for-wordFor double-slit interference, λ = ax/D, where a is the slit separation, x is the fringe separation (the distance between the centres of adjacent bright fringes) and D is the slit-to-screen distance. It is valid only when D ≫ a so the rays are effectively parallel, and a ≫ λ so the angles are small and sin θ ≈ tan θ.
λ = ax/D ⇔ x = λD/a · nth maximum at xₙ = nλD/aWhat this actually means
Know what each symbol means, because a and x are the two most swapped letters in the whole syllabus. a is between the slits (small, sub-millimetre); x is between the fringes on the screen (millimetres).
The derivation is three lines. Path difference is a sin θ, constructive needs a sin θ = nλ, and for small angles sin θ ≈ tan θ = xₙ/D. Rearranging gives xₙ = nλD/a, and subtracting successive terms gives x = λD/a.
Never measure just one fringe gap. Measure across ten fringes and divide by ten, or measure from the first to the fifth bright fringe and divide by four. Note it is four gaps, not five, between the first and fifth fringe.
The dependencies are worth reciting: fringes get wider if you increase λ or D, or decrease a. Blue light gives narrower fringes than red for the same setup.
The approximations really do fail at large angles. Far from the centre the fringes get slightly wider than λD/a predicts, which is exactly why this formula must not be reused on a grating.
Swapping a and x, or dividing by the number of fringes instead of the number of gaps between them.
Prove it — watch it be true
- Set a slit separation and screen distance, then measure the fringe spacing on the screen intensity profile.
- Compare the measured value with the λD/a readout and confirm they agree.
- Double D and confirm the measured spacing doubles, then halve a and confirm it doubles again.