Double Slits & Diffraction Gratings
- Apply the double-slit condition for bright and dark fringes
- Compute fringe spacing on a distant screen
- Explain why a grating produces sharper maxima than two slits
Path difference decides everything
Light from two slits arrives at a point on the screen having traveled slightly different distances. When that path difference is a whole number of wavelengths the waves arrive in phase and reinforce — a bright fringe. When it is a half-integer number they arrive out of phase and cancel — a dark fringe. Every interference formula in the unit is a statement about path difference.
What makes the pattern spread out
Fringe spacing is Δy = λL/d. So the fringes spread further apart with a longer wavelength, a more distant screen, or closer slits. That last one is counterintuitive and worth holding onto: bringing the slits together spreads the pattern out. It also means a measured fringe spacing determines the wavelength, which is how Young's experiment established that light is a wave and measured its wavelength without any modern equipment.
A grating is many slits, and it is sharper
A diffraction grating has thousands of slits per millimeter. The condition for a maximum is the same d sin θ = mλ, but with many slits contributing, the maxima are far narrower and brighter while everything between them cancels almost completely. That sharpness is why gratings, not double slits, are used for actual spectroscopy: they resolve wavelengths that a two-slit pattern would blur together. Note that gratings are usually specified in lines per millimeter, so d is the reciprocal of that.
Single slit is different
A single slit of width a produces a diffraction pattern whose dark fringes satisfy a sin θ = mλ for m = 1, 2, 3 — note that this is the condition for minima, the reverse of the double-slit convention, with a central bright maximum that is twice as wide as the others. Mixing up which condition gives bright and which gives dark between the single-slit and double-slit cases is a reliable source of lost marks.
Light of wavelength 550 nm falls on two slits 0.080 mm apart, with a screen 2.5 m away. Find the fringe spacing and the position of the third-order bright fringe.
- 1.Fringe spacing: Δy = λL/d = (550 × 10⁻⁹)(2.5)/(0.080 × 10⁻³).
- 2.= (1.375 × 10⁻⁶)/(8.0 × 10⁻⁵) = 1.72 × 10⁻² m, or 17 mm.
- 3.Third order means m = 3: y₃ = 3 × 17 mm = 52 mm from the center.
- 4.Check the small-angle assumption: sin θ ≈ 0.052/2.5 = 0.021, so θ ≈ 1.2°. Small angles are safe here.
Convert units before substituting. Nanometers, millimeters and meters appear in the same problem, and mixing them produces answers wrong by factors of a thousand. Putting everything in meters first costs nothing.
In a double-slit experiment, the slit separation is halved. The fringe spacing on the screen:
At a point on the screen where the path difference is exactly 2.5 wavelengths, the result is:
A diffraction grating produces sharper maxima than a double slit because:
Check whether the question is about a single slit or a double slit before writing the condition. The same equation d sin θ = mλ gives bright fringes for two slits and dark fringes for one, and using the wrong one inverts the entire pattern.
Answer the 3 checkpoints as you read.
Sign in to save your progress