Diffraction Grating
Condition for constructive interference maxima.
Formula
Variables
How it works
A diffraction grating has many equally spaced slits, so light from neighboring slits travels different path lengths to a point on a screen. Bright maxima occur only when that path difference equals an integer number of wavelengths, making the waves arrive in phase and reinforce each other strongly. The spacing between slits is d, so the path difference at angle theta is d\sin\theta, and setting this equal to m\lambda gives the allowed bright orders.
When to use it
Use this when finding the angles of bright spectral lines from a grating, or when identifying an unknown wavelength from a measured diffraction pattern. It is especially common in spectroscopy, laser experiments, and wavelength calibration.
Worked example, step by step
A diffraction grating has 600 lines per mm. A laser of wavelength 532 nm is shone onto it. Step 1: Find the slit spacing. 600 lines per mm means d = 1/600 mm = 1.67\u00d710^-3 mm = 1.67\u00d710^-6 m Step 2: Use the grating equation for the first order maximum, m = 1. d\sin\theta = m\lambda (1.67\u00d710^-6)\sin\theta = 1(532\u00d710^-9) Step 3: Solve for sin\theta. \sin\theta = (532\u00d710^-9)/(1.67\u00d710^-6) = 0.319 Step 4: Find the angle. \theta = sin^-1(0.319) \approx 18.6^\u00b0 Final answer: the first bright maximum occurs at \theta \approx 18.6^\u00b0 from the central axis.
Common mistakes to avoid
- Using the number of lines per mm directly as d instead of taking its reciprocal
- Forgetting that m is an integer order number starting at 0 for the central maximum
- Mixing units, such as leaving wavelength in nm while d is in meters
Did You Know?
A CD acts as a diffraction grating, splitting white light into rainbows across its shiny surface.
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