gen-define-the-diffraction-grating-equation-and-give-one-short-w-1b08

Define the diffraction grating equation and give one short worked example.

# Diffraction Grating Equation ## Definition The **diffraction grating equation** describes the condition for constructive interference when light passes through a grating: $$d \sin \theta = n\lambda$$ Where: - **d** = distance between adjacent grating lines (grating spacing) - **θ** = angle of the diffracted beam from the normal - **n** = order of diffraction (0, ±1, ±2, ±3, ...) - **λ** = wavelength of light --- ## Worked Example **Problem:** A diffraction grating has 500 lines per mm. When red light (λ = 650 nm) shines on it, at what angle does the first-order maximum appear? **Solution:** First, find the grating spacing: $$d = \frac{1}{500 \text{ lines/mm}} = \frac{1}{500,000 \text{ lines/m}} = 2.0 \times 10^{-6} \text{ m}$$ Apply the grating equation with n = 1: $$d \sin \theta = n\lambda$$ $$\sin \theta = \frac{n\lambda}{d} = \frac{(1)(650 \times 10^{-9})}{2.0 \times 10^{-6}}$$ $$\sin \theta = 0.325$$ $$\theta = \sin^{-1}(0.325) = 19.0°$$ **Answer:** The first-order maximum appears at **19.0°** from the normal.