A light source emits two major spectral lines: an orange...

A light source emits two major spectral lines: an orange line of wavelength $610 \mathrm{~nm}$ and a blue-green line of wavelength $480 \mathrm{~nm}$. If the spectrum is resolved by a diffraction grating having 5000 lines $/ \mathrm{cm}$ and viewed on a screen $2.00 \mathrm{~m}$ from the grating, what is the distance (in centimeters) between the two spectral lines in the second-order spectrum?

Key Concepts

Angular Dispersion and Screen Projection

Angular dispersion describes how the angle at which light is diffracted depends on its wavelength, and it is key to achieving spectral separation. When these diffracted beams are projected onto a screen, simple geometry (using the tangent of the angle or small-angle approximations) converts these angular differences into linear distances, allowing the physical measurement of the separation between spectral lines.

Diffraction Order

The diffraction order (m) represents the sequence of intensity maxima produced by a grating. Higher order spectra (such as second order when m = 2) correspond to angles where the diffracted waves interfere constructively for longer optical paths, leading to enhanced separation between wavelengths.

Grating Equation

The grating equation, d sin(?) = m?, relates the slit separation (d), the order of the diffraction (m), the diffraction angle (?), and the wavelength of the light (?). It is crucial for predicting the angular position of spectral lines in the output pattern from the grating.

Diffraction Grating

A diffraction grating consists of a large number of equally spaced slits or lines that disperse light into its component wavelengths. The interference of light diffracted from these multiple slits produces sharp principal maxima at specific angles and is fundamental to understanding spectral resolution in optics.

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