# 27.4 Multiple slit diffraction  (Page 3/6)

 Page 3 / 6
$\text{sin}\phantom{\rule{0.25em}{0ex}}{\theta }_{\text{V}}=\frac{3\text{.}\text{80}×{\text{10}}^{-7}\phantom{\rule{0.25em}{0ex}}\text{m}}{1\text{.}\text{00}×{\text{10}}^{-6}\phantom{\rule{0.25em}{0ex}}\text{m}}=0\text{.}\text{380.}$

Thus the angle ${\theta }_{\text{V}}$ is

${\theta }_{\text{V}}={\text{sin}}^{-1}\phantom{\rule{0.25em}{0ex}}0\text{.}\text{380}=\text{22}\text{.}33º.$

Similarly,

$\text{sin}\phantom{\rule{0.25em}{0ex}}{\theta }_{\text{R}}=\frac{7\text{.}\text{60}×{\text{10}}^{-7}\phantom{\rule{0.25em}{0ex}}\text{m}}{1.00×{\text{10}}^{-6}\phantom{\rule{0.25em}{0ex}}\text{m}}.$

Thus the angle ${\theta }_{\text{R}}$ is

${\theta }_{\text{R}}={\text{sin}}^{-1}\phantom{\rule{0.25em}{0ex}}0\text{.}\text{760}=\text{49.46º.}$

Notice that in both equations, we reported the results of these intermediate calculations to four significant figures to use with the calculation in part (b).

Solution for (b)

The distances on the screen are labeled ${y}_{\text{V}}$ and ${y}_{\text{R}}$ in [link] . Noting that $\text{tan}\phantom{\rule{0.25em}{0ex}}\theta =y/x$ , we can solve for ${y}_{\text{V}}$ and ${y}_{\text{R}}$ . That is,

${y}_{\text{V}}=x\phantom{\rule{0.25em}{0ex}}\text{tan}\phantom{\rule{0.25em}{0ex}}{\theta }_{\text{V}}=\left(2.00 m\right)\left(\text{tan 22.33º}\right)=0.815 m$

and

${y}_{\text{R}}=x\phantom{\rule{0.25em}{0ex}}\text{tan}\phantom{\rule{0.25em}{0ex}}{\theta }_{\text{R}}=\left(\text{2.00 m}\right)\left(\text{tan 49.46º}\right)=\text{2.338 m.}$

The distance between them is therefore

${y}_{\text{R}}-{y}_{\text{V}}=1.52 m.$

Discussion

The large distance between the red and violet ends of the rainbow produced from the white light indicates the potential this diffraction grating has as a spectroscopic tool. The more it can spread out the wavelengths (greater dispersion), the more detail can be seen in a spectrum. This depends on the quality of the diffraction grating—it must be very precisely made in addition to having closely spaced lines.

## Section summary

• A diffraction grating is a large collection of evenly spaced parallel slits that produces an interference pattern similar to but sharper than that of a double slit.
• There is constructive interference for a diffraction grating when $d\phantom{\rule{0.25em}{0ex}}\text{sin}\phantom{\rule{0.25em}{0ex}}\theta =\mathrm{m\lambda }\phantom{\rule{0.25em}{0ex}}\text{(for}\phantom{\rule{0.25em}{0ex}}m=\text{0,}\phantom{\rule{0.25em}{0ex}}\text{1,}\phantom{\rule{0.25em}{0ex}}\text{–1,}\phantom{\rule{0.25em}{0ex}}\text{2,}\phantom{\rule{0.25em}{0ex}}\text{–2,}\phantom{\rule{0.25em}{0ex}}\dots \right)$ , where $d$ is the distance between slits in the grating, $\lambda$ is the wavelength of light, and $m$ is the order of the maximum.

## Conceptual questions

What is the advantage of a diffraction grating over a double slit in dispersing light into a spectrum?

What are the advantages of a diffraction grating over a prism in dispersing light for spectral analysis?

Can the lines in a diffraction grating be too close together to be useful as a spectroscopic tool for visible light? If so, what type of EM radiation would the grating be suitable for? Explain.

If a beam of white light passes through a diffraction grating with vertical lines, the light is dispersed into rainbow colors on the right and left. If a glass prism disperses white light to the right into a rainbow, how does the sequence of colors compare with that produced on the right by a diffraction grating?

Suppose pure-wavelength light falls on a diffraction grating. What happens to the interference pattern if the same light falls on a grating that has more lines per centimeter? What happens to the interference pattern if a longer-wavelength light falls on the same grating? Explain how these two effects are consistent in terms of the relationship of wavelength to the distance between slits.

Suppose a feather appears green but has no green pigment. Explain in terms of diffraction.

It is possible that there is no minimum in the interference pattern of a single slit. Explain why. Is the same true of double slits and diffraction gratings?

## Problems&Exercises

A diffraction grating has 2000 lines per centimeter. At what angle will the first-order maximum be for 520-nm-wavelength green light?

$5\text{.}\text{97º}$

Find the angle for the third-order maximum for 580-nm-wavelength yellow light falling on a diffraction grating having 1500 lines per centimeter.

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