27.4 Multiple slit diffraction  (Page 3/6)

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

Similarly,

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

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 $\text{tan}\theta =y/x$ , we can solve for $y_{\text{V}}$ and $y_{\text{R}}$ . That is,

and

The distance between them is therefore

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 $d\text{sin}\theta =\mathrm{m\lambda }\text{(for}m=\text{0,}\text{1,}\text{–1,}\text{2,}\text{–2,}\dots )$ , 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

Problems&Exercises