# 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.

## Test prep for ap courses

Which of the following cannot be a possible outcome of passing white light through several evenly spaced parallel slits?

1. The central maximum will be white but the higher-order maxima will disperse into a rainbow of colors.
2. The central maximum and higher-order maxima will be of equal widths.
3. The lower wavelength components of light will have less diffraction compared to higher wavelength components for all maxima except the central one.
4. None of the above.

(b)

White light is passed through a diffraction grating to a screen some distance away. The n th-order diffraction angle for the longest wavelength (760 nm) is 53.13º. Find the n th-order diffraction angle for the shortest wavelength (380 nm). What will be the change in the two angles if the distance between the screen and the grating is doubled?

## 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.

how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
define variable velocity
displacement in easy way.
binding energy per nucleon
why God created humanity
Because HE needs someone to dominate the earth (Gen. 1:26)
Olorunfemi
Ali
Is the object in a conductor or an insulator? Justify your answer. whats the answer to this question? pls need help figure is given above
how do i calculate the pressure on the base of a deposit if the deposit is moving with a linear aceleration
why electromagnetic induction is not used in room heater ?
What is position?
What is law of gravition
what is magnetism
what is charging by induction
what is electric field lines
law of gravitation
Suppose a 0.250-kg ball is thrown at 15.0 m/s to a motionless person standing on ice who catches it with an outstretched arm as shown in [link] . (b) What is his angular velocity if each arm is 5.00 kg? You may treat the ball as a point mass and treat the person's arms as uniform rods (each has a length of 0.900 m) and the rest of his body as a uniform cylinder of radius 0.180 m. Neglect the effect of the ball on his center of mass so that his center of mass remains in his geometrical center.