# 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.

#### Questions & Answers

state Faraday first law
aliyu Reply
what does the speedometer of a car measure ?
Jyoti Reply
Car speedometer measures the rate of change of distance per unit time.
Moses
describe how a Michelson interferometer can be used to measure the index of refraction of a gas (including air)
WILLIAM Reply
using the law of reflection explain how powder takes the shine off a person's nose. what is the name of the optical effect?
WILLIAM
is higher resolution of microscope using red or blue light?.explain
WILLIAM
can sound wave in air be polarized?
WILLIAM Reply
Unlike transverse waves such as electromagnetic waves, longitudinal waves such as sound waves cannot be polarized. ... Since sound waves vibrate along their direction of propagation, they cannot be polarized
Astronomy
A proton moves at 7.50×107m/s perpendicular to a magnetic field. The field causes the proton to travel in a circular path of radius 0.800 m. What is the field strength?
Celedonio Reply
derived dimenionsal formula
Ajak Reply
what is the difference between mass and weight
Isru Reply
assume that a boy was born when his father was eighteen years.if the boy is thirteen years old now, how is his father in
Isru
what is head-on collision
Javaid Reply
what is airflow
Godswill Reply
derivative of first differential equation
Haruna Reply
why static friction is greater than Kinetic friction
Ali Reply
draw magnetic field pattern for two wire carrying current in the same direction
Ven Reply
An American traveler in New Zealand carries a transformer to convert New Zealand’s standard 240 V to 120 V so that she can use some small appliances on her trip.
nkombo Reply
What is the ratio of turns in the primary and secondary coils of her transformer?
nkombo
what is energy
Yusuf
How electric lines and equipotential surface are mutually perpendicular?
Abid Reply
The potential difference between any two points on the surface is zero that implies È.Ŕ=0, Where R is the distance between two different points &E= Electric field intensity. From which we have cos þ =0, where þ is the angle between the directions of field and distance line, as E andR are zero. Thus
MAHADEV
sorry..E and R are non zero...
MAHADEV
By how much leeway (both percentage and mass) would you have in the selection of the mass of the object in the previous problem if you did not wish the new period to be greater than 2.01 s or less than 1.99 s?
Elene Reply
hello
Chichi
Hi
Matthew
hello
Sujan
Hi I'm Matthew, and the answer is Lee weighs in mass 0.008kg OR 0.009kg
Matthew
14 year old answers college physics and the crowd goes wild!
Matthew
Hlo
spread

### Read also:

#### Get Jobilize Job Search Mobile App in your pocket Now!

Source:  OpenStax, College physics. OpenStax CNX. Jul 27, 2015 Download for free at http://legacy.cnx.org/content/col11406/1.9
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'College physics' conversation and receive update notifications?

 By By By OpenStax By OpenStax By Jugnu Khan By Mucho Mizinduko By OpenStax By Mldelatte By OpenStax By OpenStax By Nicole Duquette By Jessica Collett