Chapter 33: Wave Optics

A triple-slit experiment consists of three narrow slits, equally spaced by distance $d$and illuminated by light of wavelength $\lambda$. Each slit alone produces intensity $I_1$on the viewing screen at distance$L$.
$\left(a\right)$Consider a point on the distant viewing screen such that the path-length difference between any two adjacent slits is$\lambda$. What is the intensity at this point?
$\left(b\right)$What is the intensity at a point where the path-length difference between any two adjacent slits is$\raisebox{1ex}{$\lambda $}\!\left/ \!\raisebox{-1ex}{$2$}\right.$?

Because sound is a wave, it's possible to make a diffraction grating for sound from a large board of sound-absorbing material with several parallel slits cut for sound to go through. When $10kHz$sound waves pass through such a grating, listeners $10m$from the grating report "loud spots" $1.4m$on both sides of center. What is the spacing between the slits? Use $340\raisebox{1ex}{$m$}\!\left/ \!\raisebox{-1ex}{$s$}\right.$for the speed of sound.

A diffraction grating with $600\raisebox{1ex}{$lines$}\!\left/ \!\raisebox{-1ex}{$mm$}\right.$is illuminated with light of wavelength $510nm$. A very wide viewing screen is$2.0m$ behind the grating.
$\left(a\right)$What is the distance between the two$m=1$ bright fringes?
$\left(b\right)$How many bright fringes can be seen on the screen?

A $500line/mm$diffraction grating is illuminated by light of wavelength $510nm$.How many bright fringes are seen on a $2.0-m$-wide screen located $2.0m$behind the grating?

White light$\left(400-700nm\right)$incident on a $600line/mm$diffraction grating produces rainbows of diffracted light. What is the width of the first-order rainbow on a screen $2.0m$behind the grating?

A chemist identifies compounds by identifying bright lines in their spectra. She does so by heating the compounds until they glow, sending the light through a diffraction grating, and measuring the positions of first-order spectral lines on a detector $15.0cm$behind the grating. Unfortunately, she has lost the card that gives the specifications of the grating. Fortunately, she has a known compound that she can use to calibrate the grating. She heats the known compound, which emits light at a wavelength of $461nm$, and observes a spectral line $9.95cm$from the center of the diffraction pattern. What are the wavelengths emitted by compounds $A$and $B$that have spectral lines detected at positions $8.55cm$and $12.15cm$, respectively?

$\left(a\right)$Find an expression for the positions $y_1$of the first-order fringes of a diffraction grating if the line spacing is large enough for the small-angle approximation $\tan \theta \approx \sin \theta \approx \theta$to be valid. Your expression should be in terms of $d,L$and$\lambda$.
$\left(b\right)$. Use your expression from part a to find an expression for the separation$∆y$on the screen of two fringes that differ in wavelength by$∆\lambda$.
$\left(c\right)$Rather than a viewing screen, modern spectrometers use detectors-similar to the one in your digital camera-that are divided into pixels. Consider a spectrometer with a $333lines/mm$grating and a detector with $100pixels/mm$located $12cm$behind the grating. The resolution of a spectrometer is the smallest wavelength separation $∆{\lambda }_{min}$that can be measured reliably. What is the resolution of this spectrometer for wavelengths near localid="1649156925210" $550nm$, in the center of the visible spectrum? You can assume that the fringe due to one specific wavelength is narrow enough to illuminate only one column of pixels.

For your science fair project you need to design a diffraction grating that will disperse the visible spectrum $\left(400-700nm\right)$over$30.0^{\circ }$ in first order.
$\left(a\right)$ How many lines per millimeter does your grating need?
$\left(b\right)$What is the first-order diffraction angle of light from a sodium lamp $\left(\lambda =589nm\right)$?

$FIGUREP33.49$shows the interference pattern on a screen $1.0m$behind an $800line/mm$diffraction grating. What is the wavelength (in $mm$) of the light?

FIGURE shows the light intensity on a viewing screen behind a single slit of width $a$. The light’s wavelength is $\lambda$. Is $\lambda <a$,$\lambda =a$, $\lambda >a$, oris it not possible to tell? Explain.