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Chapter 33: Q. 13 (page 955)

The two most prominent wavelengths in the light emitted by a hydrogen discharge lamp are 656nm (red) and 486nm (blue). Light from a hydrogen lamp illuminates a diffraction grating with 500lines/mm, and the light is observed on a screen 1.50m behind the grating. What is the distance between the first-order red and blue fringes?

Short Answer

Expert verified

The distance between red and blue fringes is14.5cm.

Step by step solution

01

Formula for distance

The center of them-th fringe is given by,

Ym=Ltanθm

=Ltansin-1mλd

localid="1649218383524" dis the distance between gratings

localid="1649218434130" λis wavelength

localid="1649218442071" θmis diffraction angle

02

Calculation for distance

The grating constant is500lines/mm.

There will be a space between two succeeding gratings.

d=1×10-3500

=2×10-6m¯

So,

ΔY=Ltanθ1,red-Ltanθ1,blue

=Ltansin-1λredd-tansin-1λblued

In terms of numbers,

ΔY=1.5tansin-10.6562-tansin-10.4862

=14.5cm

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Most popular questions from this chapter

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 10kHzsound waves pass through such a grating, listeners 10mfrom the grating report "loud spots" 1.4mon both sides of center. What is the spacing between the slits? Use 340msfor the speed of sound.

FIGURE P33.56 shows the light intensity on a screen behind a circular aperture. The wavelength of the light is 500nmand the screen is 1.0mbehind the slit. What is the diameter (in mm) of the aperture?

Your artist friend is designing an exhibit inspired by circular-aperture diffraction. A pinhole in a red zone is going to be illuminated with a red laser beam of wavelength 670nm, while a pinhole in a violet zone is going to be illuminated with a violet laser beam of wavelength 410nm. She wants all the diffraction patterns seen on a distant screen to have the same size. For this to work, what must be the ratio of the red pinhole’s diameter to that of the violet pinhole?

A diffraction grating has slit spacing d. Fringes are viewed on a screen at distance L. Find an expression for the wavelength of light that produces a first-order fringe on the viewing screen at distanceLfrom the center of the screen.

FIGURE shows two nearly overlapped intensity peaks of the sort you might produce with a diffraction grating . As a practical matter, two peaks can just barely be resolved if their spacing ∆yequals the width w of each peak, where wis measured at half of the peak’s height. Two peaks closer together than wwill merge into a single peak. We can use this idea to understand the resolution of a diffraction grating.

a. In the small-angle approximation, the position of the m=1peak of a diffraction grating falls at the same location as the m=1fringe of a double slit: y1=λL/d. Suppose two wavelengths differing by ∆lpass through a grating at the same time. Find an expression for localid="1649086237242" ∆y, the separation of their first-order peaks.

b. We noted that the widths of the bright fringes are proportional to localid="1649086301255" 1/N, where localid="1649086311478" Nis the number of slits in the grating. Let’s hypothesize that the fringe width is localid="1649086321711" w=y1/NShow that this is true for the double-slit pattern. We’ll then assume it to be true as localid="1649086339026" Nincreases.

c. Use your results from parts a and b together with the idea that localid="1649086329574" Δymin=wto find an expression for localid="1649086347645" Δλmin, the minimum wavelength separation (in first order) for which the diffraction fringes can barely be resolved.

d. Ordinary hydrogen atoms emit red light with a wavelength of localid="1649086355936" 656.45nm.In deuterium, which is a “heavy” isotope of hydrogen, the wavelength is localid="1649086363764" 656.27nm.What is the minimum number of slits in a diffraction grating that can barely resolve these two wavelengths in the first-order diffraction pattern?
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