Monochromatic laser light is shown on a diffraction grating, and the pattern is

Question

Monochromatic laser light is shown on a diffraction grating, and the pattern is viewed on a screen 0.5 m away. While the setting is in the air, the 2^nd (second) bright fringes occur at plusminus 10.00 cm on either side of the central bright fringe. The entire apparatus is now immersed in a transparent liquid. When the experiment is repeated, the 2^nd (second) bright fringes now occur at plusminus 6.65 cm from the central bright fringe. (a) Compute the angles of the 2^nd (second) bright fringes (in the air and in the liquid.) (b) What is the index of refraction of the liquid? (c) Predict the location of the 3^rd (third) bright fringes (while the apparatus A in the liquid.) (d) Predict the angle angles theta(relative to the centerline coming outward from the midpoint between the slits) of the bright fringe that farthest from the central bright fringe (while the apparatus is in the liquid.) (e) Finally, you look up the specifications for the laser and find that its operating frequency is 5.20 times 10^14 Hz. What is the specification for this diffraction grating (that is the number of slits per mm)? (c = 3.00 times 10^8 m/s)

Explanation / Answer

part a:

in the air, angle=arctan(10 cm/0.5 m)=arctan(0.1/0.5)=11.31 degrees

in the lquid, angle=arctan(6.65 cm/0.5 m)=arctan(0.0665/0.5)=7.5759 degrees

part b:

distance of mth order bright fringe from central fringe=m*lambda*D/d

where m=order of the bright fringe
lambda=wavelength

D=distance of the screen

d=slit width

so as fringe distance is directly proportional to wavelength, if wavelength of the wave in air is lambda1 and in water is lambda2,

lambda1/lambda2=10/6.65

==>index of refraction=10/6.65=1.5038

part c:

while in liquid, using m=2, 2*lambda*D/d=6.65 cm

then for m=3, 3*lambda*D/d=(6.65/2)*3=9.975 cm

location of third order bright fringe is 9.975 cm

part d:

farthest will theoretically at an angle of 90 degrees

part e:

frequency=5.2*10^14 Hz

speed=3*10^8 m/s

then wavelength in air=speed/frequency=5.7692*10^(-7) m

then for the 2nd order bright fringe in air,

2*lambda*D/d=10 cm=0.1

==>2*5.7692*10^(-7)*0.5/d=0.1

==>d=5.7692*10^(-6) m

number of slits per mm=0.001 m/d

=173.33

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