Deck 25: Interference and Diffraction

Question 1

Two coherent waves interfere constructively. One wave has an intensity of $8.00 I_{0}$ and the other has an intensity of $4.00 I_{0}$ . What is the intensity of the superposed wave?

A) $23.3 I_{0}$
B) $8.00 I_{0}$
C) $12.0 I_{0}$
D) $4.00 I_{0}$
E) $24.0 I_{0}$

Answer 1

$23.3 I_{0}$

Question 2

If two coherent waves, one with intensity $8.0 I_{0}$ and the other with intensity $4.0 I_{0}$ , undergo destructive interference, what is the resulting intensity?

A) $4.0 I_{0}$
B) $0.69 I_{0}$
C) $-4.0 I_{0}$
D) $1.9 I_{0}$
E) $6.0 I_{0}$

Answer 2

$0.69 I_{0}$

Question 3

Wave 1 has intensity $9.0 I_{0}$ while wave 2 has intensity $4.0 I_{0}$ . What is their ratio of amplitudes, $A_{1} / A_{2}$ ?

A) 20
B) 1.3
C) 2.3
D) 1.5
E) 5.1

Answer 3

1.5

Question 4

Wave 1 has amplitude of $9.0 A_{0}$ and wave 2 has amplitude $4.0 A_{0}$ . What is the ratio of their intensities, $I_{1} / I_{2}$ ?

A) 5.1
B) 1.3
C) 2.3
D) 1.5
E) 20

Question 5

Using a Michelson interferometer and a light source of $400 \mathrm{~nm}$ , a change from bright to dark indicates a change in the difference of arm lengths of

A) $1.0 \times 10^{-4} \mathrm{~mm}$ .
B) $4.0 \times 10^{-4} \mathrm{~mm}$ .
C) $2.0 \times 10^{-4} \mathrm{~mm}$ .
D) $3.0 \times 10^{-4} \mathrm{~mm}$ .

Question 6

What minimum thickness of oil $(\mathrm{n}=1.50)$ on the surface of water $(\mathrm{n}=1.33)$ would give destructive interference for $550 \mathrm{~nm}$ light at normal incidence?

A) $78.0 \mathrm{~nm}$
B) $92.0 \mathrm{~nm}$
C) $80.0 \mathrm{~nm}$
D) $275 \mathrm{~nm}$
E) $183 \mathrm{~nm}$

Question 7

What minimum thickness of oil ( $\mathrm{n}=1.50)$ on the surface of water $(\mathrm{n}=1.33)$ would give constructive interference for $550 \mathrm{~nm}$ light at normal incidence?

A) $92.0 \mathrm{~nm}$
B) $183 \mathrm{~nm}$
C) $80.0 \mathrm{~nm}$
D) $275 \mathrm{~nm}$
E) $78.0 \mathrm{~nm}$

Question 8

If light of wavelength $690 \mathrm{~nm}$ is used in a double-slit experiment with slit separation $0.206 \mathrm{~mm}$ , what angle separates the two fourth-order maxima?

A) $0.768^{\circ}$
B) $3.07^{\circ}$
C) $0.154^{\circ}$
D) $0.307^{\circ}$
E) $1.54^{\circ}$

Question 9

Substituting $\tan \theta$ for $\sin \theta$ results in what percent error for an angle of $5.00^{\circ}$ ?

A) 5.00
B) 0.191
C) 0.0333
D) 0.0166
E) 0.382

Question 10

A grating is made with 600 slits per millimeter. What is the slit separation?

A) $916 \mathrm{~nm}$
B) $160 \mathrm{pm}$
C) $600 \mathrm{~nm}$
D) $3.33 \times 10^{-6} \mathrm{~m}$
E) $1.67 \times 10^{-6} \mathrm{~m}$

Question 11

A grating with 8000 slits over $2.54 \mathrm{~cm}$ is illuminated by light of a wavelength of $546 \mathrm{~nm}$ . What is the angle for the third-order maximum?

A) $10.5^{\circ}$
B) $15.1^{\circ}$
C) $31.1^{\circ}$
D) $26.3^{\circ}$

Question 12

What is the limiting angle of resolution, in degrees, for a microscope with an objective $0.50 \mathrm{~cm}$ in diameter when viewing with light of wavelength $590 \mathrm{~nm}$ ?

A) $2.1 \times 10^{-3}$
B) $4.1 \times 10^{-3}$
C) $7.2 \times 10^{-5}$
D) $1.4 \times 10^{-4}$
E) $8.2 \times 10^{-3}$

Question 13

If a laser printer prints at 600 dots per inch, what is the angular separation of the dots when viewed from 40 $\mathrm{cm}$ ?

A) $1.1 \times 10^{-4} \mathrm{rad}$
B) $2.7 \times 10^{-4} \mathrm{rad}$
C) $4.2 \times 10^{-5} \mathrm{rad}$
D) $3.7 \times 10^{-3} \mathrm{rad}$
E) $5.5 \times 10^{-8} \mathrm{rad}$

Question 14

What is the angular resolution of the eye at a wavelength of $500 \mathrm{~nm}$ for a pupil diameter of $2.8 \mathrm{~mm}$ ?

A) $3.3 \times 10^{-5} \mathrm{rad}$
B) $2.2 \times 10^{-4} \mathrm{rad}$
C) $4.5 \times 10^{-5} \mathrm{rad}$
D) $8.9 \times 10^{-5} \mathrm{rad}$
E) $1.8 \times 10^{-4} \mathrm{rad}$

Question 15

A Michelson interferometer is used to measure the index of refraction of a gas by counting interference fringes as the gas is slowly leaked into a transparent vessel placed in one arm of the interferometer. The vessel initially contains a vacuum, and as the gas is let in, 397 bright fringes are counted. That is, the central region of the screen changes from bright to dark and back to bright 397 times. If the wavelength of light used is $560 \mathrm{~nm}$ , and the vessel containing the gas is $25.0 \mathrm{~cm}$ long, what is the index of refraction of the gas?

A) 1.00178
B) 1.000111
C) 1.000889
D) 1.000445
E) 1.000222

Question 16

A Michelson interferometer can be used to measure very small distances by adjusting the length of one arm while keeping the other arm fixed. As the arm length is slowly changed, the central region of the screen changes from bright to dark and back to bright a number of times, and these bright fringes are counted. Suppose the interferometer uses light of wavelength $560 \mathrm{~nm}$ , and as one arm is lengthened by an amount $\Delta \mathrm{x}$ , exactly 79 bright fringes are counted. What is the distance $\Delta x$ ?

A) $44.2 \mu \mathrm{m}$
B) $22.1 \mu \mathrm{m}$
C) $7.09 \mathrm{~nm}$
D) $22.4 \mu \mathrm{m}$
E) $44.8 \mu \mathrm{m}$
F) $11.1 \mu \mathrm{m}$

Question 17

A Michelson interferometer is used to measure the index of refraction of a gas by counting interference fringes as the gas is slowly leaked into a transparent vessel placed in one arm of the interferometer. The vessel initially contains a vacuum, and as the gas is let in, the number of bright fringes is counted-i.e., the number of times the central region of the screen changes from bright to dark and back to bright. Suppose the index of refraction of the gas is 1.0003 . If the wavelength of light used is $560 \mathrm{~nm}$ , and the vessel containing the gas is $25 \mathrm{~cm}$ long, how many bright fringes will be observed?

A) $1.78 \times 10^{6}$
B) 133
C) $2.23 \times 10^{5}$
D) $4.47 \times10^{5}$
E) 134
F) 267

Question 18

White light is shone on a very thin layer of mica ( $\mathrm{n}=1.57)$ , suspended in air, at normal incidence. When the mica layer is observed from the illuminated side, constructive interference is observed for wavelengths of violet $(400 \mathrm{~nm})$ and yellow $(560 \mathrm{~nm})$ light. What is the thickness of the mica layer?

A) $446 \mathrm{~nm}$
B) $892 \mathrm{~nm}$
C) $700 \mathrm{~nm}$
D) $382 \mathrm{~nm}$
E) $764 \mathrm{~nm}$
F) $600 \mathrm{~nm}$

Question 19

Light from a HeNe laser (632.8 nm wavelength) is directed upon a straightened strand of human hair, $7.5 \mathrm{~m}$ from the wall. By examining the diffraction pattern produced on the wall, the diameter of the hair can be determined (the diffraction pattern thus produced has minima in the same locations as if the hair were a slit of the same width in an otherwise opaque barrier). If the central diffraction maximum is measured to be $14.6 \mathrm{~cm}$ wide, what is the diameter of the hair?

A) $65 \mu \mathrm{m}$
B) $86.6 \mu \mathrm{m}$
C) $43.3 \mu \mathrm{m}$
D) $32.5 \mu \mathrm{m}$

Question 20

Two planes of atoms in a crystal are separated by $3.25 \times 10-10 \mathrm{~m}$ . Light of what wavelength would show constructive interference at an angle of 17.5 degrees relative to the crystal surface?

A) $1.95 \times 10-10 \mathrm{~m}$
B) $9.75 \times 10^{-10} \mathrm{~m}$
C) $1.46 \times 10^{-10} \mathrm{~m}$
D) $3.9 \times 10-10 \mathrm{~m}$
E) $4.9 \times 10^{-9} \mathrm{~m}$

Question 21

When scattering $x$ -rays with a wavelength of $3.28 \times 10-10 \mathrm{~m}$ from a crystal, the first interference maximum occurs at an angle of 23.0 degrees relative to the crystal surface. What is the crystal spacing?

A) $16.8 \times 10^{-10} \mathrm{~m}$
B) $6.3 \times 10-10 \mathrm{~m}$
C) $8.4 \times 10^{-10} \mathrm{~m}$
D) $4.2 \times 10^{-10} \mathrm{~m}$