Click here to return to Phyllis Fleming's homepage Phyllis Fleming Physics

Physics 106

Review - Physical Optics

     *Optional Problems


The circles in Fig. 1 below represent cross-sections of spherical wave fronts emanating from a source S. All points on a wave front are in phase. (a) What do the concentric circles pass through for each of the sinusoidal waves shown in Fig. 1? Find the distances (b) SP, (c) SP’, and (d) SP” in terms of wavelength λ. (e) If you wished to draw wave fronts through troughs in Fig. 1, where and how would you draw them?


In Fig. 2 below, two sources S1 and S2 emit waves.  The two waves overlap at P.  Find (a) the length of the path from S1 to P, S1P,  (b) the length of the path from S2 to P, S2P, and (c) the path difference (S1P - S2P) in terms of the wavelength λ.  (d) Will the disturbances of the two waves at P produce constructive or destructive interference? Explain your answer.


Write a mathematical equation that shows the relation among the quantities: frequency, wavelength, and velocity of a wave?


A person is at equal distances from two speakers of a stereo hi-fi system and hears a note of single frequency equal to 275 Hz (275 s-1). He moves sideways until he hears the note fade to a minimum.  At this position he is 10 feet from the left speaker and 8 feet from the right speaker. Find the speed of sound.


Figure 3 below shows two point sources, S1 and S2 emitting waves that are detected at a distant point P from the two sources. When the distances r1 and r2 from sources 1 and 2 to the point P are large compared with the separation d of the sources, the two rays along the lines of sight from the two sources to point P are nearly parallel, both being essentially at the same angle Θ from the X axis as shown in the figure. Since I am unable to draw such large distances and at the same time show the path difference, S1P - S2P = S1P, as a reasonable length, I have made a "break" in the drawing as the two "parallel" rays come together and arrive at P.

(a) If point P is one of maximum intensity (constructive interference),  find S1P in terms of the wavelength λ.

(b) Use triangle BS1S2 to find an expression in terms of the separation d of the sources and a function of Θ that you can equate to the path difference found in (a).

(c) Repeat (b) for the case in which P is a point of minimum intensity
    (destructive interference).


Let us take the frequency f of the two sources in Fig. 3 above to be the same and say the waves travel in the same medium so the wave produced by S1 has the same wavelength λ as the wave produced by S2. The equation of the wave produced by S1 is y1(r1,t) = A1 sin (2 πvt - 2 πr1/ λ) and the equation of the wave produced by S2 is y2(r2,t) = A2 sin (2πνt - 2 πr2/ λ).  At P in Fig. 3, the phase difference between the two waves is ΔΦ = (2 π/ λ)(r1 - r2), where r1 is greater than r2.

In Fig. 4 below we take the amplitudes of the waves A1 = A2 = Ao and draw "phasor diagrams." While amplitudes are not vectors, you can use the phasors to find the resultant amplitude.

In Fig. 4b, (a) find the resultant amplitude in terms of Ao, and (b) the values of ΔΦ in radians that would produce this amplitude.

In Fig. 4c, (c) find the resultant amplitude in terms of Ao, and (d) the values of ΔΦ in radians that would produce this amplitude.


You probably found that the resultant amplitude in Fig. 4b was 2Ao and
ΔΦ = 2m π where m = 0, 1, 2, 3, 4, . .

(a) Since it is always true that ΔΦ = (2 π/ λ)(r1 - r2), use your result to find the path difference (r1 - r2).

(b) Now use your results of Problem 6 part d to find the path difference when point P in Fig. 3 above is one of destructive interference.


  In figure 5 above, you found the path difference (r1 - r2) = d sin Θ = m λ,
  where m = 0, 1, 2, 3, . . . . for constructive interference.

(a) In Fig. 5 below, point P is a point on the screen for the mth maximum. The distance from the center of the screen to P is ym.  L is the distance of the slits to the screen. For cases in which ym is much smaller than L,  find ym in terms of m,  λ,  and d.

(b) Find the distances between maxima or Δy = ym+1 - ym.

(c) Repeat (a) and (b) when point P is a point on the screen for the mth minimum.


(a) Use Fig. 4a above to show that in general the square of the resultant amplitude A2 = 4Ao2 cos2 ΔΦ/2.  Hint: Find the components of A1 along the X and Y-axes and then the X and Y components of the resultant A of A1 and A2. Use the trigonometric identity cos2 ΔΦ/2 = 1/2 + 1/2 cos ΔΦ.

(b) The intensity I is proportional to the square of the amplitude. The resultant intensity I = 4Io cos2 ΔΦ/2, where Io is the intensity due to one source and ΔΦ/2 = πd sin Θ/ λ.  Sketch I as a function of d sin Θ.


Three radio antennas arranged as shown in Fig. 6, emit waves of wavelength λ =180 m, all in phase, with the same amplitude Ao and intensity Io. Find the intensity of the resultant waves from the three antennas at a point far from the antennas for the directions (a), (b), and (c) shown in Fig. 6 below.


A beam of unpolarized light with an irradiance of 1000 W/m2 hits a linear polarizer whose transmission axis is vertical. The light then passes through a second linear polarizer with an orientation of 60o with the vertical. What is the irradiance of the light passed by the second polaroid?


Parallel rays of light of wavelength 655 nm fall upon a pair of slits that are 1.57 x 10-5 m apart. The interference pattern is focused on a screen 2.00 m from the slits. Find (a) the angle for the third maximum and (b) the distance of this maximum from the center of the screen.


What is the relationship between the wavelength λin a vacuum or air and λn the wavelength of light in a medium of index of refraction n?


Discuss the difference for reflected pulses on a string when the pulse goes from (a) a less dense string to a more dense string and (b) a more dense string to a less dense string. The speed of the pulse is greater in a less dense string than it is in a more dense string.


Light of wavelength λ = 500 nm is incident upon a double-slit arrangement in which the distance between the slits d = 1.00 x 10-3 m. On a screen a distance L = 1.00 m from the slits, a bright line is observed at ym = 1.50 mm from the center of the screen. What bright line is this, or in other words, what is the value of m?


In a double-slit interference pattern, the distance between positions of maximum light intensity along a distant screen is Δy when the distance between the slits is d and the wavelength of the incident light is λ. If the distance between the slits is doubled and the wavelength is halved, find the new distance between the maxima Δy’ in terms of Δy.


An interference pattern for light incident upon two slits is observed first in air and then under water. For which case will the distance between maxima be greater? Explain your answer.


A very small source of white light is placed at point A, above a mirror. Light can travel from A to B by two paths, as shown in Fig. 7 below (not to scale). Determine at least two wavelengths that will not be present at point B. There is a 180o phase shift on reflection at a mirror.


An oil drop (n = 1.20) floating on water (n = 1.33) is illuminated by yellow light ( λo = 580 nm) and is observed from above by reflected light.

(a) Will the outer (thinnest) regions of the drop appear bright or dark?

(b) Find the thickness of the film for several regions where the yellow light is observed.


A soap film with n = 1.34 immersed in air is illuminated by yellow light
( λo = 580 nm).  Find the thickness of the film for (a) a maximum and (b) a minimum for m = 2.


You are listening to a hi-fi instrument that is in another room, and note that the lower-frequency notes are accentuated. Explain the reason for this.


Why, in our everyday experience, is the wave nature of sound more obvious than the wave nature of light?


Figure 8 below is a graph of the intensity of light of wavelength λ as a function of the distance y along a screen for a single-slit diffraction pattern with slit width a.  Sketch a graph of intensity as a function of the distance along the screen for (a) light of wavelength λ/2 and slit width a and (b) light of wavelength λand slit width a/2.


A single slit 0.10 mm wide which is 2.0 m from the screen is illuminated by light of wavelength 580 nm.  Find the width of the central maximum.


What is the minimum distance between two points that will permit them to be resolved at 1.00 km using a terrestrial telescope with a 6.5 cm - diameter objective?  Take λ= 550 nm.


The wavelengths of the hydrogen alpha line and the hydrogen beta line are 653.4 nm and 580.8 nm, respectively.  If you are using a grating with 2.00 x 105 lines per meter, what is the angular separation for these two lines in the first and third order?


Light of wavelength λ= 500 nm from sources S1 and S2 meet at point P in Fig. 9 below. The path for light from source S1 is totally in air, but the light from source S2 passes through a medium of index of refraction n2 = 1.5 and length L = 0.50 µm. (a) Find the phase difference when the light beams meet at P.  (b) Is there constructive or destructive interference at point P? (c) What length L' of the medium with index of refraction n2 will give a phase difference of 2 π?


What angle is needed between the direction of polarized light and the axis of a polarizing filter to cut its intensity in half?


Show that if you have three polarizing filters, with the second at an angle of 45o with the first and the third at an angle of 90o with the first, the intensity of the light is reduced to 25% of its value.


At what angle will light incident from air and reflected from diamond be completely polarized? The index of refraction of diamond is 2.42.


Light reflected at 62.5o from a gemstone in a ring is completely polarized. Can the gem be a diamond?

Homepage Sitemap

Website Designed By:
Questions, Comments To:
Date Created:
Date Last Updated:


Susan D. Kunk
Phyllis J. Fleming
August 8, 2002
April 25, 2003