1. Interference of Waves Beats Double Slit
Physics 1B03summer-Lecture 10

2. Physics 1B03summer-Lecture 10
Beats
Two waves of different frequencies arriving together produce a fluctuation in power or amplitude.
Since the frequencies are different, the two vibrations drift in and out of phase with each other, causing the total amplitude to vary with time. y
time
1 beat
Physics 1B03summer-Lecture 10

3. Physics 1B03summer-Lecture 10
time
in phase o out of phase in phase t
Physics 1B03summer-Lecture 10

4. Physics 1B03summer-Lecture 10
The math:
Same amplitudes, different frequencies:
Trigonometry:
cos a + cos b = 2 cos [(a-b)/2] cos [(a+b)/2]
Result:
slowly-varying
amplitude
SHM at average
frequency
Physics 1B03summer-Lecture 10

5. Physics 1B03summer-Lecture 10
Note:
2 beats per cycle of
# beats/second =
The beat frequency (number of beats per second) is equal to the difference between the frequencies:
Physics 1B03summer-Lecture 10

6. Physics 1B03summer-Lecture 10
Quiz
Two guitar strings originally vibrate at the same 400-Hz frequency. If you hear a beat of 5Hz, what is/are the other possible frequencie(s) ?
10 Hz
395 Hz
405 Hz
395 Hz and 405 Hz
Physics 1B03summer-Lecture 10

7. Physics 1B03summer-Lecture 10
Interference
2 waves, of the same frequency; arrive out of phase.
Eg: y1=Asin wt y2=Asin (wt+f)
Then
yR= y1 + y2 = AR sin(wt+fR),
and the resultant amplitude is AR=2Acos(½f).
Identical waves which travel different distances will arrive out of phase and will interfere, so that the resultant amplitude varies with location.
Physics 1B03summer-Lecture 10

8. Physics 1B03summer-Lecture 10
Phase difference :
Define
Then, at detector:
(pick starting time
so initial phase is zero here)
Physics 1B03summer-Lecture 10

9. Physics 1B03summer-Lecture 10
Example:
Two sources, in phase; waves arrive by different paths: S1 P r1
detector r2
At detector P: S2
Physics 1B03summer-Lecture 10

10. Physics 1B03summer-Lecture 10
detector x
2 speakers, in phase; f = 170 Hz (so l = 2.0 m; the speed of sound is about 340 m/s)
As you move along the x axis, where is the sound:
a minimum (compared to nearby points)?
a maximum (compared to nearby points)?
Physics 1B03summer-Lecture 10

11. Physics 1B03summer-Lecture 10
Solution:
Physics 1B03summer-Lecture 10

12. Physics 1B03summer-Lecture 10
10 min rest
Physics 1B03summer-Lecture 10

13. Physics 1B03summer-Lecture 10
Interference of Light
Light is an electromagnetic (EM) wave.
Wave properties:
Diffraction – bends around corners, spreads out from narrow slits
Interference – waves from two or more coherent sources interfere
Physics 1B03summer-Lecture 10

14. Electromagnetic Waves
B (magnetic field) Eo v
Usually we keep track of the electric field E :
Electric field amplitude
Electromagnetic waves are transverse waves
Physics 1B03summer-Lecture 10

15. Physics 1B03summer-Lecture 10
Infrared
Red nm
Yellow 600 nm
Green 550 nm
Blue 450 nm
Violet nm
Ultraviolet l
Visible-Light
Spectrum
Physics 1B03summer-Lecture 10

16. The Electromagnetic Spectrum
l (m) f (Hz)
3 x
3 x
7 x x1014
4 x 10-7
3 x
3 x
Radio TV
Microwave
Infrared
Visible
Ultraviolet
X rays
g rays
Physics 1B03summer-Lecture 10

17. Our galaxy (Milky Way) at viewed at different wavelengths
Physics 1B03summer-Lecture 10

18. Physics 1B03summer-Lecture 10
Radio lobes (jets) from a supermassive black hole at the center of the galaxy NGC 4261
Physics 1B03summer-Lecture 10

19. Physics 1B03summer-Lecture 10
Double Slit
(Thomas Young, 1801)
m=2
m=1
m=0 (center)
m=-1
m=-2 θ
incident light
double
slit
separation d
screen
Result: Many bright “fringes” on screen, with dark lines in between.
Physics 1B03summer-Lecture 10

20. Physics 1B03summer-Lecture 10
The slits act as two sources in phase. Due to diffraction, the light spreads out after it passes through each slit. When the two waves arrive at some point P on the screen, they can be in or out of phase, depending on the difference in the length of the paths. The path difference varies from place to place on the screen. P r1
To determine the locations of the bright fringes (interference maxima), we need to find the points for which the path difference Dr is equal to an integer number of wavelengths. For dark fringes (minima), the path difference is integer multiples of half of a wavelength. r2 q d
Δr = r1-r2
Physics 1B03summer-Lecture 10

21. Physics 1B03summer-Lecture 10
For light, the slits will usually be very close together compared to the distance to the screen. So we will place the screen “at infinity” to simplify the calculation.
move P to infinity r1 P
r >> d,
r1 & r2 nearly parallel r2 d q θ
Δr = r1-r2 d θ Δr
Δr = d sin θ
Physics 1B03summer-Lecture 10

22. Physics 1B03summer-Lecture 10
Interference: 2 coherent waves, out of phase due to a path difference Dr:
Constructive Interference (maximum intensity)
for f = 0, ±2π, ±4π, ±6π, ………
-> Δr =0, ±l, ±2 l, ±3 l, ………
Destructive Interference (minimum intensity)
for f = ±π, ±3π, ±5π, ………
-> Δr =±λ/2, ±3λ/2, ±5λ/2, ………
Physics 1B03summer-Lecture 10

23. Physics 1B03summer-Lecture 10
Constructive Interference: (bright)
Δr = mλ or
d sin θ = mλ, m = 0, ±1, ±2, … But, if the slit-screen distance (L) is large, then sinθ~θ
and so sinθ=θ=y/L (in radians): L y θ d
So we have:
Physics 1B03summer-Lecture 10

24. Physics 1B03summer-Lecture 10
Destructive Interference: (no light)
Δr = (m + ½)λ or
d sin θ = (m + ½) λ, m = 0, ±1, ±2, …
So, we have:
Physics 1B03summer-Lecture 10

25. Physics 1B03summer-Lecture 10
Quiz
Two slits are illuminated with red light to produce an interference patter on a distant screen. If the red light is replaces with blue light, how does the pattern change?
The bright spots move closer together
The bright spots move farther apart
The pattern does not change
The patter doesn’t chance, but the width of the spots changes
Physics 1B03summer-Lecture 10

26. 2 slits, 0.20 mm apart; red light (l = 667 nm)
Example
2 slits, 0.20 mm apart; red light (l = 667 nm) y
3 m
screen
Where are a) the bright fringes?
b) the dark lines?
(give values of y)
Physics 1B03summer-Lecture 10

27. Physics 1B03summer-Lecture 10
Solution:
Physics 1B03summer-Lecture 10

28. Physics 1B03summer-Lecture 10
Example
A double slit interference patter is observed on a screen 1.0m behind two slits spaced 0.3mm apart. Ten bright fringes span a distance of 1.65 cm.
What is the wavelength of light used ?
Physics 1B03summer-Lecture 10

29. Physics 1B03summer-Lecture 10
Quiz
Which of the following would cause the separation between the fringes to decrease?
Increasing the wavelength
Decreasing the wavelength
Moving the slits closer together
Moving the slits farther apart
None of the above
Physics 1B03summer-Lecture 10

30. Physics 1B03summer-Lecture 10
10 min rest
Physics 1B03summer-Lecture 10

31. Physics 1B03summer-Lecture 10
Refractive Index
The speed of light depends on the material. We define the refractive index “n” as
n = (speed of light in vacuum)/(speed of light in a material)
material refractive index speed of light
vacuum 1 c  300,000 km/s
air
glass about ,000 km/s
water ,000 km/s
diamond ,000 km/s
Physics 1B03summer-Lecture 10

32. Physics 1B03summer-Lecture 10
Question: A beam of yellow light (wavelength 600 nm), travelling in air, passes into a pool of water. By what factor do the following quantities change as the beam goes from air into water?
speed
frequency
wavelength
Physics 1B03summer-Lecture 10

33. Reflection and Phase Change
Light waves may have a 180° phase change when they reflect from a boundary:
“optically dense” medium (larger refractive index)
no phase change at this reflection
180° phase change when reflecting from a denser medium
Just remember this : low to high, phase shift of pi !
Physics 1B03summer-Lecture 10

34. Physics 1B03summer-Lecture 10
Example: Thin film
What is the minimum thickness of a soap film
(n  1.33) needed to produce constructive interference for light with a 500nm wavelength ?
(air : n  1.00).
What about destructive interference ?
Physics 1B03summer-Lecture 10

35. Example: Antireflection coatings
To reduce reflections from glass lenses (n  1.5), the glass surfaces are coated with a thin layer of magnesium fluoride (n  1.38). What is the correct thickness of the coating for green light (550 nm vacuum wavelength)?
MgF2
air
glass
Physics 1B03summer-Lecture 10

36. Physics 1B03summer-Lecture 10
Example
A beam of 580 nm light passes through two closely spaced glass plates (nglass=1.6), as shown in the figure below. For what minimum nonzero value of the plate separation d is the transmitted light dark?
Physics 1B03summer-Lecture 10

37. Physics 1B03summer-Lecture 10
Quiz
Why do we see many colours on a soap bubble?
A) because white light is made up of different wavelengths B) because the bubble has different thickness C) both A and B D) because the bubble is round and light reflects from the other side
Physics 1B03summer-Lecture 10