Transcript Electric Potential - McMaster Physics and Astronomy

Interference of Waves
Beats
Double Slit
Physics 1B03summer-Lecture
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
time
in phase
180o out of phase
in phase
t
Physics 1B03summer-Lecture
The math:
Same amplitudes, different frequencies:
y1  A cos(1t )
y2  A cos(2t )
Trigonometry:
cos a + cos b = 2 cos [(a-b)/2] cos [(a+b)/2]
Result:
y  y1  y2
 1   2 
 1   2 
 2 A cos
t  cos
t
2
2




slowly-varying
amplitude
SHM at average
frequency
Physics 1B03summer-Lecture
Note:
1   2 

t
2 beats per cycle of cos
2


# beats/second =
f1  f 2 

2

 2 
The beat frequency (number of beats per second)
is equal to the difference between the frequencies:
f b  f1  f 2
Physics 1B03summer-Lecture
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) ?
a)
b)
c)
d)
10 Hz
395 Hz
405 Hz
395 Hz and 405 Hz
Physics 1B03summer-Lecture
Interference
2 waves, of the same frequency; arrive out of phase.
Eg:
Then
y1=Asin t
y2=Asin (t+f)
yR= y1 + y2 = AR sin(t+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
Phase difference :
(kr1  t )  (kr2  t )  k (r1  r2 )  kr
Define
f  kr  2
(or
Then, at detector:
r

r

radians
cycles)
(pick starting time
so initial phase is zero here)
y1  A sin(t )
y2  A sin(t  f )
f
f
 y R  (2 A cos ) sin(t  )
2
2
Physics 1B03summer-Lecture
Example:
Two sources, in phase; waves arrive by
different paths:
S1
P
detector
r1
r2
S2
At detector P:
y1  A sin( kr1  t )
y2  A sin( kr2  t )
Physics 1B03summer-Lecture
8m
detector
x
2 speakers, in phase; f = 170 Hz (so  = 2.0 m; the
speed of sound is about 340 m/s)
As you move along the x axis, where is the sound:
a) a minimum (compared to nearby points)?
b) a maximum (compared to nearby points)?
Physics 1B03summer-Lecture
Solution:
Physics 1B03summer-Lecture
10 min rest
Physics 1B03summer-Lecture
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
Electromagnetic Waves
  
E , B, v all 
Eo
E
B (magnetic field)
v
Usually we keep track of the electric field E :


E( x, t )  A sin( kx  t )
Electric field amplitude
Electromagnetic waves are transverse waves
Physics 1B03summer-Lecture
Infrared

Red
780 nm
Yellow
600 nm
Green
550 nm
Blue
450 nm
Violet
380 nm
Visible-Light
Spectrum
Ultraviolet
Physics 1B03summer-Lecture
The Electromagnetic Spectrum
 (m)
300
3
3 x 10-3
10-6
3x
7 x 10-7
4 x 10-7
3x
10-9
3 x 10-12
f (Hz)
Radio
TV
Microwave
Infrared
Visible
Ultraviolet
X rays
g rays
106
108
1011
1014
5x1014
1017
1020
Physics 1B03summer-Lecture
Our galaxy (Milky Way) at viewed at different wavelengths
Physics 1B03summer-Lecture
Radio lobes (jets) from a supermassive black hole at the
center of the galaxy NGC 4261
Physics 1B03summer-Lecture
Double Slit
(Thomas Young, 1801)
θ
incident
light
double
slit
separation d
m=2
m=1
m=0 (center)
m=-1
m=-2
screen
Result: Many bright “fringes” on screen, with dark
lines in between.
Physics 1B03summer-Lecture
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.
r1
To determine the locations of the
bright fringes (interference
maxima), we need to find the
points for which the path
difference r is equal to an
integer number of wavelengths.
r2
d
q
For dark fringes (minima), the
path difference is integer
multiples of half of a wavelength.
Physics 1B03summer-Lecture
P
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
r2
d
r >> d,
r1 & r2 nearly parallel
θ
q
d θ
Δr = d sin θ
Physics 1B03summer-Lecture
Interference: 2 coherent waves, out of phase due
to a path difference r:
f  phase difference  2  r ) radians
  r ) cycles
Constructive Interference (maximum intensity)
for f = 0, ±2π, ±4π, ±6π, ………
-> Δr =0, ±, ±2 , ±3 , ………
Destructive Interference (minimum intensity)
for f = ±π, ±3π, ±5π, ………
-> Δr =±λ/2, ±3λ/2, ±5λ/2, ………
Physics 1B03summer-Lecture
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):
y
d
So we have:
θ
L
dy
 m
L
Physics 1B03summer-Lecture
Destructive Interference: (no light)
Δr = (m + ½)λ
or
d sin θ = (m + ½) λ, m = 0, ±1, ±2, …
dy
1
So, we have:
 ( m  )
L
2
Physics 1B03summer-Lecture
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?
A)
B)
C)
D)
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
Example
2 slits, 0.20 mm apart;
red light ( = 667 nm)
y
3m
0
screen
Where are a) the bright fringes?
b) the dark lines?
(give values of y)
Physics 1B03summer-Lecture
Solution:
Physics 1B03summer-Lecture
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
Quiz
Which of the following would cause the separation
between the fringes to decrease?
A)
B)
C)
D)
E)
Increasing the wavelength
Decreasing the wavelength
Moving the slits closer together
Moving the slits farther apart
None of the above
Physics 1B03summer-Lecture
10 min rest
Physics 1B03summer-Lecture
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
vacuum
air
glass
water
diamond
refractive index
1
1.0003
about 1.5
1.333
2.4
speed of light
c  300,000 km/s
200,000 km/s
225,000 km/s
125,000 km/s
Physics 1B03summer-Lecture
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?
A) speed
B) frequency
C) wavelength
Physics 1B03summer-Lecture
Reflection and Phase Change
Light waves may have a 180° phase change when
they reflect from a boundary:
“optically dense” medium
(larger refractive index)
180° phase change
when reflecting from
a denser medium
no phase change at
this reflection
Just remember this : low to high, phase shift of pi !
Physics 1B03summer-Lecture
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
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)?
air
MgF2
glass
Physics 1B03summer-Lecture
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
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