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• The final will be given on Wednesday, March 18th, from 11:30 AM to 2:30 PM, in Warren 2001.
• Your student I.D. is required to take it.
•The final will be a closed-book exam and cover the whole course material, and it will be composed of multiple choice problems, just like the quizzes.
• There will be ~24 problems, 2-3 per each week of the class. About a half of the problems will be conceptual.
• You should bring a Scantron form with you.
• You may use a calculator (but not a laptop) during the final.
• You may also bring a single 8 1/2” x 11”sheet of paper of formulae and notes handwritten on the both sides. (Printed cheat-sheets are not allowed!)
•You may wish to bring some blank scratch paper as well.
Let’s translate it into the language of phases:
How do we know, whether we are in a node or an antinode?
1r
2r 1r
2r
mkmrr /221
where m is an integer
)cos( 11 tkrA The two waves at distances r1 and r2:
Constructive interference:
21 krkr
The two sources oscillate in phase: at )cos( tA 021 rr)cos( 22 tkrA
The result of interference will depend on phase difference, which does not depend on time!
mkrkr 221
The oscillations will be out of phase and the interference will be destructive if r1 - r2 = /2 or, r1 - r2 = - /2or in general:
How do we know, whether we are in a node or an antinode?
2r
1r
1r
2r
)2/1(21 mrr where m is an integer number
The phase difference:
mmkkrkr 2)2/1(21
Can we see any interference without a laser?
Some math: the slits are two coherent sources. The distances to the observation point are r1 and r2. Their difference
sin12 drr LyLy /)/(tan 1
Ldyrr /12 for small angles, small y/L
)sin()tan(
L
y1r
2r
Constructive (a bright strip)
sin12 drr Ldyrr /12
md sin
Destructive (a dark strip) )2/1(sin md
Approximation used: Ld
for small Ly /
L
y1r
2r
)sin
(cos4 20
dSS
− intensity of either wave alone
In general, the distribution of intensity on the screen:
00 22
14 SSS
0SBright and dark fringes:
md sin )2/1(sin md
L
y1r
2r
)(cos4)sin
(cos4 20
20 y
L
dS
dSS
In the case when 1/ Ly Ly /tansin
L
y1r
2r
y
)( yS
The intensity
04S02S
Positions of the bright and dark fringes (maxima and minima of interference)
d
Ly
d
Lmybright
d
Lmydark
)2/1(
The distance between the fringes:d
Ly
)(cos4)sin
(cos4 20
20 y
L
dS
dSS
Does this look any familiar?
In the case when 1/ Ly Ly /tansin
L
y1r
2r
y
)( yS
The intensity
The intensity has a minimum of 0, maximum of 4S0, and a mean value
of 2S0 – the same as for non-coherent sources (!)
04S02S
Electric field: )cos()cos(2 0 yL
dtEE
d
Ly
)(cos4)sin
(cos4 20
20 y
L
dS
dSS
Composite wave:
)sin()cos(2 tkxAy
Looks very much like a standing wave with
The intensity
L
y1r
2r
y
)( yS
04S02S
Electric field: )sin()sin(2 0 tyL
dEE
What are the differences?
L
dk
d
Ly
Composite wave:
Unlike the standing waves on a string:in the interference, the pattern of bright and dark fringes is created along the y-axis, whereas the wave itself propagates along the x-axis;
unlike the distance between regular nodes and antinodes the distance between the dark and bright fringes is not 1/2 of the wavelength, but rather
L
y1r
2r
y
)( yS
04S02S
Electric field: )sin()sin(2 0 tyL
dEE
d
Ly
What happens if we have got 3 or more slits?
The condition for constructive interference (bright fringes) does not change form the 2-slit apparatus:
md sinor
d
Lmybright
But the condition for destructive interference for N slits changes to:
N
md sin
where m is an integer but not an integer
multiple of N
d
How does the resulting interference pattern look like?
The maxima, bright fringes, become brighter and narrower as the number of slits increases. Most of the interference pattern becomes dark. For N slits:
02
max SNS Maximal intensity in bright fringes:
Average intensity:0NSS
http://id.mind.net/~zona/mstm/physics/waves/interference/twoSource/TwoSourceInterference1.html
http://www.spa.umn.edu/groups/demo/waves/3B5010.html&h=240&w=320&sz=9&tbnid=GWEiiWBVHAcJ:&tbnh=84&tbnw=112&start=163&prev=/images%3Fq%3Dslit%2Bdiffraction%26start%3D160%26hl%3Den%26lr%3D%26sa%3DN
http://vsg.quasihome.com/interf.htm
http://www.ngsir.netfirms.com/englishhtm/Diffraction.htm
http://www.walter-fendt.de/ph11e/interference.htm