• 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