The final will be given on Wednesday, March 18th, from 11:30 AM to 2:30 PM, in Warren 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? where m is an integer The two waves at distances r 1 and r 2 : Constructive interference: The two sources oscillate in phase: at The result of interference will depend on phase difference, which does not depend on time!

The oscillations will be out of phase and the interference will be destructive if r 1 - r 2 = /2 or, r 1 - r 2 = - /2  or in general: How do we know, whether we are in a node or an antinode? where m is an integer number The phase difference:

Can we see any interference without a laser?

Some math: the slits are two coherent sources. The distances to the observation point are r 1 and r 2. Their difference for small angles  , small y/L

Constructive (a bright strip) Destructive (a dark strip) Approximation used: for small

− intensity of either wave alone In general, the distribution of intensity on the screen: Bright and dark fringes:

In the case when The intensity Positions of the bright and dark fringes (maxima and minima of interference) The distance between the fringes:

Does this look any familiar? In the case when The intensity The intensity has a minimum of 0, maximum of 4S 0, and a mean value of 2S 0 – the same as for non-coherent sources (!) Electric field:

Composite wave: Looks very much like a standing wave with The intensity Electric field: What are the differences?

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 Electric field:

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: or But the condition for destructive interference for N slits changes to: where m is an integer but not an integer multiple of N 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: Maximal intensity in bright fringes: Average intensity:

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