MIT OpenCourseWare http://ocw.mit.edu 8.02 Electricity and Magnetism, Spring 2002 Please use the following citation format: Lewin, Walter, 8.02 Electricity and Magnetism, Spring 2002 (Massachusetts Institute of Technology: MIT OpenCourseWare). http://ocw.mit.edu (accessed MM DD, YYYY). License: Creative Commons Attribution-Noncommercial-Share Alike. Note: Please use the actual date you accessed this material in your citation. For more information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/termsMIT OpenCourseWare http://ocw.mit.edu 8.02 Electricity and Magnetism, Spring 2002 Transcript – Lecture 33 I'm very proud of you. You did very well on the last exam. Class average is a little bit above 70. Congratulations. There were 22 students who scored 100. Many of you are interested in where the dividing line is between C and D. If I take only the three exams into account, forget the quizzes, forget the homework, forget the motor, and you add up the three grades of your three exams, the dividing line between C and D will be somewhere in the region 135 to 138. So you can use that for your calibration where you stand. The controversy between Newton and Huygens about the nature of light was settled in 1801 when Young demonstrated convincingly that light shows all the characteristic of waves. Now in the early twentieth century, the particle character of light surfaced again and this mysterious and very fascinating duality of being waves and particles at the same time is now beautifully merged in quantum mechanics. But today I will focus on the wave character only. Very characteristic for waves are interference patterns which are produced by two sources, which simultaneously produce traveling waves at exactly the same frequency. Let this be source number one and let this be source number two.And they each produce waves with the same frequency, therefore the same wavelength, and they go out let's say in all directions. They could be spherical, in the case of water surface, going out like rings. And suppose you were here at position P in space at a distance R1 from source number one and at a distance R2 from source number two. Then it is possible that at the point P the two waves that arrive are in phase with each other. That means the mountain from two arrives at the same time as a mountain from one, and the valley from two arrives at the same time as the valley from one. So the mountains become higher and the valleys become lower. We call that constructive interference. It is also possible that the waves as they arrive at point P are exactly 180 degrees out of phase, so that means that the mountain from two arrives at the same time as the valley from one. In which case they can kill each other, and that we call destructive interference. You can have this with water waves, so it's on a two-dimensional surface. You can also have it with sound, which would be three-dimensional. So the waves go out on a sphere. And you can have it with electromagnetic radiation as we will also see today, which is of course also three dimensions. If particles oscillate then their energy is proportional to the square of their amplitudes. So therefore since energy must be conserved, the amplitude of sound oscillations and also of the electric vector in the case ofelectromagnetic radiation, the amplitude must fall off as one over the distance, 1 / R. Because you're talking about 3-D waves. You're talking about spherical waves. And the surface area of a sphere grows with R squared. And so the amplitude must fall off as 1 / R. Now if we look at the superposition of two waves, in this case at point P and we make the distance large, so that R1 and R2 are much, much larger than the separation between these two points, then this fact that the amplitude of the wave from two is slightly smaller than the amplitude from the wave from one can then be pretty much ignored. Imagine that the path from here to here is one-half of a wavelength longer than the path from here to here. That means that this wave from here to here will have traveled half a period of an oscillation longer than this one. And that means they are exactly 180 degrees out of phase and so the two can kill each other. And we call that destructive interference. And so we're going to have destructive interference when R2 - R1 is for instance plus or minus one-half lambda, but it could also be plus or minus 3/2 lambda, 5/2 lambda, and so on. And so in general you would have destructive interference if the difference between R2 and R1 is 2N + 1 times lambda divided by 2 whereby N is an integer, could be 0, or plus or minus 1, or plus or minus 2, and so on. That's when you would have destructive interference. We would have constructive interference if R2 - R1 is simply N times lambda. So then the waves at point P are in phase and N is again, could be 0, plus or minus 1, plus or minus 2, and so on.If the sum of the distance to two points is a constant you get an ellipse in mathematics. If the difference is a constant, which is the case here, the difference to two points is a constant value, for instance one-half lambda, then the curve is a hyperbola. It would be a hyperbola if we deal with a two-dimensional surface. But if we think of this as three-dimensional, so you can rotate the whole thing about this axis, then you get hyperboloids, you get bowl-shaped surfaces. And so if I'm now trying to tighten the nuts a little bit, suppose I have here two of these sources that produce waves and the separation between them is D, then it is obvious that the line right through the middle of them and perpendicular to them is always a maximum if the two sources are oscillating in phase. So this line is immediately clear that R2 - R1 is 0 here. If the two are in phase. And they always have to generate the same frequency, of course. So this line would be always a maximum. Constructive interference. It's this 0, substitute there. And in case that we're talking about three-dimensional, this is of course a plane. Going perpendicular to the blackboard right through the middle. The different R2 - R1 equals lambda would again give me constructive interference. That would be a hyperbola then, R2 - R1 equals lambda, that would again be a maximum, and you can draw the same line on this side, and then R2 - R1 being 2 lambda again would be a maximum.And again, if this is three-dimensional, you can rotate it about this line and you get bowls. And so in
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