Question
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Wave interference in the double slit experiment: For a sketch, please consult figure 35.5 in Young and Freedman. This is the same sketch as used in class, except that in class we labeled the top slit as "A" and the bottom slit as "B". Let P be a generic point on the screen, and let rA and rB be the distances between P and slits A and B respectively. An incident plane wave with wave vector passes through the double slit, and then has the form Denote the magnitude of the wave vectors as and What is the relation between the magnitudes k, kA kB and omega and upsilon (where upsilon is the speed of the wave)? I of a light wave is proportional to its energy. The expression for I for a light wave is similar to the expression for the energy in a string (which we studied earlier) and is given by I = omega 2f2/(2c). By substituting in f = fA + fB into this formula for I, write I at P as the sum of three terms: of a contribution IA from wave A, a contribution IB from B, and an interference term IAB Compute the average of IA over one period of the wave. This will be independent of where P is on the screen. Finally, compute the average over one period of the interference contribution IAB, which will be proportional to cos k(rA - rB)/2. This is the term that shows the interference pattern! Adding the terms, what is the total intensity I? Fourier Series: The last two problems are about Fourier series. Recall that a Fourier series is a series expansion for any periodic function f(x). Letting the period be 2L, then the expansion has the form where am, bm are constant coefficients that depend on the function f being expanded. Fourier series-verifying an integral: In deriving the expressions for the coefficients one needs to evaluate a set of integrals. Verify that Again, please note that the length of the interval is 2L. A Fourier expansion: The function is defined by f(x) = -H, - L le x
Explanation / Answer
Please show work Wave interference in the double slit experiment: For a sketch,