p37_013

p37_013 - 13. (a) The intensity for a single-slit...

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Unformatted text preview: 13. (a) The intensity for a single-slit diﬀraction pattern is given by I = Im sin2 α α2 where α is described in the text (see Eq. 37-6). To locate the extrema, we set the derivative of I with respect to α equal to zero and solve for α. The derivative is sin α dI = 2Im 3 (α cos α − sin α) . dα α The derivative vanishes if α = 0 but sin α = 0. This yields α = mπ , where m is a nonzero integer. These are the intensity minima: I = 0 for α = mπ . The derivative also vanishes for α cos α − sin α = 0. This condition can be written tan α = α. These implicitly locate the maxima. (b) The values of α that satisfy tan α = α can be found by trial and error on a pocket calculator or computer. Each of them is slightly less than one of the values (m + 1 )π rad, so we start with these 2 values. The ﬁrst few are 0, 4.4934, 7.7252, 10.9041, 14.0662, and 17.2207. They can also be found graphically. As in the diagram below, we plot y = tan α and y = α on the same graph. The intersections of the line with the tan α curves are the solutions. The ﬁrst two solutions listed above are shown on the diagram. y .. . .. . .. . . .. . .... ... . . . . . .. ... . . . . ... . . ... . . .. . . . . ... . ... . . . ... ... . . . . .. .. . . . . . . ... ... . . . . ... ... . . . . ... . . ... . . . . ... . ... . . . . . ... ... . . . . ... . . ... . . .. .. . . . . . . ... ... . . . . ... . . ... . . .. . . ... . . ... . . ... . . . . ... . . ... . . ... . . .. . . . . ... . . ... . . .. .. . . . . ... . . ... . . . . ... ... . . . . . ... . ... . . . . ... . . ... . . . . ... ... . . .. . . .. . . . . ... ... . . . . . ... . ... . . . . ... . . ... . . . ... . . . . ... . . ... . . . . ... . . ..... . . .... . . . . . . .... . . ..... . . .. .. .. ... .. ...... .. ...... .. .... . .. ..... .. .... . . .. .... . . ... .. ... .. ... .. ... ... ... ... ... ... ... ... ... ... ... ... ... .. .. .. .. y = tan α y = tan α y=α 0 π/2 π 3π/2 α (rad) (c) We write α = (m + 1 )π for the maxima. For the central maximum, α = 0 and m = − 1 . For the 2 2 next, α = 4.4934 and m = 0.930. For the next, α = 7.7252 and m = 1.959. ...
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This note was uploaded on 11/12/2011 for the course PHYS 2001 taught by Professor Sprunger during the Fall '08 term at LSU.

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