University Physics with Modern Physics, Volume 2 (Chs. 21-37); Mastering Physics with Pearson eText -- ValuePack Access Card (14th Edition)
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Chapter 36, Problem 36.68CP

CALC It is possible to calculate the intensity in the single-slit Fraunhofer diffraction pattern without using the phasor method of Section 36.3. Let y′ represent the position of a point within the slit of width a in Fig. 36.5a, with y′ = 0 at the center of the slit so that the slit extends from y′ = −a/2 to y− a/ 2. We imagine dividing the slit up into infinitesimal strips of width d y′ each of which acts as a source of secondary wavelets, (a) The amplitude of the total wave at the point O on the distant screen in Fig. 36.5a is E0. Explain why the amplitude of the wavelet from each infinitesimal strip within the slit is E0(dy/a), so that the electric field of the wavelet a distance x from the infinitesimal strip is dE = E0(dy/a) sin (kxωt). (b) Explain why the wavelet from each strip as detected at point P in Fig. 36.5a can be expressed as

d E = E 0 d y a sin [ k ( D y sin θ ) ω t ]

Where D is the distance from the center of the slit to point P and k =2πλ.. (c) By integrating the contributions dE from all parts of the slit, show that the total wave detected at point P is

E = E 0 sin ( k D ω t ) sin [ k a ( sin θ ) / 2 ] k a ( sin θ ) / 2         E 0 sin ( k D ω t ) sin [ π a ( sin θ ) / λ ] π a ( sin θ ) / λ

(The trigonometric identities in Appendix B will be useful.) Show that at θ = 0°, corresponding to point O in Fig. 36.5a, the wave is E = E0 sin(kDωt) and has amplitude E0, as stated in part (a), (d) Use the result of part (c) to show that if the intensity at point O is I0, then the intensity at a point P is given by Eq. (36.7).

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Chapter 36 Solutions

University Physics with Modern Physics, Volume 2 (Chs. 21-37); Mastering Physics with Pearson eText -- ValuePack Access Card (14th Edition)

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