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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 (kx – ωt). (b) Explain why the wavelet from each strip as detected at point P in Fig. 36.5a can be expressed as
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
(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(kD − ωt) 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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