Chapter 32, Problem 68P

The intensity of the single-slit diffraction pattern can be calculated by summing the amplitudes of infinitely many field amplitudes corresponding to waves from every infinitesimal part of the slit. (a) Referring to Fig. 32.20, show that the field from an element of slit width dy, a distance y from the bottom edge of the slit, is dE = (E_pdy/a) sin(ωt + ϕ(y)), where ϕ(y) = (2πy/λ)sin θ. (b) Integrate dE over the entire slit from y = 0 to y = a, and use trig identities from Appendix A. to find the total amplitude and from there show that the average intensity is given by Equation 32.10.