Intensity Pattern of N Slits. (a) Consider an arrangement of N sli...

Intensity Pattern of N Slits. (a) Consider an arrangement of N slits with a distance d between adjacent slits. The slits emit coherently and in phase at wavelength λ. Show that at a time t, the electric field at a distant point P is

where E₀ is the amplitude at P of the electric field due to an individual slit, Φ = (2πd sin θ)/λ, u is the angle of the rays reaching P (as measured from the perpendicular bisector of the slit arrangement), and R is the distance from P to the most distant slit. In this problem, assume that R is much larger than d. (b) To carry out the sum in part (a), it is convenient to use the complex-number relationship e^iz = cos z + i sin z, where In this expression, cos z is the real part of the complex number e^iz, and sin z is its imaginary part. Show that the electric field E_P(t) is equal to the real part of the complex quantity

(c) Using the properties of the exponential function that e^Ae^B = e^(A+B) and (e^A)n = e^nA, show that the sum in part (b) can be written as

Then, using the relationship e^iz = cos z + i sin z, show that the (real) electric field at point P is

The quantity in the first square brackets in this expression is the amplitude of the electric field at P. (d) Use the result for the electric-field amplitude in part (c) to show that the intensity at an angle θ is

where I₀ is the maximum intensity for an individual slit. (e) Check the result in part (d) for the case N = 2. It will help to recall that sin 2A = 2 sin A cos A. Explain why your result differs from Eq. (35.10), the expression for the intensity in two-source interference, by a factor of 4. (Hint: Is I₀ defined in the same way in both expressions?)