Recall that the intensity of the single slit with width a is given by I = I0 [(sin^2 ) /^2 ] , =[ (a)/ ] sin

a) Show that when = 0, I = I0.

b) Using calculus, show that in general critical points (local maximums and minimums) will appear when satisfies either of the following two equations:

(i) sin = 0

(ii) = tan

c) Equation (i) given in part b) is fairly easy to solve, what types of fringes do its solutions correspond to?

d) Equation (ii) given in part b) cannot be solved in a simple closed form (except the simple solution of = 0), although one can find very accurate solutions using numerical methods, but let's not pursue this method. Instead, do the following:

Show that the first nonzero positive solution is almost halfway between two dark spots. You can do this however you want. For example, you could plot the left and right hand sides and look for their intercept, you could just plug numbers into your calculator to check, etc.

Using the value of you found above, what is the corresponding intensity? How does it compare to the intensity at the center?

Using properties of tan , argue that for larger and larger values of , bright spots should become more and more evenly spaced, and show that they will approach being halfway between dark spots.