Phy 122 - Assignment 12:

A. 1. Angle between sources barely resolved through a circular opening:

Now from θ, find s. Technically this is an isosceles triangle. However, because it is so long and thin, you can take a shortcut. The base angles are very close to 90°, so it makes only a very small error if you treat it like a right triangle.

2. With n higher on one side of the film and lower on the other, there is destructive interference when

2 n t = (m + ½) λ

For the minimum thickness, m = 0.

2 (1.30) t = ½ (500 nm)

t = = 96.2 nm

B. (a) Grating equation: m l = d sinq m = 3 (3^rd order)

(3) (5.00 x 10^-7 m) = d sin 32°

d = 1.5 x 10^-6 = 2.83 x 10^-6 m = 2.83 x 10^-4 cm

sin 32°

Slits per cm = 1 / (cm per slit) = 1 / (2.83 x 10^-4) = 3.53 x 10³ ans.

(b) From the grating equation: sin q = m l / d

Try larger and larger values of m until sinq passes 1. The sine function can’t ever be more than 1, so that shows you the largest possible m.

sin q = (5)( 5.00 x 10^-7 m)/(2.83 x 10^-6 m) = .883 ok

sin q = (6)( 5.00 x 10^-7 m)/(2.83 x 10^-6 m) = 1.06 nope.

ans: 5^th order

C. 1. Oil’s n must be less than water’s. (and more than air’s.) If both rays reflect off a higher n, both undergo the same 180° phase shift. That way, interference is constructive when the thickness is nearly zero.

2.

From the dotted triangle: tan θ =

Bright fringes are at d sin θ = m λ, so sin θ =

Because θ is small, tan θ approximately equals sin θ.

= m = 1 because first order

A = (1.2 m) = = .00262 m

D. 1. Destructive. The reason for coating a lens is to make it nonreflective. That is accomplished by having the two reflected rays destroy each other.

2. One ray reflects off a higher n (the soap film), the other off a lower n (the air under the film). This gives one ray a 180° phase shift and the other ray none. So, with a path difference of about zero, they are 180° out of phase.

3. Single slit maxima are where

a sin θ = (m + ½) λ

a = slit width = .8 mm

m = 2 because second order

Because θ is very small, sin θ ≈ tan θ = = .00175

So, (.0008 m)(.00175) = (2 + ½) λ

λ = = 5.60 x 10^-7 m = 560 nm

E. 1. i. a is the width of the slit.

ii. d is its reciprocal. (cm per line = 1 / lines per cm.)

2. Destructively. The path difference of 2λ would normally give constructive interference, but ray A has a 180° phase shift when it is reflected, while B does not. Therefore, the waves are out of phase and cancel.

3. Constructive interference if path difference = mλ.

First order means m = 1.

(1)λ = 16.12 – 14 = 2.12 cm

F.