Physics 1502: Lecture 32Today’s Agenda

• Announcements:

– Midterm 2: graded after Thanks Giving

– Homework 09: Friday December 4Homework 09: Friday December 4

• Optics – Eye

– interference

~fe

I1

eyepiece

I2

~fo

objectiveL

The

EYE

Compound Microscope

o1

h

O

I2h2

feye

h1

I1

i1

Objective(fob< 1cm)

fob

L

Eyepiece(feye~5cm)

Magnification:

Refracting Telescope

Star

feye

I2h2

fob

Objective(fob~ 250cm)

Eyepiece(feye~5cm)

i1

I1h1

AngularMagnification:

Interference

Superposition• What happens when two waves collide ?

– They add point by point

Why?

Because the wave equation is linear. This is the principle of superposition.

Lecture 32 – Act 1• If you added the two sinusoidal waves shown in the top plot,

what would the result look like ?

A wave through a slit

Wavefronts: slit acts like point source

Rays

A wave through two slits (two coherent point sources)

Intensity

What happens when two light waves are present at the same point in space and time?

What will we see? Intensity! Add Amplitudes! (electric fields or magnetic

fields)

Brightness ~ <Amplitude2> ~ ½ E02

Lecture 32 – Act 2• Suppose laser light of wavelengthis incident on the two-slit

apparatus as shown below.

Which of the following statements are true?

(A) There are new patterns of light and dark.

(B) The light at all points on the screen is increased (compared to one slit).

(C) The light at all points on the screen is decreased (compareed to two slits).

A wave through two slits

Screen

L

Assume L is large, Rays are parallel

d

A wave through two slits

Screen

P=d sin

d

In Phase, i.e. Maxima when P = d sin = nOut of Phase, i.e. Minima when P = d sin = (n+1/2)

A wave through two slitsIn Phase, i.e. Maxima when P = d sin = n

Out of Phase, i.e. Minima when P = d sin = (n+1/2)

+

+

Waves and Interference• Note that you could derive the reflectance equation

(i=R) using a particle model for light. Bouncing balls.

• You could also derive Snell’s Law for particles.

n1sin (i)=n2sin(2)

The particles change speed in different media(Newton did just this)

• You cannot get a particle model for these interference effects. You would have to magically create particles at the bright spots and annihilate them at the dark spots.

• Interference effects mean that light must be made up of waves.

The AmplitudesWhat determines the wave

amplitude at P?The difference in the path lengths!

(ie = S1P - S2P)

If the is an integral number of wavelengths, the phase difference is zero and we get constructive interference.

If is l/2, 3 l/2, 5 l/2, etc, we get destructive interference.

The general case is given by:

The amplitude for the wave coming from S1:

The amplitude for the wave coming from S2:The amplitude for the total wave at P :

with

The IntensityWhat is the intensity at P?

The only term with a t dependence is sin2( ).That term averages to ½ .

If we had only had one slit, the intensity would have been,

So we can rewrite the total intensity as,

with

The Intensity

We can rewrite intensity at point Pin terms of distance y

Using this relation, we can rewrite expression for the intensity at point P as function of y

Constructive interference occurs at

where m=+/-1, +/-2 …

d spacing• Note that the angle between bright spots is given by,

– sin = n/d

• To see effect we want d.

>d means no bright spot, <<d means bright spots too close together.

• For an x-ray, = 1 Å, E = 10 keV d ~ 1-20 Å.

– Interference patterns off of crystals

• For an electron, = 1 Å, E = 100 meV (deBroglie - 1925)

– 100 meV means an electron is accelerated through a voltage of 0.1 V

– Interference patterns off of crystals

– Davisson and Germer, (1927)

• So, electrons are waves ??

Phasor Addition of Waves

Consider a sinusoidal wave whose electric field component is

Consider second sinusoidal wave

The projection of sum of two phasors EP is equal to

E0E1(t) t

E2(t)E0

EP(t)

ER

/2

E0

tE1(t)

t+E0

E2(t)

Phasor Diagrams for TwoCoherent Sources

ER=2E0

E0 E0 E0

E0

ER

450

E0

E0ER

900

ER=0

E0 E0

E0

E0ER

2700 ER=2E0

E0 E0

SUMMARY2 slits interference pattern (Young’s experiment)

How would pattern be changed if we add one or more slits ?(assuming the same slit separation )

3 slits, 4 slits, 5 slits, etc.

Phasor: 1 vector represents 1 traveling wave

single traveling wave 2 wave interference

N=2 N=4N=3

N-slits Interference Patterns