Met

al

e-

e-

e-

e-

e-

e+

12x p

Faraday’s experiment (1833)• Dissolve one mole of some substance in water• Let an electric current run through it• Measure how much charge runs through before it

stops Na+ Cl-

+–

96,500 CQ 1 Faraday AN e

•All ions have the same charge (or simple multiples of that charge)•Avogadro’s number was not known at this time

J.J. Thomson discovers electron: 1897

e-

•Charged particle, bends in presence of magnetic field

F q qu

Be R p mu•Relativity not discovered until 1905

mu

Be um R

•Velocity measured with help of electric field

+–

0F E

Bu

•Ratio of charge to mass now known

2

EB

em R

The Plum Pudding Model: 1904•Electrons have only tiny fraction of an atom’s mass•Atoms have no net charge

1904: J.J. Thomson proposes the “Plum Pudding” Model•Electrons “imbedded” in the rest of the atom’s charge•Rest of charge is spread throughout the atom

e

a

mm

1 Faradayeme

e a

a a

m eNe m N

Millikan measures charge e: 1909•Atomizer produced tiny drops of oil; gravity pulls them down•Atomizer also induces small charges•Electric field opposes gravity•If electric field is right, drop stops falling

E gF F Eq gm 343g r 3

2

E 218 kV/m1 m

851 kg/m

9.80 m/s

r

g

34

3Eg rq

32 3 6

5

4 9.8 m/s 851 kg/m 10 m

3 2.18 10 V/m 1 J/C/V

191.60 10 C

+

•Millikan always found the charge was an integer multiple of e191.602 10 Ce

Millikan measures charge e: 1909

The atom in 1909:•Strong evidence for atoms had been found•Avogadro’s number, and hence the mass of atoms, was now known•Electron mass and charge were known•Atoms contained negatively charged electrons•The electrons had only a tiny fraction of the mass of the atom•Distribution and nature of the positive charge was unknown

Meanwhile . . .

Statistical MechanicsThe application of statistics to the properties of systems containing a large number of objects

Mid – late 1900’s, Statistical Mechanics successfully explains many of the properties of gases and other materials•Kinetic theory of gases•ThermodynamicsThe techniques of statistical mechanics:•When there are many possibilities, energy will be distributed among all of them•The probability of a single “item” being in a given “state” depends on temperature and energy BE k TP E e

23 51.3806 10 J/K 8.6173 10 eV/KBk

gravity

Gas molecules in a tall box:

AnnouncementsASSIGNMENTS

Day Read Quiz HomeworkToday Sec. 3-2 Quiz H Hwk. HFriday Study For Test noneMonday Sec. 3-3 & 3-4 Quiz I Hwk. I

Equations for Test:•Force and Work Equations added•Lorentz boost demoted

9/16

2

x x vt

y yz z

t t vx c

dpFdt

W E F d

Test Friday:•Pencil(s)•Paper•Calculator

Black Body Radiation: Light in a boxConsider a nearly enclosed container at uniform temperature:

•Light gets produced in hot interior•Bounces around randomly inside before escaping•Should be completely random by the time it comes out•Pringheim measures spectrum, 1899

u() = energy/ volume /nm

Black Body Radiation

Can statistical mechanics predict the outcome?

•Find effects of all possible electromagnetic waves that can exist in a volume•Two factors must be calculated:

•n(): Number of “states” with wavelength •E: Average energy

Finding n()How many waves can you fit in a given volume?

•Leads to a factor of 1/4

•What are all the directions light can go?

•Leads to a factor of 4•How many polarizations?

•Leads to a factor of 2

Goal - Predict:

0

Energy/volume/wavelength

Energy/volume

u

U u d

u n E

4

8u E

How to find E What does E mean?•It is an expectation value

Example: Suppose you roll a fair die. If you roll 1 you win $3, if you roll 2 or 3 you win $1, but if you roll 4, 5, or 6, you lose $2. What is the expectation

value of the amount of money you win?

1 1 16 3 2$ 3 1 2

E

E P E E

B

B

E k T

E k T

P E e

P E Ce

Sum of all probabilities must be 1 1 BE k T

E

Ce1

BE k T

E

Ce

B

B

E k T

EE k T

E

Ee

e

BE k T

E

E CEeWhat do we do with

these sums over energy?

16$

What do we do with the sums?B

B

E k T

EE k T

E

e EE

e

•Energy can be anything•Replace sums by integrals?

0

0

B

B

E k T

E k T

e E dEE

e dE

Waves of varying strengths with

the same wavelength

2B

B

k Tk T

4

40

8

8

B

B

k Tu

k TU d

! The ultraviolet catastrophe

Bk T

Comparison Theory vs. Experiment:Theory

Experiment

B

B

E k T

EE k T

E

e EE

e

What went wrong?•Not truly in thermal equilibrium?•Possible state counting done wrong?•Sum Integral not really valid?

Max Planck’s strategy (1900):•Assume energy E must always be an integer multiple of frequency f times a constant h

•E = nhf, where n = 0, 1, 2, …•Perform all calculations with h finite•Take limit h 0 at the end

4

8u E

4

8 Bk Tu

Math Interlude:2 3 11

1x x x

x

Take d/dx of this expression . . .

2 1

20

10 1 2 31

n

n

x x nxx

Multiply by x . . .

2 3

20 1 2 31

xx x xx

0

n

n

x

Some math notation: exp xx eexp

exp

E B

E B

E Ek T

EE

k T

0

n

n

nx

Planck’s computation:E nhf f c

0,1, 2,n

0

0

exp

exp

n B

n B

hc hcnnk T

hcnk T

0

0

exp

exp

n

n Bn

n B

hcnk Thc

hck T

2

exp

1 exp1

1 exp

B

B

B

hc k T

hc k Thc

hc k T

1exp 1B

hchc k T

exp

exp

E B

E B

E Ek T

EE

k T

E nhc

From waves:

Planck’s Black Body Law

4

8u E

5

8 1exp 1B

hc uhc k T

Max Planck’s strategy (1900):•Take limit h 0 at the end•Except, it fit the curve with finite h!

34

15

6.626 10 J s

4.136 10 eV s

h

h

E nhf

Planck Constant

“When doing statistical mechanics, this is how you count states”

Total Energy Density 5

8 1exp 1B

hcuhc k T

0

U u d

50

8exp 1B

dhchc k T

Let = hc/xkBT

5

0

8exp 1

B

B

d hc xk Thc

hc xk T x

4 3

0

81

Bx

k T x dxhchc e

45

3

8

15Bk T

Uhc

4

15

Wien’s Law 5

8 1exp 1B

hcuhc k T

For what wavelength is this maximum?

0 d ud

26 5 2

exp5 1 10 8exp 1 exp 1

B

B B B

hc k Thchchc k T k T hc k T

exp0 5

exp 1B

B B

hc k Thck T hc k T

5 1 exp BB

hc hc k Tk T

4.96511B

hck T

32.8978 10 m K4.96511 B

hcTk

Planck constant

42

315Bk T

Uc

Often, when describing things oscillating, it is more useful to work in terms of angular frequency instead of frequency f

2 f E hf2

h

2h

This ratio comes up so often, it is given its own name and symbol. It is called the reduced Planck constant, and is read as h-bar

2h

Units of Planck constant•h and h-bar have units of kg*m^2/s – same as angular momentum

E

Photoelectric Effect: Hertz, 1887•Metal is hit by light•Electrons pop off•Must exceed minimum frequency

•Depends on the metal•Brighter light, more electrons•They start coming off immediately

•Even in low intensity

Met

al

e-

e-

e-

e-

Einstein, 1905•It takes a minimum amount of energy to free an electron•Light really comes in chunks of energy hf•If hf < , the light cannot release any electrons from the metal•If hf > , the light can liberate electrons

•The energy of each electron released will be Ekin = hf –

Photoelectric Effect•Will the electron pass through a charged plate that repels electrons?•Must have enough energy

•Makes it if:

Met

al

e-

+––+V

kinE eV

hf eV maxeV hf

f

Vmax

slope =

h/e

Nobel Prize, 1921

Sample ProblemmaxeV hf

When ultraviolet light of wavelength 227 nm strikes calcium metal, electrons are observed to come off which can penetrate a barrier of potential up to Vmax = 2.57 V.

1. What is the work function for calcium?2. What is the longest wavelength that can free electrons from calcium?3. If light of wavelength 312 nm were used instead, what would be the

energy of the emitted electrons?

We need the frequency:f c

cf

8

9

3.00 10 m/s227 10 m

15 11.32 10 s

maxhf eV 15 15 14.136 10 eV s 1.32 10 s 2.57 Ve

5.46 eV 2.57 eV 2.89 eV= Continued . . .

Sample Problem continued

maxeV hf

2. What is the longest wavelength that can free electrons from calcium?3. If light of wavelength 312 nm were used instead, what would be the

energy of the emitted electrons?

f c 2.89 eV •The lowest frequency comes from Vmax = 0 min0 hf

minfh 15

2.89 eV4.136 10 eV s

14 16.99 10 s

•Now we get the wavelength:cf

8

14 1

3.00 10 m/s6.99 10 s

74.29 10 m 429 nm

•Need frequency for last part:cf

8

9

3.00 10 m/s312 10 m

14 19.61 10 s

maxeV hf 15 14 14.136 10 eV s 9.61 10 s 2.89 eV 1.08 eV

X-rays•Mysterious rays were discovered by Röntgen in 1895

•Suspected to be short-wavelength EM waves•Order 1-0.1 nm wavelength

•Scattered very weakly off of atoms•Bragg, 1912, measured wavelength accurately

ddcos dcos

•Scattering strong only if waves are in phase•Must be integer multiple of wavelength

2 cosd m

Atom

The Compton Effect•By 1920’s X-rays were clearly light waves•1922 Arthur Compton showed they carried momentum

e-

e-

e-

Photon in

Photon out

•Conservation of momentum and energy implies a change in wavelength

1 coshmc

Meanwhile . . .

Photons carry energy and momentum, just

like any other particle