Exam Review MTE3Phys208

MTE 3 Wed Nov 28 5:30-7pm Ch 2103

! Alternate exams: Wed 6:30-8:00pm in Ch 2103 and Thu 5:30-7:00pm in the Lab room

! Contents:

! Ampere’s Law (32.6)

! Faraday’s Law (ch 33, no Inductors 33.8, no LC Circuits 33.9, LR Circuits 33.10)

! Maxwell equations (ch 34, no 34.2)

! EM waves (34.6-7)

! Polarization (34.8)

! Photoelectric effect (38.1-2-3)

! Matter waves and De Broglie wavelength (38.4)

! Atom (37.6, 37.8, 38.5-7)

! Wave function and uncertainty (Ch 39)

2

3

Ampere’s law

!

B• ds" = µoI

Example: field produced by an infinitelylong wire

!

B " ds = B2#a = µ0I$

Example: infinite solenoid

lBdsBdd1path 1path

=! ! !="=" sBsB

NIBdo

µ==! " lsB

Remember last lecture Faraday’s Law

4

!

" = E• ds# = $d

dt%

B= $

d

dtB# • dA

Magnetic flux through surface bounded by path

Integral over closed path

E is not conservative!!!

!

E• ds

Lenz law

If N loops with same area:

!

" = #Nd$

B

dt

changing the magnetic flux

5

1) Magnetic flux change: the magnitude of B changes in time

2) the area crossed by B lines changes with t (motional emf)

3) The angle ! between B and

normal to loop changes with t

flux increases flux decreases

12/09/2002 U. Wisconsin, Physics 208, Fall 2006 6

Answer

Code:

1

2

3

Ampere’s / Faraday’s & Lenz’s laws

X BB flux increasessince B = µ0I/(2"r)

and the loop goes towards smaller rvalues

Loop moving in a B-field and Lenz law

7

http://physics.bu.edu/~duffy/semester2/c19_quiz.html

� A rectangular loop of wire is pulled at a constant velocity from a region of

zero magnetic field into a region of a uniform magnetic field. When the

loop enters the field region the current induced in the wire

� A) will be zero

� B) will be ccw

� C) will be cw

v

Iind

x

!

I ="

R=d#

B

dt= BL

dx

dt= BLv

L

12/09/2002 U. Wisconsin, Physics 208, Fall 2006 8

Ampere-Maxwell’s law

ABCBE

9

The 4 Maxwell’s Equations

!

E " dA =q

#0

S$ (Gauss' Law) B " dA = 0

S$

E " ds = %d&B

dt (Faraday - Henry) B " ds

L$ = µ

0I +

L$ µ

0#0

d&E

dt (Ampere - Maxwell law)

! Currents create a magnetic field

! A changing electric field can create a magnetic field

charged particles create an electric field

An electric field can be created by a changing magnetic field

Consequence: induced current

There are no magnetic monopoles

Lorentz force

!

F = q(E + v "B)

• E and B are perpendicular oscillating vectors

•The direction of propagation is perpendicular to E and B

#

Facts on EM waves

! EM waves are solutions of Maxwell’s equations.

! In empty space: sinusoidal wave propagating along x with velocity

! E = Emax cos (kx – $t)

! B = Bmax cos (kx – $t)

E ! B = 0

E x B direction of c

Transverse waves

In many kitchens, a microwave oven is used to cook food. The frequency of the microwaves is of the order of 1010 Hz. The wavelengths of these microwaves are on the order of

(a) kilometers

(b) meters

(c) centimeters

(d) micrometers

Quick Quiz on EM spectrum

!

" = c / f =3#10

8m /s

1010/s

= 3cm

E-B relationship (important!)! You measure the amplitude of the electric field for an electromagnetic plane wave to be

E1, and the amplitude of the magnetic field to be B1. If the wave is adjusted so that the

amplitude of the electric field is now 2E1, the amplitude you will measure for the

magnetic field will be

! a) 2B1,

! b) 4B1,

! c) B1/2,

! d) B1/4,

! e) the amplitude of the magnetic field is independent of the amplitude of the electric field.

12

! (a). The amplitude of the magnetic and electric field are related through this equation:

(Emax/Bmax) = c. Therefore, if the electric field amplitude increases by a factor of two, so

must the magnetic field amplitude since c is constant.

Quick Quiz on EM waves

x

z

y

c

EB

Energy carried by EM waves

!

uE

=1

2"0E2 = u

B=B2

2µ0

=E2

2c2µ

0

! Total instantaneous energy density of EM waves

u =uE + uB = 1/2 %oE2 + B2 /(2µo)

! Since B = E/c and

In a given volume, the energy is shared equally by the two fields!

uE

= uB

!

Pav

=U

av

"t=uavAL

"t= u

avAc

Radiation properties

! This is the power per unit area (J/s.m2 = W/m2)

!

Iav

=Pav

A= u

avc

I & E2

E/B=c

!

prad =F

A=Power /A

c=I

c

! Complete absorption on a surface: total transported energy U in time interval 't ( total momentum p = U / c and prad=Sav/c

! Perfectly reflecting surface: momentum of incoming and reflected light p = U/c ( total transferred momentum p = 2U/c

and prad = 2Sav/c

Which of the following is constant for a plane electromagnetic wave?

(a) magnitude of the Poynting vector

(b) energy density uE

(c) energy density uB

(d) wave intensity

Quick Quiz on Poynting vector

Answer: (d). The first three choices are instantaneous values and vary in time. The wave intensity is an average over a full cycle.

!

uE

=1

2"0E2

!

uB

=B2

2µ0

!

S =EB

µ0

=E2

cµ0

Measuring Radiation Pressure: Quiz

High vacuum

In the apparatus in the figure, suppose the black disk is replaced by one with half the radius. Which of the following are different after the disk is replaced? (a) radiation pressure on the disk; (b) radiation force on the disk; (c) radiation momentum delivered to

the disk in a given time interval.

Answer:A. (a)B. (b)C. (b),(c)D. None

Answer

! (b), (c).

! (a) The radiation pressure P = S / c does not change because

pressure is force per unit area.

! (b) the smaller disk absorbs less radiation, resulting in a smaller force (F = P x A)

! (c) For the same reason, the momentum delivered to the disk in 't is reduced.

!

F ="p

"t

Polarization by selective absorption

transmission axis

Polaroid sheet

Long-chain hydrocarbon molecules

If linearly polarized light of intensity I0 passes through a polarizing

filter with transmission axis at an angle ! along y

Einc = E0sin! i + E0 cos! j

After the polarizer Etransm = E0cos! j

So the intensity transmitted isItransm = E0

2 cos2! = )0cos2!

y

x

!

E0cos!

If upolarized light of intensity I0 passes through it the intensity

becomes I0/2

Question on Polarization

Circularly polarized light of amplitude Ei crosses 3 polarizers.What is the ratio If/Ii?

Polarizer 1 passes the component along the transmission axis that has amplitude Ei. In the region between 1 and 2 the field oscillates up and down along the first polarizer axis and it is linearly polarized.After 2 only the component Eicos($2-$1) and after 3:Eicos($2-$1) cos($3-$2)) => If/Ii =1/2 cos2($2-$1) cos2($3-$2)What happens if light were linearly polarized initially?

How many green photons are in a flash of 't =1 ms of power

P = 4x10-14 Watt?

Green photons: # = 500 nm, E = hc/# = 2.5 eV

A. 10 photons

B. 100 photons

C. 1,000 photons

D. 10,000 photons

Number of photons and Intensity

•The intensity of the beam can be increased by increasing the number of photons/second.

•Photons/second = energy/second = power

N = Etot/E = P 't/E =

4 x 10-14 x 10-3 / (1.6 x 10-19 x 2.5) = 100

Work Function

! Max kinetic energy of electrons ejected from the surface of the metal and not making collisions with other metal atoms before escaping

Kmax = hƒ – !

! ! is called the work function and depends on metal

! minimum energy needed to free electron from the substance.

! Silver: 4.26 eV! Alluminium: 4.28 eV! Copper: 4.65 eV! Gold 5.1 eV cutoff frequency is

ƒc = ! / h

Particle wavelength

23

An electron and a proton both moving at nonrelativistic

speeds have the same de Broglie wavelength. Which of the

following are also the same for the two particles?

(a) speed

(b) kinetic energy

(c) momentum

(d) frequency

!= h/pe = h/pp => pe = pp

if pe = pp since me " mp also ve " vp

if ve " vp => Ke " Kp

if ! is the same and ve " vp => f = v/! is different

If a proton is accelerated in #V = 10kV which is the de Broglie wavelength? mp =

1.67 x 10-27 and mc2 = 938 MeV and notice e#V = 1.6 x 10-19 x 10x103 J but

convenient to use 10 keV!! (1 eV = 1.6 x 10-19 J)

!

" =hc

2mc2e#V

=1240eVnm

2 $ 938 $106eV $10 $10

3eV

= 2.9 $10%4nm

A stone is dropped from the top of a building.

What happens to the de Broglie wavelength of the stone as it falls?

1. It decreases

2. It stays the same

3. It increases

KE=mv2/2 = mgh speed increases from 0 to v = sqrt(2gh) momentum, p=mv, increases.

De Broglie Question

25

Bohr’s model of H-like atoms

! Electron in circular allowed orbits around nucleus with

! Integer number of de Broglie wavelengths must fit on circumference of the orbit:

!

This is quantization angular momentum (L=mvr) and comes from de Broglie!!

2"r = n# = nh

p= n

h

mv

!

mvr = nh

!

L = nh

!

r =n2

Za0

!

E =p2

2m"kZe

2

r# E = "

1

2

k2Z2me

4

n2h2

= "13.6eVZ2

n2

26

Question

! How much longer is the electron wavelength in the n=2 state compared to n = 1?

!

"2/"

1=1

!

"2/"

1=1.5

!

"2/"

1= 2

!

"2/"

1= 4

A.

B.

C.

D.

!

2"r = n# = nh

p= n

h

mvr1=ao, r2=4ao

Circumference is 4 times as long.

Twice as many wavelengths -> wavelength twice as long

!

2"a0

= #1

8"a0

= 2#2

27

Compare the wavelength of a photon produced from a transition from n=3 to n=1 with that of a photon produced from a transition n=2 to n=1.

Spectral Question

n=2

n=3

n=1

A. #31 < #21

B. #31 = #21

C. #31 > #21

E31 > E21 so #31 < #21

Wavelength is smaller for larger

jump!

!

"E = hf =hc

#

Let’s demonstrate E= -13.6eVZ/n2

28

!

F = kZe

2

r2

= mv2

r

mvr = nh

!

kZe

2

r= mv

2

Total energy = kinetic energy + potential energy = kinetic energy + eV

!

r = kZe

2

mv2

!

v =nh

mr

Substitute (2) in (1):

(1)

(2)

!

r = kZe

2m2r2

mn2h2" r =

n2

Z

h2

me2k

=n2

Z

(1.055 #10$34)2

9.11#10$31# (1.6 #10

$19)2# 9 #10

9=n2

Z0.53#10

$10m

!

r =n2

Za0

!

E =p2

2m" k

Ze2

r=mv

2

2" k

Ze2

r= "k

Ze2

2r= "k

Ze2

2#Zme

2k

n2h2

= "Z2

n2

k2me

4

2h2

= "13.6eVZ2

n2

!

E = "Z2

n2

(9 #109)2# 9.11#10

"31# (1.6 #10

"19)4

2 # (1.055 #10"34)2

= "Z2

n22.2 #10

"18J = "

Z2

n213.6eV