Download Study Guide For Final Exam

Study Guide For Final Exam
Physics 2220 – Albretsen
Updated: 04/12/2016
ALL PREVIOUS STUDY GUIDES SINCE THE FINAL IS COMPREHENSIVE
Module 16
The Principle of Relativity
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To describe an event, it is necessary to establish a frame of reference
Recall Newton's Laws are valid in all inertial frames
• Inertial frame defined as one in which Newton's First Law is valid – free body
experiences no acceleration
• Any system moving with constant velocity with respect to an inertial system must also
be an inertial system
There is NO preferred frame
• Results of an experiment performed in a frame with constant velocity will be the same as
the same experiment done in a stationary frame
Principle of Newtonian Relativity
“The laws of mechanics must be the same in all inertial frames of reference”
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Example – Pickup truck moving at constant velocity – throw ball up
Consider observer on truck and observer at rest with respect to the truck
Observers disagree on certain aspects of the experiment, however they agree on validity of:
• Newton's Laws
• Conservation of Energy
• Conservation of Momentum
No mechanical experiment can tell the difference
The notion of absolute motion in space is meaningless
Say an event occurs in an inertial frame
• Location and time of event described by (x,y,z,t)
• Want to transform these coordinates to another frame
Consider to frames, S and S'
• S' moves with constant velocity v along the x-axis and x'-axis relative to S
• Event occur at point P and origins of S' and S coincide at t = 0
• S (x, y, z, t) – at rest
• S' (x', y', z', t') – moving
Related by Galilean Transformation of coordinates
x ' = x − vt
y '=y
z '=z
t ' =t
•
Galilean Addition Laws of Velocity
• ux = Instantaneous velocity in S frame
• u'x = Instantaneous velocity in S' frame
u ' x = ux − v
The Speed of Light
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Galilean transformations contradicts with EM waves
Assume Maxwell's Laws are the same in all inertial frames
• Paradox: c = speed of light always
• Contradicts what is expected based on Galilean law – speed of light not the same in all
inertial frames
So either:
• Maxwell's Laws are not the same for all inertial frames, or
• Galilean Law of addition for velocities is incorrect
Einstein's Principle of Relativity
1. All the laws of physics are the same in all inertial reference frames
2. The speed of light has the same value in all inertial frames, regardless of the velocity of the
observer or the velocity of the source emitting the light
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No preferred inertial reference frame exists
It is impossible to detect absolute motion
Be prepared for some rather bizarre consequences
Simultaneity and the Relativity of Time
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Two events that are simultaneous in one reference frame are in general not simultaneous in a
second frame moving relative to the first
Simultaneity is not an absolute concept, but one that depends upon the state of motion of the
observer
Time Dilation
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Δt = Time interval measured by an observer moving with respect to a clock (event)
Δt' = Time interval measured by an observer at rest with respect to a clock (event)
• Proper Time
 is always greater than unity
Δt > Δt'
“A moving clock runs slower than a clock at rest by a factor of “
1
 t'
=
 t=
=  t '
v2
v2
1
−
1− 2
c2
c


Length Contraction
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Lp = Proper length – length of the object measured by someone who is at rest relative to the
object (distance)
In an object has a length Lp when at rest, then when it moved with a speed v in a direction
parallel to its length, it contracts to length L
L = Lp
•

v2 Lp
1− 2 =

c
Length contraction takes place only along the direction of motion
Lorentz Transformation Equations
•
Replaces Galilean transformation equations
x ' =   x − vt 
y '=y
z '=z
v
t '= t− 2 x
c


x =  x '  vt ' 
y= y '
z=z '
v
t = t '  2 x '
c


Lorentz Velocity Transformation Equations
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Replaces Galilean Addition Laws of Velocity
u 'x =
ux − v
u v
1− x2
c
ux =
u 'x  v
u' v
1  x2
c
Relativistic Momentum
Generalize Newton's Laws and define momentum to conform with Lorentz transformations and
relativity
m ux
px =
=  m ux
u2x
1− 2
c
Relativistic Energy
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
•
Generalize Newton's Laws and define energy to conform with Lorentz transformations and
2
relativity
mc
K=
− mc 2 =  mc2 − mc2
Kinetic energy(K):
2
u
1− 2
c
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mc2 = Rest energy (independent of speed)
mc2 = Sum of kinetic and rest energies = Total energy (E)
•

E=
m c2

1−
u2
c2
=  mc2
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Shows mass is a property of energy
Energy – Momentum relationship:
E2 = p2 c 2  mc2 2
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For particles of zero mass (photons):
E = pc
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Remember:
−19
1 eV = 1.60 × 10
J
Module 17
Introduction to Quantum Mechanics
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Applying laws of classical physics to explain behavior of matter at an atomic scale fails for:
• Blackbody Radiation
• Photoelectric Effect
• Spectral Lines by Atoms
Revolution in physics between 1900 and 1930 – Quantum Mechanics
The Photoelectric Effect
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Light incident on certain metal surfaces causes electrons to be emitted from those surfaces
Using a photoelectric apparatus – Classical Physics cannot explain:
1. No electrons are emitted if frequency of light is below a cutoff frequency fc, classical
predicts the effect should occur at any frequency
2. Maximum kinetic energy of photoelectrons is independent of light intensity
3. Maximum kinetic energy of photoelectrons increases with increasing frequency
4. Electrons are emitted almost instantaneously – classically it should take some time to
absorb radiation to acquire enough kinetic energy
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Explained by Einstein in 1905 by extending Planck's concept of quantization
• Assumed light of frequency f as a stream of photons, each with energy E = hf
• Photon is so localized it gave all its energy to a single electron
• Maximum kinetic energy of the electron is:
K max = hf − 
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 = work function of the metal = minimum energy with which the electron is bound
1. Energy of the photon must be equal to or greater then the work function. If not, the
electron is not ejected
2. Kinetic energy depends only on light frequency and the work function, not the intensity
3. Equation shows why Kmax increases as f increases
4. Energy is in small packets and there is a one-on-one interaction
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Wavelengths greater than c do not result in emission of photoelectrons
c =
c hc
=
fc 
The Uncertainty Principle
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Quantum Theory predicts it is fundamentally impossible to make simultaneous measurements
of a particle's speed (momentum) and position with infinite accuracy
ℏ
 x  px ≥
2
ℏ
 E t≥
2
Module 18
Blackbody Radiation
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An object at any temperature emits radiation (light) known as thermal radiation
• As temperature of an object increases, thermal radiation emits a continuous distribution
of wavelengths from IR, visible, and UV
Classical viewpoint
• Thermal radiation originates from accelerated charges near surface
• Distribution of accelerations account for continuous spectrum
Problem by end of 19th century
• Blackbody
• Ideal object that absorbs all radiation incident on it
• Only light seen coming from blackbody is thermal radiation
• Experimental vs. Theoretical data did not agree
• Experimental
• Radiated energy varies with wavelength (λ) and temperature (T)
• As temperature increases, amount of emitted energy increases
• Peak of distribution (λmax) shifts according to Wien's Law:
−2
 max T = 0.2892 × 10
•
• Classical theories fail to explain the blackbody curve
Classical spectrum radiation equation – Rayleigh-Jeans Law
I  , T  =
•
mK
2  ck B T
4
• At long wavelengths, Rayleigh-Jeans agrees with experiment
• At short wavelengths, radiation from any blackbody should become infinite
In 1900, Planck discovered a formula that was in agreement with observations
2  hc 2
I  , T  =
5

 e
hc
 kB T
−1

• h = Planck's Constant = 6.626  10-34 Js
1. Makes to assumptions concerning the nature of the oscillating molecules:
• Molecules can only have discrete units of energy
En = nhf
• n = positive integer called a quantum number
• f = frequency of the vibration of the molecules
2. Molecules emit or absorb energy in discrete packets called photons
• Molecules emit / absorb photons by “jumping” from one quantum state to
another (ex. n=3 to n=2)
Atomic Spectra
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All substances at a given temperature emit thermal radiation in a continuous spectrum
Can also have
• Emission Line Spectra
• Absorption Line Spectra
Properties of spectra
• Wavelengths in a given spectra is characteristic of the given element
• Each element has a different spectra
Formula discovered by Balmer in 1885 which correctly predicts wavelengths of four visible
emission lines for hydrogen
1
1
1
= RH 2 − 2
n = 3, 4, 5, ...
(Balmer Series)

2
n

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
• RH = Rydberg Constant = 1.0973732  107 m-1
Other line series:






1
1
1
= RH 2 − 2

1
n
1
1
1
= RH 2 − 2

3
n
1
1
1
= RH 2 − 2

4
n
n = 2, 3, 4, ...
(Lyman Series)
n = 4, 5, 6, ...
(Paschen Series)
n = 5, 6, 7, ...
(Bracket Series)
Bohr's Quantum Model of the Atom
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Basic Ideas:
• Electrons more in circular orbits around the proton under the influence of the Coulomb
force of attraction
• Only certain orbits are stable. Stable orbits are ones in which the atom does not emit
energy in the form of radiation. Total energy of atom remains constant
• Radiation is emitted by the atom when the electron “jumps” from a more energetic orbit
to a lower orbit. Frequency of the photon emitted is related the the change in the atom's
energy
E − E = hf
i
f
Size of the allowed electron orbits is determined by a condition imposed on the
electron's orbital angular momentum
mvr = n ℏ
n = 1,2,3. ...
h
ℏ=
2
Quantized orbits:
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rn =
•
n 2 ℏ2
mk e e2
n = 1,2,3. ..
For n =1, define the Bohr Radius (a0)
2
a0 =
•
ℏ
= 0.0529 nm
2
mke e
Quantized energy:
2
 
k e 1
En =− e
2 a0 n2
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n = 1,2,3. ..
n = 1, Ground State
n = 2, First excited state
Energy between states, in terms of wavelength:
k e2 1
1 f
1
= = e
− 2
2
 c 2a 0 hc n f ni

•

Model can be extended to other elements where all but one electron is removed
• Z = atomic number of the element
a
r n = n2  0
n = 1,2,3. ..
Z
ke e2 Z2
En =−
n = 1,2,3. ..
2 a 0 n2
 