Special Relativity…continued, Diffraction
Grating Equation and Photo-electric effect
•Relativistic Doppler Shift
•Relativistic Momentum and Energy
•Diffraction Grating Equation
•Photo-electric effect
•Homework Hints…
Relativistic Doppler Shift
• Doppler Shift for Sound,
vs=speed of sound
vr=radial velocity of source
• Relativistic Doppler Shift
– Time Dilation
– Varying distance
n rest 1- u 2 /c 2 n rest 1- u2 /c 2
n obs =
=
1+ (u /c)cos q
1+ v r /c
v r = ucosq
For =0 or 180, then:
n obs = n rest
1- v r /c
1+ v r /c
(radial _ motion)
Transverse Doppler Shift =90
n
n obs = rest
g
Relativistic Doppler Shift
Redshift/Blueshift parameter z
c = ln
lobs = lrest
z=
1+ v r /c
1- v r /c
1+ v r /c
1- v r /c
(radial motion)
-1
(radial motion)
Measure redshift  Measure recession velocity!!!
Relativistic Doppler Shift
Redshift/Blueshift
Relativistic Velocity Transformations
v x¢ =
vx - u
1- uv x /c 2
2
2
v
1u
/
c
v¢y = y
1- uvx / c 2
v z 1- u /c
¢
vz =
2
1- uv z /c
2
2
Relativistic Headlight Effect
example 4.3.3
Relativistic Momentum and Energy
mv
Momentum
p=
Kinetic Energy
K = mc (g -1)
Total Energy
E = gmc
Rest Energy
Erest = mc
1- u 2 /c 2
2
Momentum Energy Relation
E = p c +m c
2
2 2
2 4
2
2
= gmv
Relativistic Momentum and Energy
The Derivation of E=mc2
Relativistic Momentum and Energy
The Derivation of E=mc2
Relativistic Momentum and Energy
Newtonian Mechanics “breaks down”
at high speed v-->c.
F = ma
does not hold.
F=
However
dp
dt
does still hold if the momentum
becomes….
p=
mv
1- u 2 /c 2
= gmv
Derivation pages 105-106
Four-Vectors
Space-time four-vector
Invariant length
Invariant length
Lorentz Transformation
éct ¢ù é g
ê ú ê
ê x ¢ú = ê-bg
ê y ¢ú ê 0
ê ú ê
ë z¢ û ë 0
Energy-Momentum four-vector
-bg
g
0
0
ùéctù
úê ú
0 0úê x ú
1 0úê y ú
úê ú
0 1ûë z û
0
Lorentz Transformation
Spectral Lines
Application of Spectral Measurements
• Stellar Doppler Shift
• Galactic Doppler Shifts
• Quasar Doppler Shifts
Radial Velocities
Spectral Lines
Spectrographs
• Spectroscopy
• Diffraction grating equation
dsinq = nl
(n=0,1,2,…)
n = order
• Resolving Power
Photoelectric Effect
Classical Expectations
•
•
•
Kinetic energy of ejected electrons should depend on strength of electric field
and therefore intensity of light and not the number of ejected electrons.
Maximum kinetic energy of ejected electrons should not depend on frequency
of light.
Any frequency light should be capable of ejecting electrons.
Observations
•
•
•
•
Kinetic energy of ejected electrons does not depend on intensity of light!
Increasing intensity will produce more ejected electrons.
Maximum kinetic energy of ejected electrons depends on frequency of light.
Frequency must exceed cutoff frequency before any electrons are ejected
Photoelectric Effect
Einstein took Planck’s assumption of quantized energy of EM waves seriously.
Light consisted of massless photons whose energy was:
E photon = hn =
hc
l
K max = E photon - F = hn - F =
hc
l
-F
Einstein was awarded the Nobel Prize in 1921 for his work on the photo-electric
effect
Photo-electric Effect
Inertial reference frame
Remember that the clocks are located at every point in space
Example 4.3.2 useful
Time Dilation…Light Clock
Worked Problems
Worked Problem