What kinds of forces can we have in relativity?
•Instantaneous action at a distance?
kqQrˆ
F 2
r
GmMrˆ
F 
r2
These are no good because instantaneous effects violate relativity
•No problem, we already know about electric and magnetic fields
F  q  qu  
Q
q
Electric force
Magnetic force
Gravity is very hard:
Einstein’s General Theory of Relativity
Newton’s Laws and Force
Newton’s Second Law in non-relativistic physics:
dv d
dp
F
  mv  
F  ma  m
dt dt
dt
Newton’s Third Law:
Relativity
F1  F2
dp1 dp2
d

  p1  p2  ,
0  F1  F2 
dt
dt
dt
p1  p2  constant
Conservation of momentum comes from Newton’s Third Law and One
Version of Newton’s Second Law
dp
F
dt
F  ma
0  F1  F2
Cyclotron Motion
Consider a charged particle moving in a magnetic field:
•Magnetic field points out of the slide
du
dp d
F  qu   
  m u   m
dt
dt dt
•Force keeps speed the same
• is constant
u
q
F
•Centrifugal acceleration:
R

quB  m  u 2 R 
qBR  m u
p  qBR
du u 2

dt
R
B in Tesla
R in meters
q in Coulombs
p in kgm/s
Work
E  c p   mc
2
2
2

2 2
•Take time derivative of the first one
•Solve for dE/dt
•Substitute the second
dE
•Substitute the third

•Rewrite the velocity
dt
•Integrate over time
u pc

c E
dp
F
dt
dE
dp
2
2E
 2c p 
0
dt
dt
dp
dr
c 2 p dp
u

uF  F
dt
dt
E dt
W  E 
 dE   F  dr
W  E  F  d
E  F  d
Sample Problem
F  dp dt
An electron (m = 511.0 keV/c2) at rest is placed in an electric field of
magnitude 100.0 V/cm. How long does it take, and how far does it go,
before it reaches a velocity of v = 0.500c?
F  q  10 eV / m  10 keV / m
4
•Work formula:
 i  1,  f 
1
1  0.5002
Fd  E    f   i  mc 2  0.155mc 2
0.155  511.0 keV 
 7.94 m
d
10 keV/m
•Momentum formula:
Ft  p  p f  0   f mu  0.577mc
0.577  511 keV 
0.577mc 2 
 98.4 ns
t
8
10 keV/m   3.00 10 m/s 
Fc
 1.155
Composite Objects and Invariant Mass
Suppose I have a box containing two objects of mass m
moving with equal and opposite velocities u. What
does momentum and energy look like for this whole
object?
u
m
m
Etot  E1  E2   mc 2   mc 2  2 mc 2
ptot  p1  p2   mu iˆ  iˆ  0


•Looks exactly like a particle at rest of mass M = 2m.
•If you Lorentz boost this object, the Energy/momentum will transform exactly the
same way as a single object.
•If we can’t see inside this object, we can’t tell it’s not a single object with this
mass.
Mass of composite object
M  2 m
M  2m
is not the sum of its parts
u
Finding the Invariant Mass
Suppose I have a box containing many objects of
various masses, moving at various velocities.
What mass object M can have the same
momentum and energy as the whole mess?
Etot   mi i c 2  E
m1 u1
u2
m2
u4
m4
E 2  c 2 P2  M 2c 4
i
ptot   mi i ui  P
i
 Mc

2 2
2
2
 Etot
 c 2 ptot
Effective Mass
u3
m3
Internal Energy and Heat
Suppose I have a (solid) containing many atoms.
Now I heat it up. Does the mass change?
•The atoms start to move around
Heat
•This increases the energy E of each atom
•But the total momentum is still 0
•The total energy E of the whole object increases
•The invariant mass of the whole object increases
Na Cl Na Cl Na Cl
Cl Na Cl Na Cl Na
Na Cl Na Cl Na Cl
Cl Na Cl Na Cl Na
1 kg of water (specific heat 4.184 J/g/C) is heated from
0C to 100C. How much does the mass increase?
E  103 g  100C  4.184 J/g/C   4.184 105 J
E  mc 2
4.184 105 J
E
 4.655 ng
m  2 
2
c  3.00 108 m/s 
Field Energy and Mass
Suppose I have a charged particle surrounded by
electric fields. Does the field contribute to the mass?
q
•Electric fields have energy
•Electric fields contribute to total energy
•Electric fields contribute to total mass
•The mass listed for a given particle includes this mass
Consider a hydrogen atom
•Proton and electron have cancelling charges
p
e
•Partly eliminates the electric field
•Decreases total energy
Mass(H) < Mass(p) + Mass(e)
•Decreases invariant mass
Binding energy counts like negative mass
Potential energy
Any change in the potential energy of
an object changes its mass
•Heat
2
E

mc
•Electric energy
0
•Chemical energy
•Nuclear energy
•[Gravitational energy is hard]
There is nothing particularly special
about nuclear energy
•Other types of energy are too little to
significantly affect the mass
•For nuclear energy, calculating the
mass difference may be the easiest way
to find the total energy produced.
p   mu
dp
F
dt
Summary: Formulas you need
u pc
E   mc
E  c p   mc 

c E
2
E0  mc 2
W  E  F  d
2
Massless:
E c p
u c
2
2 2
2
E  c p   Mc
2
tot
2
2
tot

2 2
px    px  vE c 2 
p  qBR
F  q  qu  
py  p y
pz  pz
E     E  vpx 
What’s the name of this Song?
This day and age we're living in
Gives cause for apprehension
With speed and new invention
And things like fourth dimension.
“When a man sits with a pretty girl for an hour,
it seems like a minute. But let him sit on a hot
stove for a minute and it's longer than any hour.
That's relativity.”
Attributed to A. Einstein
Yet we get a trifle weary
With Mr. Einstein's theory.
So we must get down to earth at times
Relax relieve the tension
And no matter what the progress
Or what may yet be proved
The simple facts of life are such
They cannot be removed.
“As Time Goes By”
Music and Lyrics by
Herman Hupfield
End of material
for Test 1