Download E=mc squared - UMD Physics

Something out of nothing:
E=mc squared
What we’ve learned…
…that length and time are not absolute but depend on the point of view of the observer
T'
T
1 v
L
2
Lp
1 v
c2
time dilation
2
c2
length contraction
…that the speed of light is absolute and is the same for all observers in all reference
frames.
…that length and time are inextricably linked in order to keep the speed of light the
same in all reference frames.
…that this relationship between length and time can be represented in a four
dimensional space called “spacetime”.
…that we can transform the spacetime coordinates in one frame to another moving
along the x axis at speed v using the following relations.
x   ( x  vt)
y  y
 v2
z   z t     t  2
 c

1
x  where  
1  (v 2 / c 2 )

…that spacetime preserves the “distance” between two points.
s  ct  x   c t2  t1  x 2  x1
2
2
2
2
2
s
2
 s
2
…that from the spacetime coordinates we can
derive a relationship between the velocity in
one frame and the velocity in a another
ux 
ux  v
1  (ux v / c 2 )
We’ve skirted this issue of a speed limit
(v< or =c)
Okay, but what does this “speed” limit
do to Newton’s laws of motion??
To the conservation of energy?
Constant force applied - leads to acceleration - spaceship goes faster. As
it’s speed increases it begins to resist acceleration.
F=ma…if F is constant, does that mean the mass of the spaceship is increasing?
Is momentum really conserved?:
In a frame S:
v
v
Sv=0
pbefore  mv  m(v)  0
pafter  0
In a frame S’ moving to the right at speed v:
v1  0
v2
V
First use relativistic velocity transformation:
v1
v1  v
v v

0
2
2
1 (v1v /c ) 1 (v)(v) /c 
v 
2 
v2  v
v  v
2v


1 (v 2v /c 2 ) 1 (v)(v) /c 2  1 (v 2 /c 2 )
V' 
V v
0v

 v
1 (Vv /c 2 ) 1 (0)(v) /c 2 
ux 
ux  v
1  (ux v / c 2 )
pbefore 
 2mv
1  (v 2 / c 2 )
pafter  2mv
A gedankenexperiment:
In the rest frame of A:
B
A
In the rest frame of B:
B
A
A is at rest, B comes along from the
right at a significant fraction of the
speed of light, and in a glancing
collision, imparts a tiny fraction of
it’s momentum to A (so it hardly
slows down) and A (relatively slowly)
rolls off in a direction perpendicular
to the incident momentum.
B is at rest, A comes along from the
left at a significant fraction of the
speed of light, and in a glancing
collision, imparts a tiny fraction of
it’s momentum to B (so it hardly
slows down) and B (relatively slowly)
rolls off in a direction perpendicular
to the incident momentum.
If velocity appears smaller by a factor of  then the mass must appear larger by a
factor of !
Energy apparently is being assimilated into the mass of the object…
The mass of an object is it’s own rest frame is called the proper mass of the
object, or more often, the rest mass.
Relativistic momentum:

p
F

mu
1  (u 2 / c 2 )
du 
m 
dt 
dp

dt 1 (u2 /c 2 ) 3 2


Acceleration of a particle decreases under the action of a constant force, as we
observed
it would at the beginning of the lecture.

Relativistic energy
The change in the kinetic energy of an object is equal to the net work done on the object.
W

Insert:
x2
x1
Fdx 
F

x2
x1
dp
dx
dt
du 
m 
dt 
dp

dt 1 (u2 /c 2 ) 3 2


Noting that dx=udt
du 

m 
x2
u
udu
dt 
W  x

m

3
0
1
2
2
2
1 (u /c ) 2
1 ( u 2 )
c

W  KE 
mc2
2
1  (u
 mc2
c2
)

3
2
Note that there is a term
that is independent of the
speed…the rest energy!
This term comes from the lower edge of the integration
interval, u=0, it had energy before it started to move!
"It followed from the special theory of relativity that
mass and energy are both but different
manifestations of the same thing -- a somewhat
unfamiliar conception for the average mind.
Furthermore, the equation E is equal to m c-squared,
in which energy is put equal to mass, multiplied by
the square of the velocity of light, showed that very
small amounts of mass may be converted into a very
large amount of energy and vice versa. The mass and
energy were in fact equivalent, according to the
formula mentioned before. This was demonstrated by
Cockcroft and Walton in 1932, experimentally."
Mass Energy Equivalence
W  KE 
mc2
2
u
1 (
 mc2
c2
)
We found that an object has energy
while at rest.
The total energy therefore includes this “rest energy”:
E
mc2
2
1  (u
 mc2
c2
)
Let’s think of a classical analogy-you can convert potential energy into kinetic
energy (i.e. gravitational potential when you pedal up a hill)-you can convert
electrostatic potential into electric power (i.e. when you charge, then discharge a
capacitor).
Does this suggest that energy can be converted into mass, or that high energies
can make mass materialize?
The experimental proof….
In Paris in 1933, Irène and
Frédéric Joliot-Curie took a
photograph showing the
conversion of energy into mass.
A quantum of light, invisible
here, carries energy up from
beneath. In the middle it
changes into mass -- two
freshly created particles which
curve away from each other.
A convenient energy unit, particularly for atomic and nuclear processes, is the
energy given to an electron by accelerating it through 1 volt of electric potential
difference. The work done on the charge is given by the charge times the voltage
difference, which in this case is:
The abbreviation for electron volt is eV.
Room temperature thermal energy of a molecule............................……....0.04 eV
Visible light photons....................................................................................1.5-3.5 eV
Energy for the dissociation of an NaCl molecule into Na+ and Cl- ions:....4.2 eV
Ionization energy of atomic hydrogen ........................................................13.6 eV
The masses of elementary particles are frequently expressed in term of electron
volts by making use of Einstein's famous equation , where m is the mass of the
particle and c is the speed of light.
Real world
modern
physics!
The top
quark has a
mass of 175
GeV. (as
much as a
gold atom!)
They are
not stable,
but decay
almost
instantly, so
they cannot
be found in
nature. So
how do we
produce
them?
You can
accelerate a
proton which
has a mass of 1
GeV (only
1/175 the rest
mass of the top)
to a kinetic
energy of 1000
GeV and slam it
head on with an
anti-proton-and
you have
enough energy
to produce a
top quark!
“It's as if two tennis balls collided and a bowling ball flew out… “
Grave applications: the dawn of the nuclear
age…
difference of ~208 MeV
Is relativistic energy conserved?
We’ve convinced ourselves that relativistic momentum is conserved.


mu
pafter  pafter
p
2
2
1  (u / c )
E
mc2
2
1  (u
 mc2
c2
)
E  p c  (mc )
2
2 2
2 2
If momentum is conserved then
energy must be conserved.
Okay, now you’re convinced that relativistic energy is conserved, so here’s yet
another paradox…if a light particle (photon) can turn into an electron and a
photon and vice versa, how do you reconcile that with the fact that the photon
is massless??
Tune in next time when we talk about….
…the quantum theory of light.