The Einstein mass-energy relation
In 1905 Albert Einstein published his
Special theory of relativity. One of the conclusions of this theory is that matter and energy are
equivalent (see below). In fact the history of the Universe is a good example of this. In the early
stages of development of the Universe there was a great deal more energy in the Universe and
this has been converted into matter as time has passed.
Matter and energy are related
by the famous equation proposed by Einstein:
The quantity E is the amount of
energy produced when a mass m is completely converted to energy and c is the speed of light
(3x10
8 ms
-1).
For example if a mass of say 1 kg of any matter
could be converted into energy a huge amount of energy could be produced. By Einstein's
equation:
E = 1 x [3x10
8]
2 = 9x10
16 J
This amount
of energy (equivalent to 2.5x10
10 kWh) would heat each of the 10 000 houses in a
medium sized town with a 1 kW fire for nearly 300 years! Think of how much useful energy just
by being able to convert household waste into energy.
Unfortunately no means has yet
been devised for doing this on a large scale. However the conversion of mass to energy and
energy to mass can be carried out at a nuclear level. Pair production (energy to matter) and
annihilation (matter to energy) have both been observed.
(See: 16-
19/Nuclear physics/Nuclear structure/Text/Antimatter)
If we write m
o as the rest mass
of the object then E = m
ogc
2 where
g= (1 - v
2/c
2)
1/2
Relativity and the
equivalence of mass and energy
The Einstein equation (E = mc
2) can be
deduced from special relativity as follows:
m = m
o/[1 - v
2/c
2]
1/2
therefore m
2c
2 – m
2v
2 – m
oc
2 = 0 where m
o is the rest
mass of the object.
Differentiating with respect to time gives: c
2 2m[dm/dt] –
2mv[d(mv)/dt] = 0
Rearranging this equation gives: c
2 2m[dm/dt] =
2mv[d(mv)/dt]
and therefore c
2[dm/dt] = v[d(mv)/dt]
However dE/dt = Fv
and so dE/dt = Fv = v[d(mv)/dt] = c
2dm/dt
And so E =
mc
2
