SPECIAL
RELATIVITY - The Equivalence of Mass and Energy
Empirical observation
in the 19th century led to the conclusion that although energy can be transformed,
it cannot be created or destroyed. This concept, known as the conservation
of energy, constitutes one of the basic principles
of classical mechanics. The principle, along with the parallel principle
of conservation of matter, holds true only
for phenomena involving velocities that are small compared with the
velocity of light. At higher velocities close to that oflight, as in nuclear
reactions, energy and matter are interconvertible .Einstein’s celebrated
relation
E
= mc2
providing the link . In
modern physics the two concepts, the conservation of energy and of mass,
are thus unified.
When two particles
are in a bound state under an interaction,for example the Electromagnetic,Gravitational
or Nuclear forces,the Total Energy , E for the system is negative.If this
seems
strange then you might like to think of a bound system as being one in
which work has to be done to separate the constituents.
As usual the Potential
energy Ep is defined to be zero at infinite distance from the field source
and is Negative at any finite distance from the source for a body in a
bound state.If Kinetic Energy Ek is positive, then for ANY system
in ANY BOUND STATE we have:
E = Ep + Ek
< 0 with ( l Ek l < Ep )
With this in mind we
now introduce the concept of BINDING ENERGY , B. The absolute value of
the total energy of the bound system being equal to B. The binding energy
may be converted to mass by use of E= mc2 with the new concept
of MASS DEFECT , Dm
thus being calculated from
where c is the speed
of light. and E = - B < 0
It is shown
elsewhere that for a Single Unbound particle that its Total
Energy E is given by
where mo
is the Rest mass of the particle.This relationship is compounded
from the rest energy
and any kinetic energy
the particle might have.
Extending to a system
of N particles in a bound state this becomes:
QUESTIONS
1) The above equation
applies to all particles in a bound system . Show that it reduces
to
U = mc2
for a single particle
at rest at infinity and to the familiar 1/2mv2 for the kinetic
energy of a single unbound particle.
2) Show that for speeds
considerably
less than that of light that
Thus we see the total
energy as the sum of rest mass-energy, kinetic energy and potential energy.
3) Thus show
that if E = Ek + Ep = - Dmc2
that
we have
4) Explain what this
last equation means physically, remembering thatDMis
the 'mass defect'.
APPLICATION
We can now apply this
theory to particular physical systems and try to determine whether the
mass defect can be physically observed.
We proceed by analysing
the kinetic and potential energy of two systems,
calculating the binding
energy for each and thus the mass defect.By ratioing the mass defect to
the mass of the bound system i.e DM
/ Mbwe
can see whether the defect is large enough to be measured.
|
? |
The two systems in
question are the Earth and Sun as an isolated bound system and likewise
the hydrogen atom as a bound electron and proton.A number of simplifying
assumptions are made and classical physics is used throughout.
HYDROGEN
ATOM
1) As usual the potential
energy of the lone electron is given by the formula
From this and using
Newton's second law for its motion, show that the total energy is equal
to HALF this value.
2) Show that if the
radius of the hydrogen atom is 0.53 x 10-10 m then the binding
energy B is approximately 14eV and the mass defect is 2.5 x 10-35
kg.
3) Hence show thatDM
/ Mbis
roughly 1.5 x 10-8. Comment on the physical significance of this.
SOLAR
SYSTEM
1) Show that the Binding
energy is given by B = GMm /2 r
2) Hence show that
the mass defect is given by DM = GMm / 2rc2
3) If one astronomical
unit is approximately 1.5 x 10 11 m then show that
B = 2.6 x 1033 J
and DM =2.9
x 1016kg
4) Hence show that
the ratio of the mass defect DM
to the solar
mass Mb
is 1.5 x 10-14.
Now consider the following
table for DM
/ Mb and comment on why only
the mass defect for the Nuclear Force is actually observed.
|
EGGLESCLIFFE SCHOOL
PHYSICS DEPARTMENT
|
|
RETURN TO RELATIVITY
| RETURN TO HOME PAGE
