Electrostatic Potential

Electrostatic Potential


This is a colossal number! Suppose we were studying a physics problem
involving the motion of particles under the action of two forces with the
same range, but differing in magnitude by a factor 1042. It would seem
a plausible approximation (to say the least) to start the investgation by
neglecting the weaker force altogether. Applying this reasoning to the
motion of particles in the Universe, we would expect the Universe to
be governed entirely by electrical forces. However, this is not the case.
CHAPTER3 TIME-INDEPENDENT MAXWELL EQUATIONS 51
The force which holds us to the surface of the Earth, and prevents us
from floating off into space, is gravity. The force which causes the Earth
to orbit the Sun is also gravity. In fact, on astronomical length-scales
gravity is the dominant force, and electrical forces are largely irrelevant.
The key to understanding this paradox is that there are both positive and
negative electric charges, whereas there are only positive gravitational
“charges.” This means that gravitational forces are always cumulative,
whereas electrical forces can cancel one another out. Suppose, for the
sake of argument, that the Universe starts out with randomly distributed
electric charges. Initially, we expect electrical forces to completely dominate
gravity. These forces try to make every positive charge get as far
away as possible from the other positive charges in the Universe, and as
close as possible to the other negative charges. After a while, we expect
the positive and negative charges to form close pairs. Just how close is
determined by Quantum Mechanics, but, in general, it is fairly close:
i.e., about 10?10 m. The electrical forces due to the charges in each pair
effectively cancel one another out on length-scales much larger than the
mutual spacing of the pair. However, it is only possible for gravity to be
the dominant long-range force in the Universe if the number of positive
charges is almost equal to the number of negative charges. In this situation,
every positive charge can find a negative charge to team up with,
and there are virtually no charges left over. In order for the cancellation
of long-range electrical forces to be effective, the relative difference in
the number of positive and negative charges in the Universe must be
incredibly small. In fact, positive and negative charges have to cancel
one another to such accuracy that most physicists believe that the net
electrical charge of the Universe is exactly zero. But, it is not sufficient
for the Universe to start out with zero charge. Suppose there were some
elementary particle process which did not conserve electric charge. Even
if this were to go on at a very low rate, it would not take long before the
fine balance between positive and negative charges in the Universe was
wrecked. So, it is important that electric charge is a conserved quantity
(i.e., the net charge of the Universe can neither increase or decrease). As
far as we know, this is the case. To date, no elementary particle reactions
have been discovered which create or destroy net electric charge.
In summary, there are two long-range forces in the Universe, electricity
and gravity. The former is enormously stronger than the latter, but
is usually “hidden” away inside neutral atoms. The fine balance of forces
due to negative and positive electric charges starts to break down on
atomic scales. In fact, interatomic and intermolecular forces are all electrical
in nature. So, electrical forces are basically what prevent us from
52 MAXWELL’S EQUATIONS AND THE PRINCIPLES OF ELECTROMAGNETISM
falling though the floor. But, this is electromagnetism on the microscopic
or atomic scale—what is usually termed Quantum Electromagnetism. This
book is about Classical Electromagnetism. That is, electromagnetism on
length-scales much larger than the atomic scale. Classical Electromagnetism
generally describes phenomena in which some sort of “violence”
is done to matter, so that the close pairing of negative and positive
charges is disrupted. This allows electrical forces to manifest themselves
on macroscopic length-scales. Of course, very little disruption is necessary
before gigantic forces are generated. Hence, it is no coincidence
that the vast majority of useful machines which humankind has devised
during the last century or so are electrical in nature.
Coulomb’s law and Newton’s law are both examples of what are usually
referred to as action-at-a-distance laws. According to Equations (3.1)
and (3.3), if the first charge or mass is moved then the force acting on
the second charge or mass responds immediately. In particular, equal and
opposite forces act on the two charges or masses at all times. However,
this cannot be correct according to Einstein’s Special Theory of Relativity,
which implies that the maximum speed with which information can
propagate through the Universe is the speed of light in vacuum. So, if the
first charge or mass is moved then there must always be a time delay (i.e.,
at least the time needed for a light signal to propagate between the two
charges or masses) before the second charge or mass responds. Consider
a rather extreme example. Suppose the first charge or mass is suddenly
annihilated. The second charge or mass only finds out about this some
time later. During this time interval, the second charge or mass experiences
an electrical or gravitational force which is as if the first charge
or mass were still there. So, during this period, there is an action but
no reaction, which violates Newton’s third law of motion. It is clear that
action at a distance is not compatible with Relativity, and, consequently,
that Newton’s third law of motion is not strictly true. Of course, Newton’s
third law is intimately tied up with the conservation of linear momentum
in the Universe. This is a concept which most physicists are loath to
abandon. It turns out that we can “rescue” momentum conservation by
abandoning action-at-a-distance theories, and instead adopting so-called
field theories in which there is a medium, called a field, which transmits
the force from one particle to another. Of course, in electromagnetism
there are two fields—the electric field, and the magnetic field. Electromagnetic
forces are transmitted via these fields at the speed of light,
which implies that the laws of Relativity are never violated. Moreover,
the fields can soak up energy and momentum. This means that even
when the actions and reactions acting on charged particles are not quite
equal and opposite, momentum is still conserved. We can bypass some
CHAPTER3 TIME-INDEPENDENT MAXWELL EQUATIONS 53
of the problematic aspects of action at a distance by only considering
steady-state situations. For the moment, this is how we shall proceed.
Consider N charges, q1 though qN, which are located at position
vectors r1 through rN, respectively. Electrical forces obey what is known
as the principle of superposition: i.e., the electrical force acting on a test
charge q at position vector r is simply the vector sum of all of the Coulomb
law forces exerted on it by each of the N charges taken in isolation. In
other words, the electrical force exerted by the ith charge (say) on the
test charge is the same as if all of the other charges were not there. Thus,
the force acting on the test charge is given by
f (r) = q

N
i=1
qi
4?0
r ? ri
|r ? ri|3
. (3.8)
It is helpful to define a vector field E(r), called the electric field, which is
the force exerted on a unit test charge located at position vector r. So,
the force on a test charge is written
f = q E, (3.9)
and the electric field is given by
E(r) =

N
i=1
qi
4?0
r ? ri
|r ? ri|3
. (3.10)
At this point, we have no reason to believe that the electric field has any
real physical existence. It is just a useful device for calculating the force
which acts on test charges placed at various locations.
The electric field from a single charge q located at the origin is purely
radial, points outward if the charge is positive, inward if it is negative,
and has magnitude
Er(r) =
q
4?0 r2
, (3.11)
where r = |r|. We can represent an electric field by field-lines. The direction
of the lines indicates the direction of the local electric field, and the
density of the lines perpendicular to this direction is proportional to the
magnitude of the local electric field. It follows from Equation (3.11) that
the number of field-lines crossing the surface of a sphere centered on a
point charge (which is equal to Er times the area, 4? r2, of the surface) is
independent of the radius of the sphere. Thus, the field of a point positive
charge is represented by a group of equally spaced, unbroken, straight
54 MAXWELL’S EQUATIONS AND THE PRINCIPLES OF ELECTROMAGNETISM
E
q
Figure 3.2: Electric field-lines generated by a positive charge.
lines radiating from the charge—see Figure 3.2. Likewise, field of a point
negative charge is represented by a group of equally spaced, unbroken,
straight-lines converging on the charge.
The electric field from a collection of charges is simply the vector
sum of the fields from each of the charges taken in isolation. In other
words, electric fields are completely superposable. Suppose that, instead
of having discrete charges, we have a continuous distribution of charge
represented by a charge density ?(r). Thus, the charge at position vector
r
is ?(r
) d3r
, where d3r
is the volume element at r
. It follows from a
simple extension of Equation (3.10) that the electric field generated by
this charge distribution is
E(r) =
1
4?0

?(r

)
r ? r

|r ? r
|3
d3r

, (3.12)
where the volume integral is over all space, or, at least, over all space
for which ?(r
) is non-zero. We shall sometimes refer to the above result
as Coulomb’s law, since it is essentially equivalent to Equation (3.1).