Deducing The Newton-Coulomb Law

So far I have assumed in these essays that bodies exert forces upon each other, but I have said very little about the nature of those forces. Now I want to consider in more detail how bodies exert forces and to see how Relativity affects those details. As a matter of convenience I will consider the electric force as my example, though this analysis applies equally well to the other known forces of Nature.

Let's begin by declaring as a self-evident axiom the proposition that bodies exert electric forces by virtue of possessing a property that we will call electric charge, the magnitude of which property determines the amount of force that the body exerts. We have that axiom as self-evident because I have stated it more in the way of a definition of electric charge: if a body exerts a force, it possesses charge and if a body exerts no force, it possesses no charge. In asserting the existence of electric charge we have tacitly assumed that the charge on one body surrounds that body with a kind of aura and that the aura's contact with the charge possessed by another body produces the force exerted upon that body by the first body. We have thus tacitly assumed the existence of forcefields (the proper name for the force-producing aura). Because we have asserted nothing about the nature of forcefields, we may include them in our axiom; that is, we may take as our axiom the statement that a body exerts an electric force by virtue of possessing an electric charge, which charge exerts its force through an electric forcefield emanating from the charge. If I have guessed wrong in that matter, the deduction, properly carried out, will correct my error.

Now we want to work out the rules that govern the relationship between the force a body exerts and the charge it possesses. We know immediately from Newton's third law of motion that if charge determines how much force Body-1 exerts upon other bodies, then it also determines how much force other charge-carrying bodies exert upon Body-1. That means that we must describe the force exerted upon Body-A by an algebraic formula that includes the Body-1's own charge and the charges of all other bodies that can affect Body-1.

We represent electric charge with the letter Q
and we consider the forces mutually exerted upon two bodies that possess charges
Q_{1} and Q_{2} respectively. Using functional notation, we have
for the first body

(Eq'n 1)

and for the other body

(Eq'n 2)

But Newton's third law of motion requires that F_{1}
= -F_{2}, so we must have as true to Reality

(Eq'n 3)

Conservation of angular momentum gives us another constraint with which to shape the form of the law governing the forces exerted between two charged bodies. Conservation of linear momentum already obliges the structure of Reality to orient the two forces antiparallel to each other; that is, the forces must point in opposite directions parallel to the same straight line. If we choose to make some arbitrary point in space a center of revolution, then those antiparallel forces must exert equal and oppositely directed torques about that point if they are to conserve the angular momentum of the system comprising the two charge-carrying bodies. Because the forces themselves are equal and oppositely directed, the conservation of angular momentum requires simply that they act upon identically long moment arms. Because we decide the location of the center of revolution as a purely arbitrary choice, that requirement can only be satisfied by the forces both pointing along the same straight line, the line passing directly from one of the bodies to the other, thereby giving both forces the same moment arm about any point we may choose.

We can see another way in which the conservation of linear and angular momenta shapes the force law by constraint. As an aid to the imagination envision the charged body as a small pebble enveloped in a thin fog. A second pebble, carrying its own cloud of fog, comes into the first pebble's fogbank and has a force exerted upon it in consequence. Now imagine that there's a place in the first pebble's fogbank where the fog is thicker than it is elsewhere: if the second pebble enters that place, the force that the first pebble exerts upon it will increase as the thickness of the fog (the intensity of the forcefield) increases. The conservation laws require that, if that happens, the force that the second pebble's fogbank exerts upon the first pebble must increase by the same amount. That requirement necessitates that the second pebble's fogbank thicken in exactly the same way in which the first pebble's fogbank thickens; that is, the two forcefields must have the same dependence upon the distance between the two charge-carrying bodies. Further, the thickness of each fogbank must be spherically symmetric with its pebble at the center of the sphere. If that proposition were not true to Reality, then two charge-carrying bodies could be brought together with their forcefields so oriented that the forces exerted between the bodies, while oppositely directed, would not be equal in strength. Indeed, if any forcefield were to change its intensity in a direction perpendicular to the direction in which it is pointing (as would be true of a spherically asymmetric field), then a small electrically charged wheel placed in that field would spin up spontaneously, creating angular momentum in violation of the conservation law and creating energy from a static forcefield in violation of the law of conservation of energy. Thus, the force exerted between two bodies can depend only upon the distance between the bodies and upon no other geometric factors.

Does the presence of an electric charge Q_{2}
alter the forcefield emanating from the charge Q_{1}? In order to answer
that question we must first revisit the idea of forcefields. We tacitly
postulated the existence of forcefields in order to avoid using the concept of
action at a distance: we conceive a forcefield as simply a space-filling entity
that extends from its generating body and, through direct contact, exerts a
force upon any second charge-carrying body immersed within it. Because action at
a distance means action at any distance, however small, the forcefield must be
continuous; that is, it must have a value at every point in space and it must
touch every body immersed in it, including the body that generates it. That last
item gives us another reason why the forcefield must be spherically symmetric:
if it were not so, it would cause its generating body to undergo
self-acceleration, in blatant violation of the law of conservation of linear
momentum.

Now we can consider three possible outcomes of Q_{2}'s
forcefield touching Q_{1} (aside from exerting a force upon Q_{1}):
Q_{1}'s forcefield may become stronger; it may become weaker; or it may
remain unchanged. If contact with Q_{2}'s forcefield makes Q_{1}'s
forcefield grow stronger, then contact with any forcefield, including its own,
will make Q_{1}'s forcefield grow stronger. But contact with its own
forcefield would then make Q_{1} subject to a positive feedback process
in which the augmented forcefield then augments itself and then augments its
augmented self endlessly. Because the distance over which this process occurs
begins at zero, the process immediately creates a forcefield of infinite
strength spreading away from Q_{1} at the speed of light. That field
would give any charge-carrying body that it contacts infinite linear momentum,
in violation of the finite-value theorem; therefore, the proposition upon which
that infinite field is based must be false to Reality, so contact with another
body's forcefield won't make Q_{1}'s forcefield grow stronger. If
contact with another body's forcefield makes Q_{1}'s forcefield become
weaker, then Q_{1} becomes subject to a positive feedback process that
instantly reduces its forcefield to zero strength. That possibility violates our
postulated existence of forces, so we dismiss it as false to Reality (pending a
proof that forces can exist without forcefields to mediate them). We are thus
left with the proposition, necessarily true to Reality, that the forcefield
emanating from a body is not affected by the contact between that body and any
other forcefield.

We add to that proposition the further statement that where two forcefields commingle neither forcefield alters the other. The reasoning supporting that statement is similar to the reasoning used above. If one field causes the other to become stronger where the two fields overlap, then the second field will strengthen the first and we will have a positive feedback process that will instantly bring both fields up to infinite strength. Likewise, if one field weakens the other, we will have a positive feedback process that will immediately extinguish both fields.

Thus we obtain the superposition principle:
because it is necessarily true to Reality that the presence of an electric
charge or an electric forcefield does not alter an electric forcefield, then it
must therefore be true to Reality that if charge-carrying Body-1 is in the
presence of any number of other charge-carrying bodies, then the net force
exerted upon Body-1 is equal to the vector sum of the forces that the other
bodies would exert if each of them were acting on Body-1 alone. If Body-1
carries a charge Q_{1} and another body, Body-2, carries charge Q_{2},
then the force exerted upon Q_{1} will have some strength F. If we add
to Body-2 enough extra charge to multiply its original charge by some number N,
then the force exerted upon Body-1 has strength NF because the charge NQ_{2}
on Body-2 has the same effect as do N bodies, each carrying a charge Q_{2},
exerting their forces upon Body-1 from the same location. Likewise, the force
that Body-1 exerts upon Body-2 increases from F to NF. From the increase in the
force exerted upon Body-1 we infer that the formula representing that force must
be directly proportional to the amount of charge residing on Body-2 and from the
increase in the force exerted upon Body-2 we infer that the formula representing
that force must also be directly proportional to the amount of charge residing
on Body-2. Those two inferences tell us that the formula representing the
electric force that one body exerts upon another body must be proportional to
the amount of charge on the forcing body and proportional to the amount of
charge on the forced body, which means that the formula must be proportional to
the product of the two charges.

We can now represent our force law as

(Eq'n 4)

If we divide out the charge on the forced body, say Body-1, then we obtain the mathematical description of the electric field of the forcing body, Body-2 in this case;

(Eq'n 5)

We now want to determine the form of the geometric factor, f[r].

The electric field can be regarded as a kind of
potential force: at each point in space the charge Q_{2} potentially
exerts a force, the amount and direction of the force to be determined by
multiplying the electric field strength at that point, **E**_{2},
by the charge on the forced body positioned at that point. With this concept we
can now restrict our discussion to forcefields in terms of their sources only
and ignore any other bodies that they may affect. I want to deepen our
understanding of the forcefields themselves so that we may discern subtleties
that we may apply to further understanding of the forces that they may exert
upon other bodies.

Let's contemplate conservation of energy. In particular, let's contemplate the way in which energy is conserved when we do work on a body immersed in a forcefield. If we move the body from one point to another, the net amount of work that we must do upon it does not depend upon the particular path that we make the body follow. If we are moving a camping stove up a mountain in Earth's gravity, for example, we must do the same amount of work to get it from base camp to the summit, regardless of whether we haul it straight up the steepest cliff or carry it up the gentlest slope. If that proposition were not true to Reality, then we could invest a certain amount of work in taking the stove to the summit by one path and then get back even more work by bringing it down to base camp by a different route, thereby creating energy and violating the conservation law. Further, it makes no difference whether we take the stove apart and carry the components up the mountain separately or carry the whole stove up at one time: the total amount of work that we must do to get the stove up the mountain does not depend in any way upon how we move it. The amount of work that we must do to move the stove depends only upon the mass of the stove and on the gravitational potentials at the position of the base camp and at the position of the summit. That is true of any static forcefield and it can be used to tell us something very interesting about the electric field.

I'll start by expanding my basic axiom pertaining to the existence of the electric charge and its associated forcefield by adding as a postulate the empirically derived facts that electric charge comes in two forms, positive and negative, that negate each other and that like charges repel each other and unlike charges attract each other. We see the first of those facts reflected in the observation that on the large scale matter in the Universe is electrically neutral because it consists of equal amounts of positive and negative electric charges that have attracted each other and committed mutual neutralization. That means that we can generate electric charge by separating the charges in an initially neutral piece of matter. But might there be some other way in which we could produce an electric charge?

In the generation of electric charge we have three possibilities: 1) electric charge may be created and uncreated at will (equivalently, one kind of charge may be created and then the other kind may be created to negate it), 2) one kind of electric charge may be created at will but it cannot be uncreated (equivalently, one kind of charge may be created but the opposite kind cannot be created), and 3) electric charge may be neither created nor uncreated (equivalently, both kinds of electric charge may be created or uncreated, but only in equal amounts of both: the creation of electron-positron pairs from gamma rays in certain subatomic processes is a perfect example of such a creation of electric charge).

Let's assume that Possibility #1 is true to Reality. We don't know whether the presence of a forcefield will suppress the creation of charges, so we will conduct our imaginary experiment in a region of space in which we have created a field-free zone by so arranging electric charges that all fields in the zone cancel each other out. In that zone we create a charge +Q and then we move that charge to a point well outside the zone. We may be obliged to do some work to move the charge against any forcefields that exist outside the zone, but we will get that work back when we create the opposite charge and move it along the same path. We then create a charge -Q in the zone and move it along the same path that +Q followed, harnessing the work done by its attraction to +Q. When the two charges come together, merge, and negate each other, we have the same circumstance that we had before we created +Q. We can then repeat the procedure, each time gaining the work done by the mutual attraction between +Q and -Q. But that violates the law of conservation of energy, so we must declare that Possibility #1 is necessarily false to Reality.

Let's now assume that Possibility #2 is true to Reality. Again, we don't know whether the presence of a forcefield suppresses the creation of charge. We obtain two charges of magnitude Q (either positive or negative, bearing in mind that we can create only one kind), even if we must create them ex nihilo, and we position them with some distance between them. The point midway between the charges is a field-free point because at that point the fields emanating from the charges cancel each other perfectly. At that point we create a third charge, move it away from the point, and harness the work done upon it by the force repelling it from the first two charges. Once that third charge has moved far enough away from the first two to be effectively "at infinity" we will have regained the circumstance that we had before it was created and can thus repeat the process. (Please notice that I am tacitly assuming that the forcefield emanating from a charge diminishes in intensity as distance from the charge increases. If that assumption is false to Reality, we can nonetheless locate field-free points and create energy via a slightly more complicated version of the process described above.) Again, we have contrived a process that will violate the law of conservation of energy, so we must declare that Possibility #2 is necessarily false to Reality.

That leaves only Possibility #3 to be necessarily true to Reality. But the statement that electric charge can be neither created nor uncreated or can only be created or uncreated in equal amounts of opposite charges tells us that electric charge is subject to a conservation law of its own. That conservation law has an interesting mathematical expression in what is called an equation of continuity. Imagine that we have measured out a region of space whose volume is represented by an upper-case vee and whose surface is represented by an upper-case ess. We have subdivided the volume into infinitesimally small blocks, each with volume dV, and the surface into infinitesimally small patches, each with area dS. We now feign to know the electric charge density, represented by the Greek letter rho (D) and measured in coulombs per cubic meter, in each of the blocks that we have defined above. We calculate the total electric charge in the region V by multiplying the density in each block by the volume of the block and then summing the resulting products in the process of integration,

(Eq'n 6)

Because electric charge is conserved, that charge can only change due to charge crossing the boundary S. The net rate at which charge crosses the surface can be calculated by integrating over the whole surface the products obtained by multiplying the area of each patch by the charge density at that patch and by the component of the charge's velocity that's perpendicular to the patch (the part of the velocity that's actually carrying charge across the patch). The equation of continuity equates that rate with dQ/dt,

(Eq'n 7)

In that equation the minus sign that we would expect to see between the two integrals has been converted into a plus sign by applying the convention that the vector representing the area of each patch points out of the volume V and not into it, thereby making the second integral represent the flux of charge out of the volume (a negative number if a net positive charge is flowing into the volume).

Now I want to consider some of the ways in which
charges interact with each other in a situation of perfect spherical symmetry. I
envision two or more ultra-thin, rubber-like skins of spherical shape and of the
same radius R_{1} centered on some point in field-free space. Initially
we have no charge on those skins. Imagine that we separate the charges +Q and
-Q, so move one of those charges into the outermost skin that it covers the skin
uniformly, and expand the skin to some new radius R_{2}. We assume that
the charges exert an attraction upon each other, so we assume that we must do an
amount of work W_{1} to expand the outer skin. Now we allow the next
skin, upon which the other charge is uniformly distributed, to expand and allow
its charge to merge with the charge on the outer skin, thereby negating it. The
work done upon the inner skin is W_{2}. We then allow the two skins, now
bearing no net charge and, therefore, doing no work, to contract to their
original radius R_{1}, thereby re-establishing the original condition of
the experiment. Conservation of energy necessitates that W_{1} be equal
in magnitude to W_{2}; that is, that no net work be done by this
process.

For convenience we can define a potential
function V_{1} = W_{1}/Q and write further that V_{1} =
Qf(R). So in the above experiment the charge +Q did work against V_{1}
and then the charge -Q had work done upon it by V_{1}. Because the
charges were both of the same magnitude, we infer that V_{1} must have
been the same for both of them in order to conserve energy. Further, the
geometric factor f(R) cannot have changed between the time the first skin begins
to expand and the second skin completes its expansion and reunites with it. If
existence did not make that statement true to Reality, then we could contrive a
way to create or destroy energy, in violation of the conservation law.

Imagine that we have already expanded the first
skin, the one over which we have spread a charge +Q uniformly, to the radius R_{2}.
Let us now divide the negative charge left at radius R_{1} into n equal
increments -q so that we have nq = Q. As we did above we pull the charge -q into
the outermost skin of radius R_{1} and expand it to R_{2} to
merge -q with +Q. We have done two pieces of work; w_{1} = q^{2}f(R)
and w_{2} = q^{2}f(R)(Q-q) = q^{2}f(R)(n-1)/n. The first
contribution represents the work done by the charge's self-repulsion, its
interaction with the part of the field that it is negating as it moves, and the
second represents the work done by the charge's interaction with the field
remaining. For the i-th charge moving from R_{1} to R_{2} the
contribution w_{1} is the same as for the first charge and the
contribution w_{2} = q^{2}f(R)(n-i)/n.

We get at the end of the process W = nf(R)q^{2}
+ f(R)q^{2} n(n-1)/2 = f(R)q^{2}(n^{2} + n)/2; that is,
we obtain a little over half the amount of energy that we should have obtained
in order to conform to the conservation law. Let us therefore make a postulate:
assume that, because the expanding shells have two sides, the force acts on each
shell twice as strongly as we have assumed. That postulate gives us a little too
much energy, so, in order to make the sum come out correct, we must infer that
the force a shell exerts upon itself acts only once. That leads us, in light of
the above discussion to infer that the shell has zero forcefield on one side. We
now prove the proposition that the inside is the side in question by pointing
out that if we were to take an isolated charged skin and shrink its radius to
zero we would obtain a point charge. But the forcefield definitely lies outside
the point charge, so it must have extended outside the charged shell at all
times prior to its radius collapsing to zero.

What we have gained is the fact that if we put a point-like electric charge anywhere inside a uniformly charged spherical shell, the charge on the shell will exert no net force upon that point charge. That means, by way of Newton's third law of motion, that the point charge exerts no net force upon the shell. Likewise, the point charge will exert no net torque upon the shell, as required by the law of conservation of angular momentum. From those statements we can now derive the geometric factor in the electric force law.

Imagine that we have passed a plane through the shell in such a way that the center of the shell and the point charge both lie on the plane: that plane divides the shell into two hemispheres. Imagine further that we draw a long straight line through the point charge and then twirl it in such a way that it cuts the shell into a mosaic of minute patches in much the way in which a band saw cuts a picture into a jigsaw puzzle. For each patch on one hemisphere there is a corresponding patch of the same shape on the other hemisphere.

The force that the point charge exerts upon one
patch will point in the direction opposite that of the force that the point
charge exerts upon the corresponding patch, both forces being exerted along the
straight line passing through the point charge, but we do not know the relative
strengths of the two forces. We do know, though, that the forces exerted upon
all of the patches must add up to a net zero. Let's represent the force exerted
upon a patch in one hemisphere by a_{i} and the force exerted upon the
corresponding patch in the other hemisphere by b_{i}, for which the
index i takes all the values from one to N, the number of patches into which
each hemisphere has been cut. We then know that

(Eq'n 8)

We can obtain the corresponding equation for the
torque exerted upon the shell with just a little more effort. The line of forces
passing through the point charge and the two patches under consideration is a
chord of the sphere; thus, it is the base of an isoceles triangle whose other
sides are radii of the sphere. The line of forces thus penetrates both patches
at the same angle; that is, it crosses both of the lines tangent to the sphere
at the centers of the two patches at the same angle (it being understood that
the tangents and the line of forces lie in the same plane). That means that the
torques acting on the two patches are opposed to each other: if one acts to turn
the shell clockwise, the other acts to turn the shell counterclockwise. It also
means that the torque acting on each patch can be calculated by multiplying the
force acting on the patch by the radius of the shell and by the cosine of the
angle at which the line of forces crosses the tangent. If we represent the
product of multiplying the shell's radius by the cosine on each patch by c_{i},
then the equation for the torques is

(Eq'n 9)

One very obvious solution of both those
equations is a_{i} = -b_{i} for all values of the index i. There
could conceivably be other solutions, but we need not consider them because they
must yield the same geometric factor for the force law. If that statement were
not true to Reality, then we would have at least two different formulae for the
calculation of the force between two electric charges. At worst that result
would give us a contradiction that would negate the proposition that forces
exist. At best it would give us the possibility of violating the law of
conservation of energy, if not also the law of conservation of linear momentum.

What our solution of Equations 8 and 9 tells us is that the forces that the point charge exerts upon two corresponding patches on the shell are equal as well as being oppositely directed. We are only interested in the magnitudes of the forces on the patches, so we can ignore the minus sign in our solution and write

(Eq'n 10)

Each of those forces is equal to the product of
multiplying the magnitude of the point charge, Q, by the charge on the patch in
question and the geometric factor. If the distances from the point charge to the
two patches are represented by r_{a} and r_{b} and the charges
on the patches are represented likewise by q_{a} and q_{b}, then
Equation 4 becomes

(Eq'n 11)

The charge on one of the patches is equal to the
product of the area of the patch and the surface density of the charge on the
shell. The surface density of charge on the shell is calculated by dividing the
total charge on the shell (Q') by the surface area of the shell (4BR^{2}).
The area of the patch can be calculated as the product of the square of the
patch's distance from the point charge and a proportionality factor that encodes
the shape of the patch, the angle that its tangent makes with the line of
forces, and the maximum angle that it subtends as measured from the point
charge. I represent that proportionality factor by P and note that it is the
same for both patches, so now I write Equation 11 as

(Eq'n 12)

If we divide both sides of that equation by the
factors that they have in common and note that r_{a} and r_{b}
are not equal for most pairs of patches, then we can see that the geometric
factor must have the form

(Eq'n 13)

in which K represents a constant of proportionality that would not be discerned by our imaginary experiment. With that equation we can now write the equation describing the electric field generated by a point charge as

(Eq'n 14)

in which I have replaced the constant K by the inverse product of four pi and the letter epsilon subscripted with a zero, that epsilon representing the electric permittivity of vacuum.

That explicit use of the electric permittivity brings up a subject that I have kept implicit so far but which I now want to make explicit. In this treatise I am using the version of the metric system known as MKS (for Meter-Kilogram-Second) rather than the version known as CGS (for Centimeter-Gram-Second) favored by many physicists. I prefer the MKS system because in it the fundamental constants of Nature are written out explicitly in the equations rather than merely implied as they are in the CGS system. In large measure that preference reflects the fact that I was originally taught electromagnetic theory in MKS units. But it also reflects an aesthetic choice reminiscent of Einstein's dictum that a theory should be simple but not too simple. I express those constants explicitly in my equations because, until I am convinced otherwise, they represent properties of the vacuum and therefore should not be made invisible by a redefinition of the properties of matter. They are as fundamental to the structure of Reality and thus are as much a part of its beauty as are the electric charge of the electron or the spin of the photon.

habg