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With the phrase electric charge we denote a property of matter by virtue of which particles exert a force each upon the other. The statement that matter exists we take as a given, a postulate for now. Relativity tells us that mass, a fundamental property of matter, corresponds to energy, which obeys a conservation law. We have already deduced the fact that any quantity subject to a conservation law has a smallest, non-infinitesimal part, so matter has a smallest component. In bulk, matter thus consists of collections of such particles. We now postulate that matter can affect the motion of other matter; that is, that particles can exert and respond to forces. And they do so by virtue of the property that we call electric charge.
As James Clerk Maxwell did in his 1861 paper AOn Physical Lines of Force@, we devise a description of what he called lines of force. Imagine having an electrically charged particle so anchored in our laboratory that we can conveniently explore the force that it exerts upon some other particle. In order to receive a force from our anchored particle, the exploratory particle must also exert a force (in accordance with Newton= s third law of motion), so it must also carry an electric charge. In the laboratory of our imagination we can draw figures in the air, so wherever we put our exploratory particle we draw a minuscule arrow whose length and orientation indicate the strength and direction of the force that the anchored particle exerts upon it. When we have drawn a very large number of arrows, we pick one arrow at random and draw a line in the direction it points until we come to another arrow, then we extend the line in the direction that arrow points, and continue so onward as far as we desire. In that way we draw an example of what Maxwell called a line of force. We can thus draw as many lines of force as we wish emanating from our anchored particle. But we don=t actually have to conduct the experiment in order to know several things about those lines of force.
First, we know that all of the lines must point directly away from the anchored particle. If that statement did not stand true to Reality, then putting another charged particle into that field would produce a pair of forces that, together, would violate either the law of conservation of linear momentum or the law of conservation of angular momentum.
Second, we know that if we take a small wheel, distribute electric charge uniformly around its rim, and put the wheel motionless anywhere near the anchored particle, it will not turn. If that statement did not stand true to Reality, then the distribution of force exerted on the wheel would violate the law of conservation of angular momentum. That fact necessitates that the forcefield emanating from the anchored particle and depicted in our lines of force conform to a mathematical description that has a curl equal to zero everywhere. In another essay (Maxwellian Deduction of Coulomb=s Law) I showed how Maxwell took that fact and deduced the inverse-square law description of the electric field. More directly, let=s say, that fact necessitates that the mathematical entity encoding the strength and direction of the electric field conform to a description derived from the gradient of a scalar potential; specifically,
Third, the fundamental theorem of vector calculus (Helmholtz=s theorem) tells us that a non-constant, irrotational forcefield must originate in a non-zero divergence. We encode that fact in our mathematical description of the electric field by stating
in which Q represents the strength of the field=s source,ε0 represents a proportionality constant (the electric permittivity of vacuum) that relates the electric force to the magnitude of its source, and the three-dimensional Dirac delta (the infinitesimal limit of a distribution function) encodes our assumption that the source occupies a mathematical point. If we have the source distributed over some non-infinitesimal volume, then the product of the source magnitude and the Dirac delta segues into the source density, so we have Equation 2 as
which gives us the first of Maxwell=s Equations. Clearly we identify Q with the quantity of electric charge on our anchored particle. Combining Equation 3 with Equation 1 gives us
the Laplace-Poisson equation, whose solution gives us the inverse-square law for the forcefield.
Fourth, we know that electric charge conforms to a conservation law. We can deduce that fact from applying our understanding that Equation 2 necessitates the existence of electric charge at any point where we find a discontinuity in the electric field.
In his paper Maxwell had to assume, without deducing it, the fact that electric charge obeys a conservation law. He expressed that law in the form of a continuity equation, which we write in modern vector notation as
in which j=ρv and v represents the velocity at which the relevant element of charge moves. That equation tells us that at any point in space where we find a non-zero divergence in the electric current density we must also find an equal accumulation or dissipation of the electric charge density. We can convert that statement into an equivalent statement, one that resembles Newton= s third law of motion, by stating that for every increase or decrease that we have in the amount of positive electric charge at some point we must have an equal increase or decrease in the amount of negative electric charge at that point. Let=s now deduce that statement.
Assume that nought but a single unit of electric charge exists in the entire Universe. That charge=s electric field extends throughout all of space and, we assume, ends on the boundary of space. That abrupt ending corresponds to a non-zero divergence of the field, which necessitates, in accordance with the first of Maxwell=s Equations, that the boundary of space possess or manifest an electric charge of a polarity opposite that of the assumed charge. But because it touches Absolute Nonexistence, the boundary of space cannot possess or manifest any properties whatsoever, so we must infer that the assumed charge=s field does not reach the boundary. Because the field spreads throughout all space, we must assert the existence of a charge of polarity opposite that of our assumed charge. An observer with either charge will see it lying the same distance from the boundary of space in all directions, so the charge=s field has the same strength wherever it approaches the boundary. The two fields will then cancel perfectly at the boundary, but if and only if the asserted charge has the same magnitude as has the assumed charge. That remains true to Reality regardless of how many charges we assume, so we infer that the electric charges in the Universe must add up to a net zero at all times. That inference means that any phenomenon that creates or destroys a certain amount of positive electric charge must necessarily create or destroy, at the same time and place, an equal amount of negative electric charge. Q.E.D.
Now someone may object that if we create an electric charge ex nihilo, its field, unable to move faster than light, cannot reach the boundary of space, thereby invalidating the above argument. The above argument thus applies only to charges created with the Universe itself. Charges created later would not come under its reasoning.
Suppose, then, that we create a charge ex nihilo. Its electric field expands outward at the speed of light at fastest, so we have a discontinuity in the field moving outward. That discontinuity corresponds to an electric charge itself, one equal in magnitude to the created charge (due to the equality of the electric fluxes at both discontinuities, the one at the created charge and the one expanding outward). That moving charge creates the electric field of our created charge at every point in space as it passes that point. In so doing it creates energy, both directly in the field itself and on charged particles at the point. If we now assert the conservation law pertaining to energy, we must dismiss from our reasoning the possibility of such a field growing from a created charge. In consequence we must dismiss from our reasoning the possibility of creating electric charge ex nihilo and assert that electric charge always and everywhere conforms to a conservation law of its own.
Fifth, we know that because it conforms to a conservation law, electric charge must also conform to the finite-value theorem, which means, in turn, that it must conform to the quantum theorem. That fact necessitates that electric charge have a minuscule, but not infinitesimal, smallest value. There must exist a basic unit of electric charge that determines the minimum amount by which electric charge can change in any encounter between particles.
We can have particles carrying 2/3 and 1/3 of the basic charge so long as they obey the proviso that they must combine at all times into larger particles that carry a net charge equal to some integer multiple of the basic unit of charge. Those, of course, are the amounts of charge that we infer exist on quarks. We could also have 2 units of charge, but so far no one has found any examples of those.
And, finally, our sixth fact comes from our knowing that if we have one fundamental unit of charge on a particle, we know that we cannot subdivide it. That fact necessitates that the conditions conducive to a subdivision cannot exist, which means, in particular, that the charge cannot exist spread over some extent: it must occupy a mathematical point. But that inference creates a serious problem: combined with the inverse-square law, it tells us that the electric field at the charge has infinite strength and infinite potential, which means that a charge attracted to another will do infinite work and thereby generate infinite energy as the charges come together, in violation of the conservation law pertaining to energy.
Again we get the familiar dilemma. Either we have gone wrong in our derivation or we have missed something relevant to it. As has become our habit, we assume that we have neglected to take some factor into proper account and now we want to determine the form of that factor.
We know that we would not have this problem if the charge spread throughout some small volume and we know a way in which Reality can feign such a spread. If we put a bare charge into a region filled with equal amounts of both positive and negative charges, that bare charge will attract some charges and repel others. It will thereby polarize that dielectric medium, causing a shift in the distribution of charge that, in effect, spreads the bare charge out over some volume. Thus, we bring our quantum unit of electric charge into conformity with the law of conservation of energy by asserting that the vacuum exists as a dielectric medium, containing equal quantities of positive and negative charge in the form of ghost particles, which lack the mass-energy to exist as real particles. The bare charge polarizes the vacuum and thus, in effect , spreads itself out. In this way we lay the cornerstone of our derivation of a full description of the quantum vacuum.
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