THE BASIC ELECTRIC FORCE

Using the metaphor of American law, I compare the conservation laws to the Constitution, making them the supreme law of Reality. Continuing that metaphor, I must say that the Constitution of Reality acts as a constraint upon the form that Existence permits the other laws of physics to take. In logic constraints enable us to deduce things, so I have shown you how to use the basic conservation laws to deduce a description of the form of Universal space and then how to deduce from that description the theory of Special Relativity. Now I want to use that doctrine to deduce the laws that physical forces must obey.

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. Thus, I now assume into my premises the existence of a force emanating from a property that we call electric charge and I assume that electric charge comes in positive and negative manifestations.

As an aid to the imagination we can envision the electric charge as a speck of dust enveloped in a thin fog. I present that vision because in asserting the existence of a property, electric charge, that enables one body to exert a force upon another body anywhere in space, we have tacitly assumed that the electric charge on that first body surrounds the body with a kind of fog-like aura that extends out into space and that the aura=s contact with the electric charge on another body produces the force exerted upon that body by the first body. Physicists call such force-producing auras forcefields. In that view we can conceive the electric forcefield (or, simply, electric field) as a kind of potential force: at each point in space the electric charge whence the field emanates potentially exerts a force upon any electrically charged body that might occupy that point.

Here I must add a proviso. In devising the Map of Physics I have assumed that ultimately we can deduce all of the laws of physics from a small set of axioms. In this case, though, I must present my assumptions concerning the existence of the electric force as postulates and hope that someday someone will deduce them from assumptions fundamental enough and self-evident enough that we can rightly call them axioms.

Let=s begin by imagining that we have in the laboratory of our imaginations two particles, which we call Quiz and Query (note: physicists typically use the letter Q, in both its upper case and lower case forms, to represent electric charge in their equations), and we assume into our premises the postulate that each particle carries that property, which we call electric charge, which gives each particle the ability to exert a force upon the other. As an aid to analysis we employ the useful fiction that each particle occupies the space of a mathematical point: of course real particles must occupy some minuscule volume of space, but that fact only affects our analysis at the subatomic level, where we must also apply the quantum theory, which lies beyond the scope of these essays. Here I want to consider only the classical forces and ignore their quantum manifestations.

Imagine that Quiz and Query, each bearing a certain amount of electric charge, exert a force each upon the other. If we hold Quiz stationary and move Query around it, we can measure at each point the strength and the direction of the force that Quiz exerts upon Query. At each point, then, we can draw a minuscule arrow to represent the force exerted at that point. If we draw a line, going from point to point by following the direction that the arrow at each point gives us, we will thus trace what James Clerk Maxwell (1831 Jun 13 B 1879 Nov 05) called a line of force. If we draw enough lines of force, we will depict the forcefield emanating from Quiz and thereby see an example of what mathematicians call a vector field.

I will note here that the lines of force cannot cross each other, because that would make the force at a given point ambiguous. We cannot have that happening because any ambiguity in the force violates the certainty in the force required by the conservation laws. Thus, near any given point the lines of force must run more or less parallel to each other; perhaps in some places coming closer together and in others moving farther apart, but never crossing each other. So if we trace out enough of the lines of force, we see what looks like a representation of fluid flow. Indeed, Maxwell actually devised an imaginary model of forcefields by conceiving an incompressible fluid flowing between sources and sinks and through contemplating the mathematical properties of that model deduced the laws of electricity and magnetism.

You may also notice that I have used two metaphors to describe the electric field; that of a fogbank and that of a flowing liquid. The dream-like ability to shift our thinking from one model to the other gives physicists the ability to explore the mathematical logic that underlies the laws of physics and yields astonishing results.

Now let=s bring our constraints into play. Whatever form the forcefield takes, it must conform to the requirement that the forces exerted between the two particles obey the laws pertaining to conservation of linear momentum and conservation of angular momentum. Exploring that conformity enables us to describe fully the laws governing the actual magnitude and direction of the force acting among electrically charged particles.

Conservation of linear momentum (Newton=s third law of motion) necessitates that the particles exert equal and oppositely directed forces upon each other. Conforming to that requirement ensures that the two particles, taken together as a single system, do not generate a net force that would make the system=s linear momentum increase/decrease continuously.

Now imagine that Query, carrying its own cloud of fog, comes into Quiz=s fogbank and has a force exerted upon it in consequence. And imagine that we can find a place in Quiz's fogbank where the fog has a greater thickness than it has elsewhere: if Query enters that place, the force that Quiz exerts upon it will increase as the thickness of the fog (the intensity of the forcefield) increases. Newton=s third law requires that, if that happens, the force that Query's fogbank exerts upon Quiz must increase by the same amount. That requirement necessitates that Query's fogbank thicken in exactly the same way in which Quiz's fogbank thickens; in other words, the algebraic descriptions of the two forcefields must have the same dependence upon the distance between the two charge-carrying bodies.

Conservation of angular momentum, the rotary analogue of Newton=s third law of motion, gives us another constraint that we can invoke to determine the form of the law governing electric force. That conservation law tells us that no force can exist if it would cause a system to rotate without causing some other system to rotate in the opposite sense by the appropriate amount. Thus conservation of angular momentum necessitates that the forces exerted between the particles not push the particles into revolving spontaneously about any point. In order to satisfy that requirement, the forces that Quiz and Query exert upon each other must point along the same straight line; specifically, the line that passes through both bodies.

Further, the thickness of each particle=s fogbank must possess the symmetry of a sphere with its electrically-charged particle at the center of the sphere. If that proposition did not stand 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 have equal strength. Indeed, if any forcefield were to change its intensity in a direction perpendicular to the direction in which it points (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 that would represent and asymmetry. We call a forcefiedl that conforms to that requirement, the law of conservation of angular momentum, irrotational.

In his discussion of a magnetostatic field emanating from poles, in AOn Physical Lines of Force@ (1855 - 56), Maxwell laid out a vastly elegant derivation of the inverse-square law of static forces. More mathematician than physicist, Maxwell led his readers on a stroll through the realm of mathematics, only lightly touching the realm of physics, but matching it perfectly where he did. If you want to know what it feels like to comprehend directly a Platonic Form, reading Maxwell=s essay would take you remarkably close. This may give you some idea of how Maxwell took my breath away:

If a point-like particle extends a forcefield, then the requirement that the field be irrotational corresponds to saying that the curl of the electric field equals zero. In vector calculus curl denotes one of the processes by which mathematicians calculate a description of the way a vectorfield changes for an observer moving in that field. Oversimplifying, as we must do in describing mathematical technicalities in plain English, we can say that curl tells us how the field changes when we move in a direction perpendicular to the direction in which the field points. That calculation gives us six partial derivatives, two for each of the three dimensions of space. Calculating the curl of a vectorfield tells us whether we might find, hidden among the curving lines of force, a component that circles around some point in space. In the case of the electric field the law of conservation of angular momentum necessitates that no such component can exist. That statement means that the lines of electric force cannot go round and round and connect to themselves or to their neighbors.

It follows from that statement that each pair of components of the curl of the electric field, taken together, must equal zero precisely. In other words we have three equations stating that one component of each pair equals the other component of that pair precisely. Maxwell recognized those equations as expressing the criteria for the vector dot product of the electric field at any point with a minuscule element of movement at that point to correspond to an exact differential of a scalar (i.e. not a vector) function, which he identified with the electrostatic potential. He did that because the dot product in this case corresponds to a minuscule element of work that an electrically charged particle at that point would have to do to move, at least in part, against the electric force or the work that the particle would gain in moving, at least in part, with the force. He could thus calculate the amount of work that the particle would do in going between any two points in the electric field and he knew that the amount of work done did not depend at all upon which path the particle followed in going from one point to another. That fact, one way of expressing the law of conservation of energy, necessitates that the electric field conform to a description obtained by calculating the gradient of some algebraic function, calculating the direction in which the value of the function changes most rapidly and the rate of that change.

Put another way, we can invoke the
mathematicians= rule which says that
the curl of a gradient of any algebraic function always equals zero. Because the
curl of the electric field always equals zero, we can infer that the electric
vectorfield equals the gradient of some algebraic function: physicists actually
use the __negative__ gradient of the electrostatic potential (an algebraic
function describing the voltage that we would measure at any point in space). To
the extent that the electric field represents a force, to that same extent the
electrostatic potential represents a potential energy associated with that
force.

Next we must refer to Helmholtz=s vectorfield theorem, which mathematicians call the fundamental theorem of vector calculus. If we have a vector field such that the source and circulation densities go to zero at all points infinitely far from our field point, then the vector field has a unique representation as the linear sum of an irrotational (curl-free) component field and a solenoidal (divergence-free) component field, those component fields emanating from some arrangement of circulation densities and source densities. In the case of an electrostatic field the circulation density equals zero everywhere (because the curl of the field equals zero everywhere), so we must have at least one point where the source density does not equal zero, which tells us that at that point the divergence of the electric field does not equal zero, but stands equal to some function of the electric charge occupying that point.

Divergence denotes another vector operation, one similar to the curl. In the case of divergence we can say, oversimplifying as usual, that the divergence of a vectorfield describes how the field changes as we move in a direction parallel to the direction the field points. The divergence of a forcefield has a value different from zero only where the lines of force begin or end, so we say that the divergence of the electric field equals zero at every point in space except at those points where an electric charge exists. But in our algebraic description of the electric field we have the electric field itself equal to the negative gradient of the voltage function, so we have the divergence of the negative gradient of the voltage function (which mathematicians call the negative Laplacian of the voltage function) equal to zero at every point in space except at those points where an electric charge exists.

Maxwell recognized that statement as an example of Poisson=s equation, whose solution he knew. As he put it, ANow it may be shewn that [Poisson= s equation], if true within a given space, implies that the forces acting within that space are such as would result from a distribution of centres of force beyond that space, attracting or repelling inversely as the square of the distance.@ In that statement Maxwell reveals his inference that the mathematical description of the strength of an electrostatic field emanating from a single Acentre of force@ has the form of some number divided by the square of the distance between the Acentre of force@ and the point at which we measure the strength of the field. The number that we divide represents some proportionality factor pertaining to the ability of the Acentre of force@ to exert a force.

Now we can say that statement represents the force that two point-like particles exert each upon the other if each particle carries one unit of what Maxwell called free electricity (and we call electric charge), however big that unit may be. If we put more than one unit of charge on Quiz, each of those charges responds separately to the field emanating from the single unit of charge residing on Query, so we calculate the force acting on Quiz by multiplying the force acting on one unit of charge by the number of units of charge. In accordance with Newton=s third law of motion, the same force acts on Query, though in the opposite direction. If we now put a certain number of unit charges on Query, each one will respond to the multiplied field emanating from Quiz, so we calculate the force that Quiz exerts on Query (and vice versa, of course) as the product of the two quantities of charge on the particles and of the appropriate proportionality constant divided by the square of the distance between the particles. That force must act in the direction along the straight line passing from one particle to the other. That statement expresses Coulomb=s law of the electrostatic force.

We must feel sheer astonishment when we first learn that Maxwell worked out that derivation (though not quite in that condensed form) by describing the velocities and pressures in an incompressible fluid moving through a resistant medium, the fluid coming out of point-like sources and going into point-like sinks (Maxwell=s Acentres of force@). That a fluidic model should give us the laws of electromagnetism properly astounds us. That concepts derived from direct perception (of running water) should match the laws that govern things several steps removed from direct perception (electric and magnetic forcefields) hints at a great profundity in the relationship between mathematics and Reality yet leaves to us the task of discovering its nature.

In his paper Maxwell had to assume one proposition, one that he could not deduce from more fundamental principles. He had to assume that electric charge obeys a conservation law and he expressed that law in the form of a continuity equation, which 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 somewhere else due to a net electric current flowing into or out of that point. Let= s now deduce that statement (Note: I conceived this deduction in the middle of February 2009 while working on a more technical version of this essay).

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 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.

That conservation law has an interesting mathematical expression in what physicists call an equation of continuity. Imagine that we have enclosed a region of space within a thin film, like a soap bubble. We feign to know the total amount electric charge within the bubble. Because electric charge obeys a conservation law (that is, because we can neither create electric charge ex nihilo nor destroy it ad nihilo), we know that the amount of charge within the bubble can only change due to electric charge crossing the bubble itself. The equation of continuity tells us that the amount of electric charge inside the bubble at some given time equals the amount inside the bubble at some other time plus or minus the amount that has crossed the bubble=s skin as electric current since the given time.

Now we come to something special. Imagine that an electrically charged particle lies at the center of a soap bubble. The electric field passing through the soap film has a strength inversely proportional to the square of the radius of the bubble and the bubble has a total surface area proportional directly to the square of the radius of the bubble, so if we multiply the electric strength on the bubble=s surface by the area of the bubble, we get a number that is proportional to only the electric charge on the particle. We call that number the flux of the electric field through the bubble and note that we get the same number, regardless of how big or small we make the bubble: to calculate that number we merely divide the amount of electric charge by the electric permittivity of vacuum. But Reality makes that fact even better.

It turns out that we can put the charged particle anywhere within the bubble without changing the amount of electric flux passing through the bubble: the particle doesn=t have to lie at the center. Further, the bubble=s shape does not have to conform to that of a sphere: the bubble can have any shape at all and the electric flux still comes out the same (though we need to remember that only the part of the electric field that points in the direction perpendicular to the soap film at any given point contributes to the electric flux through the soap film around that point). We end up with a simple rule: The total electric flux passing through any closed surface stands in direct proportion to the net electric charge enclosed within that surface. We call that rule Gauss=s law of the electric field, naming it after Johann Karl Friedrich Gauss (1777 Apr 30 B 1855 Feb 23), the German mathematician who figured it out (along with a lot of other potent mathematics).

And inside that law lies a thing of exquisite beauty. In many cases Gauss=s law allows us to avoid tedious and difficult calculations. Imagine that we have distributed an electric charge uniformly upon an infinitely-long straight wire. We want to know the strength of the electric field at some point away from the wire. We pick such a point, which we call the field point, and we add up the electric fields induced at that point by all of the elemental electric charges on the wire. We know that only the part of each electric field that points directly away from the wire adds to the total: the parts pointing parallel to the wire cancel out (for every elemental charge that contributes an electric field parallel to the wire at the field point we can always find an elemental charge that contributes a parallel electric field of the same strength pointing in the opposite direction). Adding up all of the contributions from the elemental electric charges spread along the full length of the wire takes us into the realm of the calculus and the process of integration. Or we can use Gauss=s law.

Imagine that we have established an imaginary soap bubble in the shape of a tuna can (a short cylinder) in such a way that the field point lies on its side and the wire passes through the centers of its top and bottom. Because the electric field has no component perpendicular to them, the top and bottom of the Acan@ contribute nothing to the electric flux through the bubble. Looking at the symmetry of the situation tells us that the electric field has the same intensity everywhere on the side of the Acan@, so we calculate the flux through that side by multiplying its area by the presumed strength of the electric field. Gauss=s law then tells us to equate that flux to the amount of electric charge enclosed within the bubble divided by the electric permittivity of vacuum. We can then solve that equation readily to find that the electric field has a strength that we can calculate by dividing the linear density of electric charge on the wire (in Coulombs per meter) by the product of two pi, the electric permittivity of vacuum, and the distance of the field point from the wire. In using Gauss=s law we have reduced a problem that would have taken us into the realm of the calculus to one involving only simple algebra.

Thus we have the laws of electrostatics, the laws pertaining to electricity standing still. But when we consider electricity in motion we will discover something breathtakingly beautiful. We will infer the existence and form of a relativistic phenomenon that you don= t have to move near the speed of light to experience.

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