Revisiting the Force Laws
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In this essay I offer a recapitulation of the means of deducing the laws governing the two forces that emanate from electrically charged bodies. At the same time I add a few more pieces to the puzzle, new connections within the Map of Physics.
We take as a postulate the statement that there exists a property of matter, which property we call electric charge, by virtue of which a particle possessing that property exerts a force upon any other particle possessing that property. We don=t yet know why particles possess force exerting properties, so we must assume their existence into our premises as a postulate. Now we want to apply the conservation laws and other fundamental features of Reality as constraints and see how they shape the force emanating from electrically charged particles.
We begin by asserting a second postulate, one intended to avoid introducing Albert Einstein=s Aspooky actions at a distance@. Thus we posit the statement that a kind of aura, which we call a forcefield, emanates from the property of electric charge and spreads out into space. Wherever that aura touches an electric charge, it exerts a force upon that charge.
For our first inference we state that when a forcefield touches a charge it does not affect the forcefield emanating from that charge. If that statement did not stand true to Reality, then two charged particles would engage each other in a positive feedback process that would either intensify both fields toward infinite strength or diminish them to zero. The first possibility, in accordance with the finite-value theorem, permits violations of the conservation laws, or we must dismiss it. And the second possibility contradicts our postulate that charges exert forces, so we dismiss it.
Further, we state that at a given distance from the generating charge the field emanating from that charge cannot change at all. If that statement did not stand true to Reality, then we could move a charged particle from one point to another in a forcefield, doing a certain amount of work, change the strength of the field, then move the particle back to its point of origin, doing a different amount of work. That process constitutes a violation of the law of conservation of energy, so we must dismiss from our logic the proposition that enables it. Thus at a given distance from a charge the field emanating from that charge has a permanently fixed value.
That fact necessitates that where the fields emanating from two charges meet, the combined field must equal the vector sum of the two component fields. Thus we obtain the superposition principle of forcefields.
Likewise, a given amount of charge cannot change in itself. If that proposition did not stand true to Reality, then we could contrive a violation of the law of conservation of energy by moving a charged particle from one point to another in another particle=s forcefield, doing a certain amount of work, change the amount of charge on the particle, and then move the particle back to its point of origin, doing a different amount of work. Thus we must infer that we can only change the amount of electric charge on a body by bringing charge to the body or moving charge off the body, but we cannot create charge ex nihilo or annihilate it. Thus subject to a conservation law, electric charge must conform to the equation of continuity,
in whichρ represents the density of electric charge at a point and i represents the density of electric current (i=ρv) flowing through that point.
We have assumed that the forcefield provides the means by which one electrically charged body exerts a force upon another electrically charged body, so we have tacitly assumed that the strength of the force has some dependence upon the amount of charge on the two bodies. If we use q to represent quantity of electric charge, then we represent the force that body-1 exerts upon body-2 as F12(q1,q2). We now want to determine the full algebraic expression describing that force.
If we augment the charge on body-1 by bringing more charge to the body, then, in accordance with the superposition principle, we augment the strength of the field emanating from body-1 in the same proportion; that is,
though at this stage we don=t know the relationship between the field strength (E) and the exerted force.
Imagine moving a particle carrying a charge q1 from Aeffectively infinity@ to some point a distance R from a particle carrying charge q2. Because the particles each exert a force upon the other, we must do a quantity of work W12 against that force (if it=s repulsive) or extract that much work (if the force is attractive) from the movement of q1. Now move a particle carrying a charge q3=q1 from effectively infinity to the point occupied by q1. In accordance with the superposition theorem the forcefield acting on q3 consists of the vector sum of the fields emanating from q1 and q2, so the work done in bringing q3 to the point occupied by q1 corresponds to the sum of two parts (W31+W32) B the work that would be done in bringing q3 to q1 if q2 were not present (W31) and the work that would be done to bring q3 to a point a distance R from q2 if q1 were not present (W32). Because the presence of q1 does not affect the field emanating from q2, we know that W32=W12. Now move q2 to effectively infinity while holding q1 and q3 stationary. Conservation of energy obliges us to obtain the amount of work W32+W12=2W12. Doing twice the work while crossing the same distance as q1 did necessitates of q2 that it be subject to twice the force that was exerted upon q1. We have effectively doubled the charge on q1, so, by way of mathematical induction, we infer that the force exerted between two bodies stands in direct proportion to the amount of charge on the forcing body. Then, by virtue of Newton=s third law, we also infer that the force also stands in direct proportion to the charge on the forced body, so that we have as true to Reality
and then, in light of Equation 2,
Newton=s third law of motion tells us that the force that body-1 exerts upon body-2 has the same magnitude, but the opposite direction of the force that body-2 exerts upon body-1; that is,
The law of conservation of angular momentum dictates that the laws of force have such a shape that those two forces cannot exert an unbalanced torque, which would create endless angular momentum ex nihilo. Therefore, those forces must act along the straight line passing from one of the forcing bodies to the other. That fact necessitates that the field generating the force must, at every point in space, point along the radial line passing from the generating body through the point.
Conservation of angular momentum also necessitates that the forcefield have zero curl,
from which we can infer, by way of the vector relation , that the forcefield represents the gradient of some potential function,
That zero-curl criterion tells us that a uniformly charged wheel placed in a forcefield will not come under a net torque and turn of itself. That fact, in turn, necessitates that the strength of the field has the same value at all points standing a given distance from the generating body and that the field has no components in the longitudinal or latitudinal directions. So now we know that
a function of radial distance only.
Combining Equations 4 and 7 tells us that a particle carrying a charge q in a field of potentialφ carries an extra amount of energy
In accordance with Einstein=s theory of Special Relativity, that energy changes the mass of the particle by an amount
Imagine a straight wire with an electric current flowing through it. The conduction electrons move with an average drift velocity of v and the rest of the wire remains motionless for us. A potentialφe emanates from the conduction electrons and a potential φw emanates from the rest of the wire. When the current in the wire has reached equilibrium we have φe+φw=0, so a charged particle near the wire neither gains nor loses mass through its interactions with the potential fields. If the source of a potential field moves (as do the conduction electrons), then the field must also confer linear momentum upon a charged particle immersed within it,
represents what Michael Faraday and James Clerk Maxwell called the electrotonic state and we call the magnetic vector potential. We know that an applied force equals the rate at which a body=s linear momentum changes (Newton=s second law), so we expect that anything that changes a body=s linear momentum constitutes a force; thus, we expect to have an electric field
in which V represents the velocity of our charged particle relative to the wire. Because V does not represent an actual vectorfield, we must treat it as a constant, so that equation becomes
Thus we obtain the Feynman derivation of magnetism (named after Richard P. Feynman, who, as far as I know, was the first person to deduce magnetism from applying the results of Special Relativity to the description of a simple electric current).
Define the magnetic induction field as
If V=0, the Equation 14 gives us Faraday=s law of electromagnetic induction,
the third of Maxwell=s Equations. Applying the fact that the divergence of a curl always equals zero to Equation 15 also gives us the second of Maxwell=s Equations,
Gauss=s law of the magnetic field.
Point by point, the potential of the electric field, representing a potential for a potential energy, must conform to the conservation law, so, as a continuous entity, its description must conform to the equation of continuity
But, as Equation 12 shows, , so if we substitute that into Equation 18 and divide the result by the square of lightspeed, we get
which expresses the Lorentz condition.
Imagine that we have taken a charge Q and compressed it as much as possible toward occupying a single mathematical point. And imagine that we have taken a minuscule charge dq of the opposite algebraic sign from Q and spread it uniformly over a spherical shell of radius r whose center coincides with the point occupied by Q. A radially directed force, F=Edq, thus pulls the shell inward toward Q. If the radius of the shell shrinks by dr, then the system has done an amount of work equal to
In accordance with our rejection of spooky actions at a distance, we assert that the work moves energy into or out of something that occupies the place where the work is done when the work is done. Because we conceive energy as expressing a relationship between bodies, we assume that the energy comes out of the shell. But we have another possibility if we are willing to give it the reality status of a body and not merely conceive it as nothing more than a useful fiction.
As the shell shrinks it diminishes the electric field emanating from Q by the amount dE, whose form we have yet to determine; that is, as the shell passes over a point the field dE emanating from the shell cancels part of the field E emanating from Q. But if we insist that the work done on or by the shell remain local, then the change in energy comes manifest in the electric field only where the shell crosses the field while doing the work. We have already established that E is proportional to Q, but we cannot on that basis infer that dE is directly proportional to dq, because dq is spread over a wide area, not concentrated at a point. If the field intensity changes at some point, then it can only do so due to an action of the part of the shell touching that point. That fact necessitates that the surface density of the electric charge () takes the active role in making the change, so we have dE proportional to dλ. We can thus rewrite Equation 20 as
in which the factorε0 transforms the proportionality in the previous sentence into an equality. As with the speed of light, we must, as far as we know at this stage, determine the value of that proportionality constant, the electric permittivity of vacuum, through observation or experiment. Also in that equation we have ε0EdE representing an increment of the density of the energy stored in the forcefield, so for the actual energy density we have
Regardless of whether it puts energy into the shell or into the forcefield, the work done must have the same value, so we can equate Equations 20 and 21 with each other, which necessitates that
If we establish an electric field by accumulating compact charges dq at a point, the Equation 2 and the principle behind it necessitates that we have
which expresses Coulomb=s law of the electric field.
Assume that we have an array of charge of uniform densityρ organized into a sphere of radius R. From Equation 24 we calculate the strength of the electric field at each point, bearing in mind the fact that inside any spherically symmetric distribution of electric charge the contribution of that charge to the field equals zero (one consequence of Coulomb= s law), and get:
1. for r# R we have
2. for r>R we have
If we calculate the divergences of Equation 25 and 26 (keeping in mind the facts that and ), then we get for r# R
and for r>R
Recalling to mind the fact that we can represent any electric field as the superposition of fields emanating from an array of point charges and that the operations of addition and differentiation commute with each other, we can see that Equation 28 gives us Maxwell=s first equation, Gauss=s law of the electric field, so long as we remember thatρ must represent the density of electric charge occupying the point where we calculate the divergence of the electric field.
If we use Equation 28 to substitute for the charge density in the equation of continuity (Equation 1), we get
for some putative field H. Thus we have
That equation looks like Ampere=s law, but at this stage we must feign not to know that H represents the magnetic field. Calculating the inner product of that equation with the electric field gives us
In light of Equation 22, we can see that the second term on the right side of the equality sign represents the rate at which an energy density changes with the elapse of time. If we conceive Equation 32 as a kind of continuity equation expressing the conservation of energy, then we must conceive the vector product ExH (Poynting=s vector) as representing a flux of energy density.
We already have a basic description of the magnetic vector potential in the form
For the electrostatic potential of the moving charges I substitute
in which I leave the integration over volume hanging fire. I then get
in which i represents the electric current density and represents the magnetic permeability of vacuum, a number that I have here defined from the electric permittivity of vacuum and the speed of light for convenience (classical physicists defined it from a different calculation and later related it to the electric and optical constants). We want to calculate the double curl of that vector field, so we have
in whichδ(r) represents the Dirac delta and I have already carried out the implicit integration in the first term on the right side of the equality sign. Because the argument of the Dirac delta goes to zero wherever we find an element of electric current, we can integrate it and get finally
which is Ampere=s law. But that looks just like Equation 31, so now we know that H=B/μ0.
Take Maxwell= s first equation in the form
Replace of the magnetic vector potential from the Lorentz condition and multiply through by minus one to get
Now take Maxwell= s fourth equation (Ampere=s law) in the form
Applying the appropriate vector identities gives us
Taking the two gradients (first and fourth terms) under the Lorentz condition and multiplying through by minus one gives us
We have tacitly brought Maxwell=s third equation, Faraday=s law (in the form E=-M A/M t) into Maxwell=s first equation in Equation 38, so Equation 39 encodes the first and third of Maxwell=s equations together. Likewise, in Equation 40 we have tacitly brought Maxwell=s second equation (through the definition B=L xA) into Maxwell=s fourth equation, so Equation 42 encodes the second and fourth of Maxwell=s equations together. Equations 39 and 42 thus constitute Maxwell=s Equations expressed in terms of the potentials rather than of the forcefields. Those equations look like four-dimensional analogues of Poisson=s equation and show more clearly the symmetry between electricity and magnetism than do the standard four equations. That fact lets us take one final step.
We now replace space and time with Hermann Minkowski=s spacetime. We define the four-dimensional analogue of the Laplacian operator, the d=Alembertian, as
We then represent the sources and the potentials as four-vectors. By conceiving the electric charge density as the timeward component of a spatio-temporal current density we get
in which the index alpha takes the successive values 1, 2, 3, 4. By conceiving the electrostatic potential as the timeward component of a spatio-temporal vector potential we get
in which we apply our original definition A=vφ/c2 with v=c in this case. If we divide Equation 39 by the speed of light and add it to Equation 42, we get at last
which now encodes all of Maxwell=s electromagnetic theory in one simple equation.
We seem to have lost a lot of information in taking that step. I can see no way to use that equation to solve real problems in electromagnetism (Oh, there=s a challenge for someone!): the field equations give us so much more information to work with. On the other hand, we can see in that equation a clear symmetry that displays the sheer elegance of the fundamental laws governing Reality. We expect no less from a purely Rationalist physics and of the Reality that it describes.
That fact raises a new challenge. In order to perfect the Map of Physics we must now figure out how to deduce Equation 46 from the fundamentals of Existence and then how to unfold it into the familiar form of Maxwell=s Equations and the subsidiary equations of electricity and magnetism that go with them. That will have to be the topic of a future essay in this series.
Appendix: The Original Maxwell= s Equations
In 1864 James Clerk Maxwell published twenty equations that he called AGeneral Equations of the Electromagnetic Field@, which equations summed up all of electromagnetic theory as it was known at the time. Modern mathematical notation, primarily our modern way of representing vectors, reduced those equations to eight. In the following equations I have altered the notation slightly to reflect my use of MKS units rather than the CGS units that Maxwell used. The original Maxwell Equations are:
AThe relation between electric displacement, true conduction, and total current, compounded of both.@ The total electric current density (j) equals the sum of the actual density of moving electric charge (i) and the time variation of the electric displacement (D=εE, in which ε represents the electric permittivity of the medium).
AThe relation between the lines of magnetic force and the induction coefficients of a circuit, as already deduced from the laws of induction.@
AThe relation between the strength of a current and its magnetic effects according to the electromagnetic system of measurement.@ Combined with Equation I, this gives us Ampere= s law, the fourth of the modern Maxwell Equations.
AThe value of the electromotive force in a body, as arising from the motion of the body in the field, the alteration of the field itself, and the variation of the electric potential from one part of the field to another.@ If we take the curl of this equation, we get Faraday= s law, the third of the modern Maxwell Equations.
AThe relation between electric displacement, and the electromotive force which produces it.@
AThe relation between an electric current, and the electromotive force which produces it.@ This is better known as Ohm=s law of electrical resistance. Because physicists use the Greek letter rho to represent both electric charge density and electrical resistivity, I have subscripted an omega to the rho that I use for resistivity.
AThe relation between the amount of free electricity at any point, and the electric displacement in the neighborhood.@ This is simply the first of the modern Maxwell Equations.
AThe relation between the increase or diminution of free electricity, and the electric currents in the neighborhood.@ This is simply the continuity equation expressing the law of conservation of electric charge.
Appendix: Condensing ěrsted= s Missed Opportunity
Let= s consider Hans Christian ěrsted=s discovery that an electric current exerts a force upon a magnet (in the form of a compass needle). We can devise a kind of syllogism, using Newton=s third law of motion as the major premise:
A; If X exerts a force on Y, then Y exerts a force on X.
B; electric current exerts a force on a magnet; therefore,
C; a magnet exerts a force on an electric current.
Next we purify or isolate the concept under consideration: we make a Apure@ electric current by putting electrically charged particles on a silk thread and then moving the thread. If we then move the thread between the poles of a magnet, the force will deflect the thread sideways. If I move with the thread and you stay with the magnet, then we take a fundamental property of Reality as our major premise:
A; Any observable effect that exists for you also exists for me.
B; You see the thread deflected away from the magnet=s axis; therefore,
C; I see the thread deflected away from the magnet=s axis.
But for me the charged thread does not move and, thus, does not represent an electric current, so:
A; Only an electric force can act on a stationary electric charge.
B; A moving magnet exerts a force on a stationary charge; therefore,
C; A moving magnet exerts an electric force.
Note how the word Aonly@ automatically excludes other possibilities and thus Aprunes@ the logical tree. In the conclusion of that last syllogism we now have the essence of Michael Faraday=s law of electromagnetic induction. Q.E.D.
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