Electromagnetic Energy Density

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We define energy as a relationship between two bodies. If one body exerts a force upon a second body and if the second body moves, we say that the first body does work upon the second body. If the motion of the second body increases, then we say that the work becomes kinetic energy in the second body, and if the motion of the second body decreases, then we say that the work becomes potential energy in the second body vis-a-vis the first body. Application of Newton’s three laws of motion tells us that the sum of a body’s potential and kinetic energies, the body’s total energy, remains constant unless and only unless the body exchanges work with another body, which means that energy comes under its own conservation law.

In the above scenario I have tacitly assumed that the first body exerts a force upon the second body by way of an ætherial entity that we call a forcefield. I assert that the first body possesses a property that fills the space around it with a ghostly presence that we might identify as a potential force at each point. And I assert further that the second body possesses a quantity of that property, which engages the forcefield and transforms the potential force into an actual force. With this concept Michael Faraday addressed the classical manifestation of what Einstein called "spooky actions at a distance".

Faraday conceived the idea of a forcefield as what he called an "aid to the imagination". He used the concept to coordinate his thinking about the forces exerted among electric currents and magnetically charged bodies (physicists still believed in the existence of magnetic charge back then and even today they accept it as a useful fiction in many applications). But when he strewed iron filings onto a sheet of paper in the presence of a magnet or an electric current and looked at the fiber-like patterns that they formed, did he see signs of a mere useful fiction (based on the human tendency to conceive patterns where none actually exist) or did he see the surface indication of an actual thing-in-itself, a true element of Reality?

To answer that question we must ask whether a forcefield possesses any property that would mark it as a real entity in its own right. Specifically, we ask whether a forcefield possesses energy. But what can we mean when we say that a forcefield possesses any quantity of something that we have defined as a relationship between two bodies?

Start with James Clerk Maxwell. In his 1861 paper, "On Physical Lines of Force", he mathematized Faraday’s forcefield concept by drawing an analogy between magnetic lines of force and streamlines in the flow of a fluid. In Maxwell’s analogy each streamline served as the axis of a vortex whose spinning at the circumferential velocity v modifies the distribution of pressure within the fluid. Applying the known laws of hydrodynamics, Maxwell noted that the stress in the fluid would resolve into a simple hydrostatic pressure p, acting in all directions, and a simple tension, which conforms to the equation

(Eq’n 1)

In that equation the reciprocal of 4π indicates that Maxwell used the CGS system of units (rather than the MKS system that I prefer) and the Greek letter mu has incorporated into it the geometric factor and the density of the fluid. Maxwell then defined α=vl, β=vm, and γ=vn, in which l, m, and n represent the direction cosines between the axis of the vortex and the x-, y-, and z-axes of the coordinate grid respectively, and used them to write the algebraic description of the stresses acting in the fluid,

(Eq’ns 2)

Maxwell used those equations to work out a description of the force that would act on a unit volume of the medium due to variations in the internal stress in the medium. Applying the law governing the equilibrium of stresses, he calculated the x-component of the force,

(Eq’n 3)

in other words, he equated the force density to the gradient of the stress. (In that equation I have begun replacing Maxwell’s notation with modern notation, replacing Maxwell’s X with Fx.) Substituting into that equation from Equations 2 yields

(Eq’n 4)

Making note of the mathematical fact that

(Eq’n 5)

Maxwell rewrote the second term of Equation 4 as

(Eq’n 6)

in which I have continued replacing Maxwell’s notation with modern notation by using the vector H whose components are α, β, and γ. Equation 4 then becomes

(Eq’n 7)

To interpret the meaning of each term in that equation, he supposed that α, β, and γ represent the x-, y-, and z-components of the magnetic field, specifically referring to them as "the components of the force which would act upon that end of a unit magnetic bar which points to the north."

In the first term the divergence equates to, via Gauss’s law, the density of the magnetic charge at the point at which we calculate the field. We know today that magnetic monopoles do not exist, so we also know that the divergence must equal zero. Maxwell himself called that charge density "imaginary magnetic matter". As the product of a charge and the field, the first term gives us the magnetic analogue of Coulomb’s law for the electric field.

To calculate the second term we must multiply the magnetic inductive capacity of the medium (μ) by the gradient of the square of the magnetic field. As Maxwell put it, "Any body therefore placed in the field will be urged towards the place of stronger magnetic intensity with a force depending partly on its own capacity for magnetic induction, and partly on the rate at which the square of the intensity increases." In other words, if we place a magnetically neutral body in the field, then the force that the field exerts within the body polarizes it, shifting south magnetic poles in one direction and north magnetic poles in the opposite direction, and, in consequence of the field at different intensities acting on equal and opposite poles, the body drifts toward the densest part of the field.

Taken together, the third and the fourth terms give us a description of what we regard as the standard magnetic interaction. The terms enclosed within parentheses represent components of the curl of the magnetic field. In regions of space devoid of time-varying electric fields, Ampere’s law relates that curl to the electric current density at the field point,

(Eq’ns 8)

The third and fourth terms thus become

(Eq’n 9)

which gives us the x-component of the vector cross product between the current density and the magnetic induction field,

(Eq’n 10)

Thus we have a proper description of the force generated through the interaction between an electric current and a magnetic field.

Finally, the fifth term describes the negative gradient of the hydrostatic pressure in the fluid. In the magnetic analogy this term corresponds to the negative gradient of a magnetostatic potential, which has the same mathematical form as does the electrostatic potential associated with electrically charged bodies. Around any magnet that we can represent as an array of north and south magnetic monopoles we would use such a magnetostatic potential, rather than using the usual magnetic vector potential, which describes the electrotonic field associated with electric currents.

Thus Maxwell used a hydrodynamic model based on analyzing pressures within bundles of vortices in a fluid and obtained a mathematical description that mimics perfectly the forces exerted in a magnetic system. He had devised a "mechanical hypothesis as to the condition of the medium indicated by lines of force, by which the observed resultant forces may be accounted for". He added, "We found that the velocity of the circumference of each vortex must be proportional to the intensity of the magnetic force, and that the density of the substance of the vortex must be proportional to the capacity of the medium for magnetic induction." He then asked, "Why are they [the vortices] arranged according to the known laws of lines of force about magnets and currents?"

Maxwell used the concept of energy to answer that question. In his mechanical model the energy density manifested in a bundle of vortices stands in direct proportion to the density of the fluid and to the square of the circumferential velocity of a vortex. A similar term in the mathematical description of the force density in a magnetic system, as derived from experimental results, relates a contribution to the force density to the gradient in a product proportional to "the capacity of the medium for magnetic induction" (a quantity that we now call the magnetic permeability of the medium) and the square of the magnetic field intensity. Comparison of the two descriptions implies that the magnetic field contains an energy density of

(Eq’n 11)

Again, though, we ask whether the energy actually exists in the magnetic field or whether the mathematical description that implies as much merely gives us a useful fiction. In the case of the mechanical model we have a fluid consisting of particles that move relative to particles in the wall of the container holding the fluid or particles in our instruments; thus, we still have the energy in the fluid as a sum of the relationships between particles in pairs. In the case of the magnetic field, on the other hand, we have associated no particles and no motion, so what kind of relationship could we propose that we could equate with energy?

In Maxwell’s analogy we equate magnetic field strength to a velocity and, even though we define energy as a relationship between two bodies, we can attribute a kinetic energy to a single particle by relating that particle’s velocity to a stationary system, which defines a state of zero velocity. That latter fact suggests that we can assign energy to a magnetic field by relating the field’s intensity to a state of zero magnetic field. And just as the mass of a particle augments the square of velocity to yield a description of the particle’s energy, so the magnetic permeability of the medium augments the square of the magnetic field intensity to yield a description of the field’s energy density.

Anybody can apply Maxwell’s analysis to an electric field and get results comparable to those that Maxwell got for the magnetic field. That analysis yields a description of the energy density in an electric field of strength E as

(Eq’n 12)

in which epsilon represents the electric permittivity of the medium. But we also have another way to deduce the form of the electric energy density.

Imagine an electrically neutral sphere covered with a thin, infinitely-expandable rubber skin. Pull a quantity +Q of positive electric charge into the skin, leaving a quantity -Q of negative electric charge on the sphere, and then expand the skin in a way that maintains its spherical symmetry. Outside the skin the electric field intensity equals zero and between the skin and the sphere the electric field intensity, in accordance with Coulomb’s law, conforms to the description

(Eq’n 13)

in which ε0 represents the electric permittivity of vacuum and r represents the distance of the field point from the center of the sphere. The field engages the charge on the skin and exerts a force that acts to pull the skin back onto the sphere, so to expand the skin by an increment of distance dr the system must do work on the skin in accordance with

(Eq’n 14)

Of that work, the charge covering a minuscule element of area da on the skin does

(Eq’n 15)

Though the energy density in that equation comes out twice as large as the amount we have in Equation 12 (an issue to which I will return below), it has the same mathematical form.

That imaginary experiment shows an electric charge passing over a certain volume and leaving an electric field behind, which implies that work done on expanding the charged skin goes into creating the electric field. That statement implies that the energy produced by the work resides in the electric field. But the force that does the work acts on the electric charge in the skin and that fact implies that the energy exists in the relationship of force between the electric charge on the skin and the electric charge of the surface of the sphere. With that ambiguity we still have no proof of the hypothesis that forcefields exist in their own right and not as mere useful fictions. Nonetheless, we do have a proof.

In 1884 John Henry Poynting (1852 Sep 09 – 1914 Mar 30), who studied physics under the guidance of Maxwell, published "On the Transfer of Energy in the Electromagnetic Field". He wrote in the opening paragraph of that paper that "If we believe in the continuity of the motion of energy, that is, if we believe that when it disappears at one point and reappears at another it must have passed through the intervening space, we are forced to conclude that the surrounding medium contains at least a part of the energy, and that it is capable of transferring it from point to point." In vacuum the medium in Poynting’s statement consists of an electromagnetic field. Poynting then derived a simple mathematical description of the flux of energy through an electromagnetic field. In modern form we have this:

Begin with the standard representation of the energy density in an electromagnetic field, whether real or fictional,

(Eq’n 16)

Calculate the rate at which that density changes at a point due solely to the elapse of time,

(Eq’n 17)

Replace the time derivatives on the right side of that equation with their equivalents via Faraday’s law and Ampere’s law,

(Eq’n 18)

That move yields

(Eq’n 19)

A minor rearrangement of that equation yields Poynting’s theorem,

(Eq’n 20)

in which we have the Poynting vector

(Eq’n 21)

In current-free space (the current density J = 0) Equation 20 has the form of an equation of continuity. That identification makes the Poynting vector the description of the flux of energy through an electromagnetic field. But does that fact make the field real?

Consider a simple example. Imagine a long, straight copper wire. One end touches one electrode of an electric battery and the other end touches a suitable grounding surface. In accordance with Ohm’s law, V=IR, a voltage difference V must exist between the ends of the wire in order to make an electric current I flow against a resistance R. Acting over the full length L of the wire, the voltage generates an electric field of intensity E=V/L. Driven by that electric field, the current generates a magnetic field around the wire, its intensity at the wire’s surface equal to H=I/2πr (in which r represents the radius of the wire) in accordance with Ampere’s law (calculated from the integral version of the law). If someone were to wrap their right hand around the wire in such a way that their thumb points in the direction of the current, then their fingers will curl in the direction in which the magnetic lines of force circulate.

Because net electric charge cannot accumulate in or on the wire, the electric field, running parallel to the wire, must extend into the space surrounding the wire and mingle with the magnetic field. The associated Poynting vector, S=EH, points from space into the wire. If we multiply that number by the surface area of the wire, 2πrL, then we get the rate of total energy flowing into the wire from the electromagnetic field as

(Eq’n 22)

which we associate with the rate at which heat appears in the wire.

But, although Equation 22 gives us a correct calculation of the rate at which the wire gains heat, the conventional explanation tells us that the heat comes from collisions among the conduction electrons and the atomic lattice of the copper. Electric friction within the metal and not a flux of some ætherial caloric seems to give the better account of the wire’s warmth, coming closer to a properly materialistic concept of the features of Reality.

A different experiment yields a different result. Maxwell’s Equations tell us that if an electric current sloshes back and forth in a wire, that current will emit waves of electric and magnetic field. Those waves propagate through space at a speed that makes each field regenerate the other in the act of propagation. The fields also cross each other at a right angle, so a Poynting vector oriented in the direction of the wave’s propagation associates an energy flux with the wave.

Because of radiation reaction against the emitting wire, the circuit loses energy. If at some later time the wave should strike a body and get absorbed, that body will gain energy (think of the heating of a body that absorbs light coming from the sun). Invoking conservation of energy and paraphrasing Poynting, we can say that if some phenomenon makes energy disappear at one point and if energy appears later at another point in association with the same phenomenon, then we must infer that the energy travels with the phenomenon, that, in the present example, it accompanies the electromagnetic wave as one of the wave’s properties.

In order to possess real properties an entity must have a real existence as a thing-in-itself, so we necessarily infer that the electric and magnetic fields in an electromagnetic wave exist as real entities and not only as mere useful fictions. But those free-flying fields originate in the vibrations of the fields emanating from electric charges and electric currents, so we extend our inference and claim that the fields emanating from properties possessed by matter also exist as things-in-themselves.

Does that inference then necessitate that the Poynting vector give the correct accounting for electrical resistance heating in a wire? More to the point, does the energy that causes the heating travel outside the wire in the Poynting vector or does it travel with the conduction electrons within the wire? If I say both, then I have a serious problem with the conservation of energy theorem, but the above analysis doesn’t give me the option of choosing one explanation over the other.

In order to resolve that dilemma I need to look ahead to one of the great discoveries of Twentieth-Century physics – the quantum theory. Indeed, the very foundation of that discovery gives us the resolution that we need: right at the turn of the century Max Planck, Albert Einstein, and other physicists discovered that electromagnetic waves act as if they consist of flocks of discrete particles, quanta that we now call photons. So now we must reconceive forcefields from things-in-themselves to properties of particles. Emanating from electrons or other charged particles, solely electric when the particles stand still and also magnetic when the particles move, the forcefields solve the mystery of action at a distance, certainly. Real enough to account for the forces exerted among particles, they don’t exist really enough to possess properties, such as energy, of their own. Thus, in the electrically heated wire the conduction electrons carry energy and lose it to the atomic lattice while the Poynting vector simply gives us a convenient means to calculate the rate of energy deposition. When they vibrate about some point in space electric particles shed photons, which carry electromagnetic fields as one of their properties. In an electromagnetic wave the fields show, via the Poynting vector, whither the photons go, but only the photons carry properties, such as energy and momentum.

Thus we gain a little extra knowledge about the nature of forcefields.

Appendix: The Real Energy Density

Imagine a metal sphere of radius r0 enclosed in a concentric spherical metal shell of radius r1>r0. Using a wire connecting the sphere to the shell, move a charge Q from the initially uncharged sphere to the shell. To achieve that movement we must use an electric generator to apply a voltage difference along the length of the wire to do the work of moving the charge against the electric field that grows in strength as more charge gets moved. When the system has moved a quantity q of charge the electric field between the sphere and the shell has the strength

(Eq’n A-1)

To move a minuscule element of charge dq against that field from the sphere to the shell requires that the system do an amount of work

(Eq’n A-2)

If we integrate that expression from q=0 to q=Q, we get a description of the total work that the system must do in order to move the total charge Q from the sphere to the shell,

(Eq’n A-3)

On the other hand, the system can include a rubber skin on the sphere and use it to transfer the whole amount Q from the sphere to the shell in one discrete move. In this case the movement of the skin creates the electric field at its maximum strength all at once,

(Eq’n A-4)

The total force that acts to pull the charged skin back onto the sphere equals the product QE, so the work that the system must do to pull the charge from the sphere to the shell conforms to

(Eq’n A-5)

For the all-at-once method of moving Q to require twice as much work as does the trickle method calls into question the theorem pertaining to conservation of energy. If the system were to use the trickle method to move the charge Q from the sphere to the shell and then use the all-at-once method to return that charge to the sphere, it would act out a repeatable cycle that would create a net increase in energy on every repetition. That such a simple manipulation of Coulomb’s law should create energy ex nihilo, in defiance of one of the conservation laws, strikes us as incredible; thus, we assume that the above analysis contains an error in reasoning, which we must find and correct.

Here we need to look more closely at the connection between an electric charge and the electric field that emanates from it. We conceive the fundamental increment of electric charge, such as that on an electron, as a point-like entity. But we also know that nothing can occupy a single point and actually exist, so we must reconceive the unit of charge as something that occupies a minuscule but nonzero volume. We want to avoid conceiving a picture (such as that of a party balloon or a dust bunny) of an entity of which light cannot form an image; instead, we simply acknowledge the fact that the charge occupies a nonzero volume and, therefore, that it possesses the area of a closed surface, presumably a spherically symmetrical one, enclosing that volume.

From that area the electric field emanates, spreading uniformly with spherical symmetry in all directions. Where that field touches an electric charge it exerts a force upon that charge, so the field must certainly exert a force upon the charge from which it emanates. The uniformity and symmetry of the charge’s distribution and of its electric field ensure that the particle carrying the charge will not self-accelerate, in violation of the law pertaining to conservation of linear momentum (though the charge’s interaction with its own electrotonic field accounts quite nicely for the particle’s inertial reaction to a force applied to the particle from outside the particle).

For the present purpose we emphasize the fact that, as seen through any perspective, the charge has two sides. We also need to remember that when electric charge spreads uniformly across a thin spherical shell, the electric field inside the shell goes to a perfect zero. Now we see how to resolve our dilemma.

If a charge Q covers a thin spherical shell of radius rs, an electric field of strength

(Eq’n A-6)

touches the outward side of the charge and a field of strength zero touches the inward side of the charge. If a tiny sphere bearing charge -Q appears in such a way that its center coincides with the center of the shell, the electric field that emanates from it in accordance with Equation A-4 touches the inward side of the charge on the shell with the strength shown in Equation A-6 and cancels out the field touching the outward side of the charge on the shell. If the charge on the shell were to migrate across the shell to accumulate inside a minuscule volume around a point on the shell, then the field emanating from the sphere would touch both sides of the charge, generating a force in accordance with Coulomb’s law. Taken together, those facts tell us that in our derivation of Equation A-5 we should have used half as much force and thus obtained the amount of work given in Equation A-3. Thus we resolve the dilemma laid out above.

So now we know the correct form for the energy density in a forcefield. At a point where an electric field has the strength E the energy density conforms to

(Eq’n A-7)

and at a point where a magnetic field has the strength H the energy density conforms to

(Eq’n A-8)

eabf

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