THE RELATIVISTIC ELECTRIC FORCE
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Up until the beginning of the Nineteenth Century natural philosophers (as scientists were called so long ago) could only study electricity in its static form. They simply didn=t have the means to create flows of enough electricity moving for a long enough time for them to make any meaningful observations or measurements. That situation changed in 1801 when Count Alessandro Giuseppe Antonio Anastasio Volta (1745 Feb 18 B 1827 Mar 05) invented the electric pile, a crude form of what we call today an electric battery. Volta had drawn inspiration from the experiments in the 1770's conducted by Luigi Galvani (1737 Sep 09 B 1798 Dec 04), in which experiments Galvani discovered that touching the legs of dead frogs with instruments made of different metals made the frogs= legs twitch. Those experiments also inspired Mary Shelley, in 1816 (the suitably dreary AYear Without a Summer@) to compose AFrankenstein; or, the Modern Prometheus@, her own version of the golem tale.
Once they had strong, long-lasting electric currents available to them from their Voltaic electric piles scientists of the time discovered that electricity in motion displayed an amazing phenomenon. Even more amazing, we see the fact that, although the scientists originally discovered the phenomenon empirically, we can deduce the phenomenon rather easily by applying the theory of Relativity to electricity in motion.
To the best of my knowledge, Richard Feynman (1918 May 11 B 1988 Feb 15) stands in science as the first person to conceive at least the first part of the following:
In the laboratory of your imagination set up a long straight wire suspended horizontally in front of you. Attach the ends of that wire to a battery so that the wire carries an electric current flowing from left to right (in the positive x-direction in our standard coordinate grid). We know that a positive electric current flowing from left to right comes from the negatively charged conduction electrons flowing from right to left within the stationary positively charged atomic matrix of the metal comprising the wire. Let=s assume that all of those electrons move at a uniform speed, which we call the drift speed, and that the electric current has reached a long-term equilibrium, so that the wire displays no net electric charge.
In their own inertial frame the conduction electrons have a certain charge density (that is, so many coulombs of electric charge per cubic meter of volume within the wire). In the inertial frame occupied and marked by the atomic matrix of the wire the Lorentz-Fitzgerald contraction acts to increase that density by way of the lengthwise shrinkage of the moving space occupied by the ensemble of conduction electrons. But by our equilibrium criterion the negative charge density of the conduction electrons must equal in magnitude the positive charge density of the atomic matrix. That means that the conduction electrons have spread themselves out within the shrunken space that they occupy to give themselves the same charge density that the body of the wire has.
Imagine further that we have a particle carrying a small amount of electric charge moving from left to right at a set speed relative to the wire=s atomic matrix, which we call the test speed, and maintaining a fixed distance from the wire. An observer moving with that particle detects the wire=s atomic matrix moving from right to left at the same speed and the conduction electrons moving from right to left at a speed that we calculate from the test speed and the drift speed by way of the relativistic velocity addition formula. Consequently, in that frame, the Lorentz-Fitzgerald contraction will increase the charge density of the atomic matrix by an amount proportional to the Lorentz factor of the test speed. It will also increase the charge density of the conduction electrons by an amount proportional to the Lorentz factor of the test speed, the Lorentz factor of the drift speed, and a number that we calculate by multiplying together the drift speed and the test speed, both expressed as a fraction of the speed of light, and adding one to the product. If we add those charge densities together, making sure to take proper account of the positive and negative signs, then we discover that in the charged particle=s frame the wire carries a net negative charge density proportional to the product of the Lorentz factor of the drift velocity with the product of both the test and drift velocities expressed as fractions of the speed of light.
That excess charge density must generate an electric field, one that will exert a force in a direction perpendicular to the orientation of the wire. We know that such a field must exert a force upon the charged particle and make it accelerate toward or away from the wire. But if the particle accelerates in the inertial frame moving at the test speed relative to the wire= s atomic matrix, then it must accelerate in all other inertial frames, particularly the frame occupied by the wire=s atomic matrix, the frame in which, by our establishing assumptions, the wire displays no electric field.
In the light of that analysis we must infer that the electric current flowing in our wire generates some kind of field, an entity that pervades the space around the current, that displays two aspects: in some frames it displays an electric field, in other frames it exerts a force upon moving electrically charged particles, and in most frames it does a little of both. Of course, we recognize that Asome kind of field@ as a magnetic field. Given the two aspects that it displays we could do little better at this point than to review the first two paragraphs of Einstein= s original essay on Special Relativity, AOn the Electrodynamics of Moving Bodies@ :
AIt is known that Maxwell=s electrodynamics B as usually understood at this present time B when applied to moving bodies, leads to asymmetries which do not appear to be inherent in the phenomena. Take, for example, the reciprocal electrodynamic action of a magnet and a conductor. The observable phenomenon here depends only on the relative motion of the conductor and the magnet, whereas the customary view draws a sharp distinction between the two cases in which either the one or the other of these bodies is in motion. For if the magnet is in motion and the conductor at rest, there arises in the neighborhood of the magnet an electric field with a certain definite energy, producing a current at the places where parts of the conductor are situated. But if the magnet is stationary and the conductor in motion, no electric field arises in the neighborhood of the magnet. In the conductor, however, we find an electromotive force, to which in itself there is no corresponding energy, but which gives rise B assuming equality of relative motion in the two cases discussed B to electric currents in the same path and intensity as those produced by the electric forces in the former case.
AExamples of this sort, together with the unsuccessful attempts to discover any motion of the earth relatively to the >light medium,= suggest that the phenomena of electrodynamics as well as of mechanics possess no properties corresponding to the idea of absolute rest. They suggest rather that, as has already been shown to the first order of small quantities, the same laws of electrodynamics and optics will be valid for all frames of reference for which the equations of mechanics hold good. We will raise this conjecture (the purport of which will hereafter be called the >Principle of Relativity=) to the status of a postulate, and also introduce another postulate, which is only apparently irreconcilable with the former, namely, that light is always propagated in empty space with a definite velocity c which is independent of the state of motion of the emitting body. These two postulates suffice for the attainment of a simple and consistent theory of the electrodynamics of moving bodies based on Maxwell=s theory for stationary bodies. The introduction of a >luminiferous aether= will prove to be superfluous inasmuch as the view here to be developed will not require an >absolutely stationary space= provided with special properties, nor assign a velocity-vector to a point of the empty space in which electromagnetic processes take place.@
But we haven=t finished our contemplation of electricity in motion. If we ask what happens if we have an electrically charged particle moving in a direction perpendicular to our current-carrying wire, Feynman=s derivation answers ANothing@ (and I can=t say for certain that Feynman took this next step in his thought experiment). The excess electric charge that accumulates on the wire due to the differential Lorentz-Fitzgerald contraction of the wire=s components remains relatively unaffected by motions perpendicular to the orientation of the wire.
Another relativistic effect comes into play, though. Imagine that we sit in the frame occupied and marked by the wire=s atomic matrix and that for a brief instant that atomic matrix ceases to exist, leaving a thread of conduction electrons moving before us. Those electrons fill the space around them with an electrostatic potential. If we immerse a particle bearing an electric charge in that field, then the particle will gain a potential energy equal to the product of the charge and the potential and, by the mass-energy equivalence theorem, thereby gain an increment of mass equal to that potential energy divided by the square of the speed of light. Likewise, each electron in the current gains an increment of mass due to its immersion in the potential field surrounding the charged particle.
As we saw when we deduced the mass-energy equivalence theorem, whenever a body gains or loses mass by direct absorption or emission of energy, a force must act upon the body to compensate the linear momentum that the body gains or loses. Further, we saw that a given observer would attribute to the body a gain or loss of linear momentum equal to the change in the body=s mass multiplied by the body=s velocity in that observer=s frame. Thus, we must attribute to each electron in our electric current a gain of linear momentum, due to the electron=s immersion in the charged body=s electrostatic field. Of course, Newton=s third law of motion still holds true to Reality, so our charged particle must gain from that electron an equal but oppositely directed increment of linear momentum. The increment of linear momentum that the charged particle possesses due to the presence of all of the electrons in the current equals the sum of all such increments.
Now the atomic matrix of the wire snaps back into existence. The potential emanating from its excess positive charges cancels the potential emanating from the negative charges of the conduction electrons, so our charged particle hasn=t gained any mass after all. Nonetheless, it retains the electrodynamic momentum that it gained from the conduction electrons: that momentum could only be canceled out by an equal and oppositely directed contribution from the atomic matrix, which cancellation would only come about if the atomic matrix moved with the conduction electrons. So as long as the atomic matrix and the conduction electrons have some motion between them, then the charged particle will possess some electrodynamic momentum.
So now we know that an electric current in a wire confers linear momentum upon a charged particle, even if the particle does not move. But if the particle does move, as I implied above, in a direction perpendicular to the orientation of the wire, then the consequent change in that momentum will subject the particle to a force that will accelerate it in the direction parallel to the wire. If we divide that force by the amount of electric charge on our charged particle, then we get a description of the effective electric field acting on the particle.
So now we have two different phenomena by which an electric current exerts a force upon a moving charged particle and their mathematical descriptions dovetail so perfectly that we may well regard them as two aspects of one force B the magnetic force. Can we encode that fact into a mathematical description of a single forcefield?
Ignoring the geometric factor, we can see that the operative parts of our description comprise three arrows representing the velocity of the charged particle; the common velocity of the conduction electrons comprising the electric current; and the distance vector pointing from the wire to the field point on the line measuring the shortest distance between the wire and the field point, the point occupied by our single electrically charged particle. We know that the induced electric field always points in a direction perpendicular to the direction of the charged particle=s motion, so we can represent that field as the vector cross product of the charged particle=s velocity with the presumed magnetic induction field. Likewise, we know that the induced electric field always points in a direction that lies in the plane defined by the directions of the distance vector from the wire to the field point and the motions of the conduction electrons, so when we express those facts mathematically we get the same description of the magnetic induction field that Jean-Baptiste Biot (1774 Apr 21 B 1862 Feb 03) and Felix Savart (1791 Jun 30 B 1841 Mar 16) first devised by scientific induction.
Let=s go back to the electrodynamic momentum that an electric current imposes upon an electrically charged particle. I want to define a new forcefield, which Michael Faraday called the electrotonic field. If we divide the electrodynamic momentum imposed on a charged particle by the electric charge on that particle, we get a mathematical description of the electrotonic field emanating from the electric current imposing the electrodynamic momentum. If the current that generates that electrotonic field changes, either in magnitude or in orientation, then the charged particle will come under a force equal to the negative of the product of the particle=s electric charge and the rate at which the electrotonic field changes with the elapse of time. More specifically, that force comes from an inherent change in the electrotonic field and not from the particle= s motion through the field (we have already absorbed that possibility into our new magnetic force).
So now we know that if we have an electrically charged particle moving with some velocity through a region of space near flowing electric currents and standing electric charges, it will encounter an electric field that conforms to the statement that the electric field equals the negative gradient of the electrostatic potential plus the vector cross product of the particle= s velocity with the magnetic induction at that point minus the rate at which the electrotonic field changes with the elapse of time. Thus we have added two new terms to our basic description of electric fields.
And now, as I indicated at the end of the previous essay, we have a relativistic phenomenon that we can experience without moving at a speed close to the speed of light. We call that phenomenon the magnetic force.
In this appendix I want to show you the four fundamental equations of electromagnetic theory in all their mathematical glory. Scientists call these Maxwell= s Equations and they come in two forms B differential equations and integral equations. For the first of those forms we have:
We call the first of those equations Gauss=s law of the electric field and it says that the divergence of the electric field at any point in space equals the density of the electric charge at that point divided by the electric permittivity of vacuum.
We call the second of those equations Gauss=s law of the magnetic field and it says that magnetic charges do not exist. In some situations we can assert the existence of such charges as a useful fiction, but ultimately all lines of magnetic force form closed loops; they don=t begin or end anywhere.
We call the third of those equations Faraday=s law of electromagnetic induction and it says that we can create a curled electric field by making a magnetic induction field fluctuate in time.
And we call the fourth of those equations Ampere=s law of the magnetic field. It tells us that we can create a curled magnetic induction field by making electric currents flow in space and/or by making an electric field fluctuate in time.
Physicists prefer to use that form of Maxwell=s equations because in that form they provide an easier form for the proof or derivation of theorems pertaining to electric and magnetic fields. For example, we can do essentially as Maxwell did and apply the curl operation to the third equation and apply the time differentiation operation to the fourth equation, carry out a little manipulation of vector relations, and get
which physicists and engineers recognize as the equation describing the propagation of a wave of electric force. In that equation the Greek letters mu and epsilon represent the magnetic permeability of vacuum and the electric permittivity of vacuum respectively and their product stands as the inverse square of the velocity at which the waves propagate. Using values that other physicists had measured in their laboratories, Maxwell calculated the value of that propagation velocity and found that it came close to the value that Armand Fizeau had measured for the speed of light. Maxwell understood that his contemporaries had only crude methods of measurement at their disposal, so he predicted with some confidence that as the measurements improved over time, the calculated propagation velocity of electromagnetic waves and the speed of light would converge onto a single number and, in fact, they have done so. From that fact physicists infer the proposition that light and electromagnetic waves represent the same phenomenon.
Let me now note that the explicit appearance of the electric permittivity of vacuum and the magnetic permeability of vacuum in those equations brings up a minor subject that I have kept implicit so far but which I now want to make explicit. Whenever I work out a problem or derivation involving electromagnetism I use 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 we write out the fundamental constants of Nature explicitly in the equations rather than merely implicitly, as I would do if I used the CGS system. In large measure that preference reflects the fact that I originally learned electromagnetic theory in MKS units. But it also reflects an aesthetic choice reminiscent of Einstein's dictum that we should make a theory simple but not too simple. I express those constants explicitly in my equations because, until someone convinces me otherwise, they represent properties of the vacuum and therefore we should not make them invisible by a redefinition of the properties of matter. They comprise as much of the foundation underlying the structure of Reality and thus contribute as much to its elegance and beauty as do the electric charge of the electron or the spin of the photon.
We have the second form of Maxwell=s Equations as a set of four integral equations:
Engineers prefer to use these equations because they refer to the kinds of extended systems that engineers deal with. I would find it very difficult, if not impossible, to deduce the existence of electromagnetic waves with these particular equations, but I would certainly use them if I wanted to design a practical application, such as an electric transformer or an electric motor. In that case the differential form of Maxwell=s Equations would make the task difficult or impossible of achievement. Again, we can describe each of those equations in words:
First, we have Gauss=s law of the electric field. The equation tells us that, if we have a closed surface (something like a soap bubble, say) and if we multiply the area of each minuscule patch on that surface by the component of the electric field perpendicular to that patch, then adding up all the contributions over the whole surface, which we call the electric flux through the surface, equals the net amount of electric charge inside the surface divided by the electric permittivity of vacuum. We usually interpret that equation as expressing the fact that electric lines of force can only begin and end on electric charges.
Second, we have Gauss=s law of the magnetic field. The equation tells us that if we have a closed surface and we calculate the total flux of the magnetic field through that surface we must get a zero every time. We usually interpret that equation as expressing the fact that magnetic lines of force can neither begin nor end so, therefore, the magnetic analogue of electric charge does not exist.
Third, we have Faraday=s law of electromagnetic induction. The equation tells us that if we define a closed line bordering a certain surface, then multiplying each minuscule element of that line by the component of the electric field parallel to that element and adding up all the contributions around the loop, we will obtain a number equal to the negative of the rate at which the flux of the magnetic field through the bounded surface changes with the elapse of time.
And fourth, we have Ampere=s law. The equation tells us that if we have a closed loop bounding a surface and if we multiply each minuscule element of the line by the component of the magnetic induction field parallel to that element, then adding up all of the contributions to that calculation around the loop will give us a number proportional to the net amount of electric current passing through the bounded surface plus the rate at which the electric flux through that surface changes with the elapse of time.
We would use the third and fourth equations to design an electric transformer, a device to convert high-voltage electricity into low-voltage electric current (or vice versa). The third equation tells us that if we have a loop of wire and we can make a magnetic field passing through the surface enclosed by that loop vibrate, we will get an electric field in the wire, one that will cause electric current to flow in the loop. The fourth equation tells us that if we put a second loop of wire near the first and make a vibrating electric current flow through it, we will get the vibrating magnetic field that we want. The amount of voltage that we get in the first loop depends, then, on how many times the wire in the first and second loops go around their respective loops and on how much current we use to generate the vibrating magnetic field.
The sheer elegance of Maxwell=s Equations has often moved physicists to flights of fancy. When he first became aware of these four equations, Ludwig Boltzmann, the Austrian physicist primarily remembered for his contributions to the statistical mechanics version of thermodynamics, quoted Goethe= s Faust: AWar es ein Gott, der diese Zeichen schrieb?@ A fuller quote of the doomed Dr. Faust gives the full flavor of what Boltzmann meant:
AWas it a god who wrote these signs that still My inner tumult and that fill My wretched heart with ecstasy? Unveiling with mysterious potency The powers of Nature round about me here?@
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