Electricity in Motion

To the best of my knowledge, Richard Feynman was the first person to conceive at least the first part of the following:

Imagine a long straight wire of cross section suspended horizontally in front of you carries an electric current flowing from left to right (in the positive x-direction). 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 v and that the current has reached a long-term equilibrium, so that the wire displays no net electric charge.

In their own inertial frame the electrons have a
density that gives them collectively a charge density of ρ_{-} (coulombs
per cubic meter). In the inertial frame occupied and marked by the atomic matrix
of the wire the Lorentz-Fitzgerald contraction increases that density to

(Eq'n 1)

due to the lengthwise shrinkage of the electron
ensemble. By our equilibrium criterion that density must equal in magnitude the
positive charge density of the atomic matrix; that is, ρ_{+} = ρ_{-}'.

Imagine that a particle carrying electric charge
q moves from left to right at a speed V relative to the wire's atomic matrix,
maintaining a distance r from the wire. An observer moving with that particle
detects the wire's atomic matrix moving from right to left at the speed V and
the conduction electrons moving from right to left at the speed (v+V)/(1+vV/c^{2}).
Consequently, in that frame, the Lorentz-Fitzgerald contraction will increase
the charge density of the atomic matrix to

(Eq'n 2)

and increase the charge density of the conduction electrons to

(Eq'n 3)

By substituting into Equation 2 from Equation 1
via the statement that ρ_{+} = ρ_{-}' and comparing the result
to Equation 3, we can see that in the charged particle's frame the wire carries
a net negative charge density of

(Eq'n 4)

That excess charge density must generate an electric field, each volume element dl of the wire contributing

(Eq'n 5)

in which the factor **r**/R
multiplied into the standard inverse square law represents the fact that only
the component of the electric field perpendicular to the wire comes out of the
superposition of all of the contributions. We know that 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 frame moving at the speed V 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.

Now we must acknowledge explicitly that Equation 5 describes the electric field as it would be measured by an observer moving with the charged particle. An observer at rest in the frame occupied and marked by the atomic matrix of the wire moves at the speed V from right to left relative to that observer and thus must measure forces exerted perpendicular to that motion as weaker than what the first observer measures, weaker by an amount equal to the Lorentz factor between the two observers; therefore, in the frame of the wire's atomic matrix we must replace Equation 5 with

(Eq'n 6)

In the light of that analysis we must infer that the electric current 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 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 "some 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, "On the Electrodynamics of Moving Bodies":

"It is known that Maxwell's electrodynamics - as usually understood at this present time - 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 - assuming equality relative motion in the two cases discussed - to electric currents in the same path and intensity as those produced by the electric forces in the former case.

"Examples 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 "Nothing" (and I'm not sure 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 φ. If we immerse a
particle bearing an electric charge q in that field, then the particle will gain
a potential energy of qφ and, by the mass-energy equivalence theorem, an
increment of mass equal to qφ/c^{2}. Likewise, each electron in the
current (carrying electric charge e) gains an increment of mass equal to eφ_{q}/c^{2}
due to its immersion in the potential field φ_{q} 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 current a gain of linear momentum,

(Eq'n 7)

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 increment of linear momentum

(Eq'n 8)

subject to the proviso that qφ_{e} = eφ_{q}.
The increment of linear momentum that the charged particle possesses due to the
presence of all of the electrons in the current equals

(Eq'n 9)

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. Indeed, we can
now see that the **v** in Equation 9 must represent the difference
in velocity between the atomic matrix and the conduction electrons.

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. In accordance with the
definition of force we have

(Eq'n 10)

If we divide that equation by q and note that dr/dt = V, then we get a description of the effective electric field acting on the particle,

(Eq'n 11)

The electrostatic potential that the conduction electrons contribute to our calculation of the electrodynamic momentum of the charged particle comes from integrating the potentials contributed by the elemental parcels of charge ραdl over the length of the wire. Thus we have

(Eq'n 12)

in which with r representing the shortest distance from the field point to the wire (at a point we label P) and l represents the distance measured along the wire from P to the elemental parcel of charge making the contribution in question. We know that

(Eq'n 13)

and that dR/dr = r/R, so we have for Equation 11 now

(Eq'n 14)

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 - 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 Equations 6 and 14 comprise the vectors **V**,
the velocity of the charged particle; **v**, the common velocity of
the conduction electrons comprising the electric current; and **r**,
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. We know
that the induced electric field always points in a direction perpendicular to
the direction of **V**, so we can represent that field as the
vector cross product of **V** with the presumed magnetic induction
field **B**; that is

(Eq'n 15)

Likewise, we know that the induced **E**
always points in a direction that lies in the plane defined by the directions of
**r** and **v**, so now we know that

(Eq'n 16)

in which g represents the geometric factor. Referring back to Equation 14, we can make that geometric factor explicit and get

(Eq'n 17)

in which the current density and dv = αdl. That gives us the same description of the magnetic induction field that Jean-Baptiste Biot (1774 Apr 21 - 1862 Feb 03) and Felix Savart (1791 Jun 30 -- 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 by rewriting Equation 9 as

(Eq'n 18)

If the current that generates that A-field changes, either in magnitude or in orientation, then we expect that the charged particle will come under a force

(Eq'n 19)

where I have used the partial differential to indicate that the force comes from an inherent change in the A-field, such as a change in the magnitude or orientation of the electric current that generates it, 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 velocity V 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

(Eq'n 20)

Thus we have added two new terms to our basic description of electric fields.

Finally, I want to refer back to the quote above from Einstein's "On The Electrodynamics of Moving Bodies". Equation 20 gives us three different contributions to the electric field that exists at a point in space, but physicists conventionally do not include the second term. Usually we implicitly interpret Equation 20 as applying to electric charges in a pseudo-stationary state; that is, we treat the charges responding to the field as if they temporarily had no motion. In contrast, the second term clearly makes explicit reference to the motion of the charges subject to the magnetic field. With that term we have two options; either we imagine moving with the velocity v and detecting an electric field acting on a stationary charge or we imagine watching the charge move as an element of an electric current interacting with a magnetic field. Henceforth, then, I will follow the physicists' convention and take the second option. So we must have separately the electric field, which comprises the negative gradient of the electrostatic potential minus the partial time derivative of the magnetic vector potential, and the Lorentz force, which we calculate by multiplying the electric charge of the forced body by the sum of the electric field of the forcing body and the vector cross product of the forced body's velocity and the magnetic induction field at the point occupied by the forced body.

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