dl wire segment
proton(s)
------------
v <-- • dq = e-
|\
| \
| \
α = angle (r, negative y axis)
r = vector (dq, q)
Vy V
^ ^
| /
| /
• ---> Vx
q = e-
^ y
|
-----> x
Assumption about the Coulomb force. The Coulomb force on q has to be calculated in the comoving frame of q.
The above assumption claims that the laboratory frame or a comoving frame of dq are "wrong" frames. The Coulomb force would be different in those frames because simultaneousness is different. When q is at a specific location (x, y) at a time t, simultaneousness determines where the electrons in the wire are.
Assumption about the extra inertia inside an electric field. The extra inertia of q relative to dq in a tangential movement relative to dq is
|U| / c²,
where U is the potential of q in the field of dq, relative to infinity. If there is an opposite charge dq' ≤ dq "close to" dq, then the inertia of q is reduced by
|U'| / c²,
where U' is the potential lf q in the field of dq'.
In a radial movement, the extra inertia is double. However, for a "current in a wire" of charge, the doubling does not hold. The extra inertia is like in a tangential movement. See below.
Definition. A current in a wire is a stream of moving charges whose charge is essentially canceled by static charges close to them, when viewed in the laboratory frame.
In an ordinary metal wire where the drift velocity of electrons is only about v = 10⁻⁶ m/s, the protons essentially cancel the charge of the electrons when viewed in the laboratory frame. The length contraction factor γ differs from 1 only by
~ 1/2 v² / c² = 5 * 10⁻³⁰.
However, if an observer moves to the direction of v at a reasonable velocity V, he may see a significant net charge in the wire.
Assumption of extra inertia when there is a "current in a wire". The distance to the charge has to be assumed constant, that is, the distance to the wire segment dl. However, the extra inertia which q "receives" or loses still moves at the drift velocity v of the charge carriers.
The current is viewed as a "flux" and the flux itself does not move, even though its charge carriers move.
Assumption about paradoxical momentum exchange. If q and dq have the same sign, then the momentum that q receives from the inertia inside the field dq is, paradoxically, opposite to what one would expect if the extra "package" of inertia is moving with dq.
The paradoxical momentum exchange is needed to make the negative test charge q to steer to the opposite direction relative to where the electrons in the wire move.
In the diagram, the wire element dl is parallel to the x axis, as well as the average drift velocity v of the electrons. The velocity of q is V.
We assume that
|v| << |V| << c,
so that we can ignore terms of the form ~ v² and ~ V² v. The interesting terms are ~ V • v.
Question. Conducting electrons in a metal move at ~ 10⁶ m/s, while the drift velocity is only ~ 10⁻⁶ m/s. How does the almost relativistic speed of conducting electrons affect the Coulomb force and the inertia force?
The Coulomb force
The Coulomb force comes from the fact that length contraction in the comoving frame of q makes q to "see" a different amount of negative charge of electrons in the wire segment from the positive charge of protons. In the laboratory frame, the segment is essentially neutral. But in the comoving frame there is considerable length contraction.
The absolute value of the Coulomb force by dq is
F = 1 / (4 π ε₀) * dq q / r².
and the absolute value of the "residual" force, when the opposite force by the protons in the wire segment is subtracted, is approximately
Fc = F Vₓ v / c².
Let us calculate the x and y components of Fc:
Fcx = F sin(α) * Vₓ v / c²,
Fcy = -F cos(α) * Vₓ v / c².
The "inertia force"
The test charge q approaches dl at a velocity
Vr = cos(α) * Vy - sin(α) * Vₓ.
The inertia of q increases in a time t by
W / c² = Vr t F / c².
The test charge q "picks up" a momentum which corresponds to an "inertia force"
Fi = v Vr F / c².
The x and y components of the inertia force are
Fix = v * (cos(α) * Vy - sin(α) * Vₓ)
* F / c²,
Fiy = 0.
The combined force on q is
Fₓ' = Vy v cos(α) * F / c²,
Fy' = -Vₓ v cos(α) * F / c².
The magnetic field calculated from the ordinary Biot-Savart law is
B = 1 / q * F v cos(α) / c².
The magnetic force calculated from B is:
Fₓ'' = Vy v cos(α) * F / c²,
Fy'' = -Vₓ v cos(α) * F / c²,
the same as we calculated as the sum of the Coulomb force and the inertia force.
Does the Coulomb force used above make sense?
If we would calculate the residual Coulomb force in the laboratory frame, or in the comoving frame of the electrons, the Coulomb force would be essentially zero. The equations of classical electromagnetism would have
B ~ v
E ~ v²,
where E is essentially zero in the setup
|v| << |V| << c.
However, the magnetic field B would be significant.
Let us have a test charge q approaching a wire from a normal direction. The following method yields a wrong result:
1. Calculate the sideways impulse on the test charge q by the electrons, in the comoving frame of the electrons. This is essentially zero.
2. Add the sideways impulse by the protons in the laboratory frame. This is zero.
3. Conclude that q will approach the wire along a straight path.
In our example, one is not allowed to add the forces by the electrons and the protons linearly. In this sense, classical electromagnetism is not a linear theory.
Conclusions
We are able to derive the Biot-Savart law, but we have to make several ad hoc assumptions. A key question is where and how is the energy and the momentum of the magnetic field B stored if we have a current in a wire. Traditionally, we think that the electric field is zero. The Poynting vector is zero in that case.
Could the field energy and the momentum exist in the microscopic fields between electrons and protons?
Another explanation: if we cut the wire loop and let two cylinders of opposite charge slide past each other, there is a huge energy flux from the kinetic energy of the cylinders to form the electric field at the ends of the system. Could this energy flux somehow exist "hidden" in a current loop?
No comments:
Post a Comment