Let us have a circular wire loop. Let us use some kind of a mechanical device to accelerate the electrons in the loop, so that a magnetic field is built. At any moment, the electric field outside the loop is negligible, but the magnetic field B stores a lot of energy.
The Poynting vector is defined as
S = 1 / μ₀ * E × B.
If the electric field E is negligible, there is no energy flow, according to the Poynting vector. How does the energy then flow to the magnetic field B, if there is no energy flow?
protons
+++++++++++
----------------------- --> v electrons e-
+++++++++++
protons
We can make a wire loop by bending a straight wire and attaching the ends. Maybe the "correct" way to analyze the magnetic field of a wire loop is to study a straight segment of a wire? We assume that the electrons form a negatively charged cylinder which moves relative to a positively charged cylinder of protons.
The total charge of the conducting electrons in the wire is huge. When cylinder of electrons get displaced relative to the protons, even a little, the electric fields at the ends of the wire will have a large energy.
wire
+ ========== -
positive negative
charge charge
When the electrons in the diagram slide to the right, a negative net charge appears at the right end of the wire and a positive net charge appears at the left end. The energy in these electric fields is considerable. Let us analyze how the energy flows to those electric fields.
The Poynting vector for two charges
The simplest configuration is the one where we move two single charges away from each other:
force F
● <------------------> ●
+q -q
The force does work on the point charges q. How is this energy transported to the fields of the charges?
Let us assume that +q is static. The charge -q moves slowly to the right. The electric field E is large close to -q, and the movement creates a magnetic field B. The energy seems to flow from the point charge -q to all over the combined field of +q and -q.
Many charges in a wire: acceleration of electrons creates a net electric field outside the wire
Suppose that we have very many charges in a wire, and they are uniformly distributed. Can we accelerate the conducting electrons, and at the same time keep the electric field E negligible outside the wire, so that the Poynting vector S cannot carry the energy to the strengthening magnetic field B?
When we accelerate an electron, its electric field is distorted. The electric field of the protons probably cannot cancel the electric field of the electron in such a case. This allows the Poynting vector to carry the energy to the strengthening magnetic field B.
Is the energy of the magnetic field in the inertia and the movement of conducting electrons?
Since it requires a lot of energy to get the conducting electrons drifting in a wire, then, by definition, those electrons have large inertia. But what is the origin of this inertia, and where is the energy stored?
The electrons close to protons acquire inertia in the field of the protons. Could this extra inertia explain the momentum and the energy of the magnetic field? Once one has got the conducting electrons moving relative to the field of the protons, it is hard to stop the electrons since they have extra inertia inside the field of the protons.
We have a problem: the thermal movement of the electrons is very fast. Why the extra inertia does not slow them down considerably?
Inertia of a sliding charged tube
Let us use the following model: we have a cylinder of positive charge, and a tube of negative charge sliding along with the cylinder:
negative charge
-------------
++++++++++++++++++ positive charge
-------------
--> v
How do we explain the inertia of the sliding negative tube?
The tube cancels the electric field of the positive charge in a certain segment. The energy of the magnetic field of the tube is
m v²,
where m c² is the energy of the electric fiekd canceled by the tube. That is, the magnetic field contains 2X the hypothetical "kinetic energy" of the electric field of the tube.
What is the inertia of the charges in the tube? Let the tube contain 6 * 10²³ electrons. The mass of the electrons is only 6 * 10⁻⁷ kg. Let the radius of the tube be 1 mm and the length 1 meter. The tube has a charge of 10⁵ coulombs. The electric field E close to the tube is something like
E = 9 * 10⁹ * 100 / (0.001)² V/m
= 10¹⁸ V/m,
and the mass-energy density of the electric field is
1/2 ε₀ E² / c² = 5 * 10⁷ kg/m³.
The mass-energy density at a distance of 1 meter from the tube is still 50 kg/m³. We conclude that the inertia of the electrons themselves is negligible. Essentially all the inertia comes from the electric field of the tube.
In our December 17, 2023 blog post we present the assumption that the inertia of a test charge q inside the electric field of an opposite charge Q grows by |U| / c² where U is the potential between q and Q. The inertia of the tube is thus twice the mass-energy of the electric field of the tube. We have an explanation for the fact that the energy of the magnetic field of the tube is twice the kinetic energy of the electric field of the tube.
When the charged tube slides, is there energy shipping?
electric field electric field
| | | | | | | |
--------=============--------
| | | | tube --> v | | | |
In our blog we have suggested that the "energy shipping" in the electric field is behind the increased inertia of the charged tube.
________ electric field line?
/
v ______
/ \
| |
\_______/
--> v
wire loop
But if we let the tube cover the entire cylinder and join the ends of the cylinder and the tube, then we essentially have a wire loop where a current is running (i.e., the charged tube is rotating around) – and there is no substantial electric field outside the tube, even though the magnetic field stored substantial momentum. But is there still significant energy shipping?
If the cylinder is still straight and we have not joined the end, then the magnetic field of the sliding tube acts together with the electric field, and the Poynting vector ships field energy from the right side of the tube to the left side.
Joining the ends of the cylinder and the tube alters the topology of the system. It is no longer a straight wire but a loop. Could it be that the loop topology somehow retains the energy shipping?
The electrons in the loop are in an accelerating motion. Their electric field cannot cancel the electric field of the protons exactly? Could it be that the Poynting vector of this small residual electric field and the magnetic field involves enough energy shipping, so that it explains the momentum held by the electrons?
The electric field lines are "lagging" the movement of the electrons. It is as if there would be a "vortex" of electric field lines running around the loop. However, loops of electric field are forbidden. What is the electric field like?
___________ S Poynting vector?
/ • • • B magnetic field
v ______ points out of the screen
/ \
| | -----> E electric field?
\_______/
e- •--> v
wire loop
The Poynting vector makes a circle around the loop if the residual electric field points radially from the center of the loop.
Let R be the distance from the center of the loop. The magnetic field and the electric field go like
B ~ 1 / R³,
E ~ 1 / R².
The area of a sphere goes as R² and the associated angular momentum goes as R. The integral of B × E for the whole space is over ~ 1 / R². Could this integral be large enough, so that it could store the energy and the angular momentum of the magnetic field?
The energy of the magnetic field is 2X the "kinetic energy" of the electric field of the the conducting electrons, and that electric field has a large "mass" as we calculated above. How can the Poynting vector store both the energy and the momentum of the magnetic field? Maybe it only needs to store the angular momentum? If the energy is located at a large radius R from the loop, then it can store a very large angular momentum.
A paradox: the Poynting vector cannot explain the momentum stored in a magnetic field? Yes it can
Above we have the spherical coordinates used in physics. The θ coordinate vector points obliquely down in the diagram.
Marvin Zanke (2019) calculated the magnetic field of a rotating, uniformly charged spherical shell. Note that the B vector in the upper hemisphere points "down" relative to the radial vector. The electric field E points almost precisely to the radial direction. Thus, the Poynting vector S outside the shell points consistently to the same direction with respect to φ. The Poynting vector describes a vortex of energy flow around the rotating shell.
Inside the shell, E is zero and the Poynting vector is zero.
If we superpose a static sphere of the opposite charge on the rotating sphere, we can make E almost exactly zero everywhere. The Poynting vector will have a much smaller value everywhere. The angular momentum L described by the Poynting vector dropped to almost zero.
But the magnetic field B did not change at all. It still contains the same energy as before. Can the angular momentum stored by B drop to almost zero?
Let us assume that the rotating shell is positively charged. We let a concentric, negatively charged shell shrink and descend down on the rotating shell. The magnetic field tries to pull the negative shell to rotate along. To keep B constant, we must add more angular momentum L to the rotating shell.
Thus, once we have neutralized the electric field of the rotating shell, the magnetic field B stores even more angular momentum L than before. At the same time, the Poynting vector S shrinks everywhere to almost zero.
Resolution of the paradox. We confused two things: angular momentum "relative" to the charge in the static sphere and the genuine angular momentum in Minkowski space. Let E be zero. The energy in the field B can apply a torque on the rotating sphere if it simultaneously applies the opposite torque on the static sphere.
Let us have two concentric rings rotating to opposite directions. There is a spring between them, such that it stores the kinetic energy of the rings and stops their motion. The spring stored "relative angular momentum" between the rings. It did not store genuine angular momentum. A pure magnetic field B without an associated field E can be compared to the the spring.
Does the resolution of the paradox work? Suppose that we inside a wire loop have tiny robots pushing the electrons to one direction. The feet of the robots give an angular momentum -L to Earth and their hands give an opposite angular momentum L to the system electrons & protons & magnetic field B. If the angular momentum L goes back to Earth through an interaction with the protons, then B does not need to store any angular momentum.
Let us have a current flowing in a wire loop. We try to reduce the magnetic field by slowing down the electrons. The changing magnetic field B tries to keep the electrons flowing through the induced electric E field from the changing magnetic field. The induced electric field applies an equal but opposite force on the protons. Thus, Earth does not receive any net torque from the magnetic field.
We conclude that there is no paradox. A pure magnetic field B without an electric field E stores energy but not momentum. It stores momentum relative to the protons in the wire, but this is not reflected in the Poynting vector.
Conclusions
There is no paradox with the Poynting vector. However, we want to analyze what actually stores the momentum relative to the protons. It is not the Poynting vector. We will continue the analysis in future blog posts.
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