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In this blog we have presented the bottle example, where a charge inside a bottle is quickly pulled from the main part of the bottle to the neck.
Let us assume that the electric field of a charge q is determined by Coulomb's law based on the retarded position of the charge. This is not strictly true. We will look at corrections from Lienard-Wiechert potentials.
We can make the form of the bottle such that at all times, the Coulomb electric field of the charge will point out of the surface.
_________
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× ×
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\
\_________
The old position of the charge is the right × and the new position is the left ×.
Gauss's law says that a charge is the sole source of the electric field, which implies that inside any closed surface, the charge is equal to the field flux out from the surface.
Let the charge be q. Before the pull, the main body has a flux > q/2 out from the bottle. Right after the pull, the main body still has the flux > q/2, but the neck, too, has a flux out > q/2.
Information about the electric field spreads at the speed of light. That is why the main part of the bottle does not yet know that the charge has moved.
The total flux is > q and Gauss's law appears to be temporarily broken. But can the electric field induced by the changing magnetic field save the law?
Maxwell's equations say that a changing magnetic field produces curl into the electric field, and vice versa. But curl does not create a source of the electric field, it creates field lines which are closed loops. Closed loops do not help in restoring Gauss's law.
Let us put our argument another way: we showed that, assuming that the electric field is determined by Coulomb's law, our source of the electric field, q, produces a flux q' > q out of the bottle. The only way to restore Gauss's law is to assume something else (like magnetic effects) creates a flux q' - q into the bottle.
We can restore Gauss's law by assuming that some effect forces the electric field lines to return back to the bottle when they have exited the neck. If the field lines are unbroken at each moment in the global coordinate frame of our experiment, then Gauss's law is restored at least in this frame. What if we move to another frame? Does the transformation of the electric and magnetic fields keep Gauss's law in force?
Lienard-Wiechert potentials
https://en.wikipedia.org/wiki/Liénard–Wiechert_potential
The Lienard-Wiechert potentials describe the scalar and the vector potential of a moving point charge. We will look at what they say about our thought experiment of the bottle.
Momentum conservation is a problem?
We wrote in our April 2018 blog post about the Larmor formula. If we are pushing a charge and accelerating it, the energy to the electromagnetic field or radiation obviously has to come from the kinetic energy of the charge. But then the charge would lose a lot of momentum, which cannot be carried away by radiation or field energy moving at the speed of light.
It looks like a linearly accelerated charge cannot radiate or lose energy to the field. The Larmor formula must fail if we believe in conservation of momentum.
But Edward M. Purcell has presented a derivation of the Larmor formula where he draws the electric field lines as continuous after a sudden push of the charge. He calculates the electric field energy and finds the Larmor formula true.
What is the solution to this contradiction?
1. Conservation of momentum fails?
2. The field lines are not continuous and Gauss's law for the electric field fails?
3. The field energy is not what Purcell calculated?
4. Something else?
If we believe in conservation of momentum, then a linear push does not add any new energy to the field. That suggests that the field lines are not continuous and Gauss's law fails.
People have not been able to measure the radiation of a linearly accelerated charge. We do not know from experiment what Nature does.
Conservation of momentum is a pillar of physics, and the energy E^2 of the electric field manifests in electromagnetic waves. Most probably we have to abandon 246-year-old Gauss's law in Maxwell's equations and replace it with some principle about the Coulomb potential of a charge plus retardation. Electric lines of force are not continuous if a charge is accelerated linearly.
It would be strange if a charge which is static in a gravitational field would radiate. What about the electric field lines? Are they continuous for such a charge?
Purcell's derivation of the Larmor formula - how much energy is there in a disturbance of the electric field?
http://physics.weber.edu/schroeder/mrr/MRRtalk.html
On the linked page, at top right, or under the headline "Radiation as a Consequence of a Cosmic Speed Limit", we see the electric field lines which Edward M. Purcell drew to derive the Larmor formula. The Coulomb potential inside the circle differs considerably from the Coulomb potential outside the circle. In the electric potential, there is a sharp jump at the circle.
To make the electric field sourceless at the circle, Purcell draws transverse field lines which connect the field lines inside the circle to the lines outside the circle.
In principle, one can accelerate a charge from zero to half the speed of light instantaneously. There is no physical limit imposed on acceleration in special relativity. The transition area A between the inside of the circle and the outside can be made as narrow as we want. There are quite a many transverse field lines in Purcell's diagram that are confined in the transition area. We can make the electric field energy E^2 * A in the transition area as large as we want. This shows that in Purcell's diagram, the electric field certainly carries away energy at the speed of light.
But we have showed in our blog that linear acceleration of a charge cannot convert its kinetic energy to field energy which would move at the speed of light, because conservation of momentum would be broken. We conclude that the transverse field lines of Purcell do not exist.
What about the Coulomb potential? It is strange if there would be sharp jumps in the potential. Since the configuration is not static, a test charge can receive energy from the changing electric field. Can we pump energy from the changing electric field through some mechanism? If yes, then we would a have a breach of conservation of energy.
Suppose that we have an electric dipole at a (large) distance r from where the charge originally was. The dipole is strained by a Coulomb field which is proportional to 1/r^2. Let the dipole be bound with a spring for which Hooke's law is F = kx. We then have the displacement x = 1 / (k r^2) and the energy stored in the spring k / (2 k^2 r^4). After some time, the dipole suddenly sees the electric field change its direction and magnitude. Some energy is released into the vibration of the string.
We can place r^2 of such dipoles in a spherical shell around the charge. The combined energy suddenly released in the shell is at most proportional to 1/r^2. This is the well-known result that you cannot efficiently transmit energy over long distances in longitudinal waves of the electromagnetic field.
At least this thought experiment showed that it is not easy to harvest much energy from the disturbance of the electric field. But even a little bit of energy would breach conservation of energy.
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