long jumping rope
hand •--------------------------------------------● fixed point
----------------------------------------------> x
Let us take children's jumping rope as an example. The rope is very long and tense, and we start to rotate it with our hand. We are sending a circularly polarized wave into the rope.
Our hand feels a force which resists the rotation. The force exists because the rope "lags behind" our hand and its tension pulls our hand back to its earlier position.
Self-force on an electric charge
If we replace the rope with an electron in our hand and its electric lines of force, could it be that we can explain the resisting force simply as
F = E e ?
There ε₀ is vacuum permittivity, e is the charge of the electron, and E is the component of the electric field which resists the circular movement of our hand.
Let us rotate our hand at a velocity v along a circle whose radius is r. The acceleration is
a = v² / r.
The Larmor formula says that the power of the outgoing radiation is
P = 1 / (6 π ε₀) * e² a² / c³.
The energy density in the spatial volume whose radius is ~ r is then
D = P r / c * 1 / (4/3 π r³)
= P / (4/3 π r² c)
= 1 / (8 π² ε₀) * e² v⁴ / (r⁴ c⁴).
The force resisting our hand movement must be
F_v = P / v
= 1 / (6 π ε₀) e² v³ / (r² c³).
The energy density of an electric field is (the factor 1/2 comes from the fact that a half of the energy of radiation is in its electric field):
D / 2 = 1/2 ε₀ E²,
where E is the electric field strength. Within the volume whose radius is ~ r, the electric field strength of the radiation is
E² = 1 / (8 π² ε₀²) * e² v⁴ / (r⁴ c⁴),
E = 1 / (2 sqrt(2) π ε₀) * e v² / (r² c²).
The force of the field E on the electron is
F = 1 / (2 sqrt(2) π ε₀) * e² v² / (r² c²).
We have
F_v / F ≅ 1/2 v / c.
We see that the electric field E can easily explain the force F_v. It is enough that a (small) component of E is opposite to v. Most of E has to be normal to v.
The jumping rope diagram explains this. Most of the tension force is along the x axis. A small component of the force pulls our hand toward the center of the rotating movement. A very small component is against the velocity vector v of our hand and we must do work against that component. The electric field in our calculation corresponds to the last two components.
Conjecture. The electric self-force on an electron is the electric force which its own electric field imposes on the electron. We can cut the electric field lines close to the electron and calculate how much the electric field differs from a spherically symmetric field. The difference is the electric field which pulls on the electron.
On December 26, 2021 we wrote about this "tense field lines" model.
If the conjecture is true, then the electric self-force is not private. It is visible to all charges. An infinitesimal test charge, close to the electron which is generating the wave, will feel the same electric field pull it as the electron. The test charge, of course, feels a much larger force from the static field of the electron.
The steel wire model conveys a wrong impression. We visualize steel wires as attached to the electron. Then they would not affect the test charge.
Let us adopt the tense field line model from now on and forget about steel wires.
Lagging field lines versus an opposite charge nearby
\ lagging field line
\
e- -----> acceleration
/
/ lagging field line
Above we have an electron being accelerated. The lagging electric field lines resist the acceleration.
\
\
e+ e-
/
/
We get similar bent field field lines if we place a positron behind a static electron. It may be that the electric force on an electron can be explained by the tense field lines model, and it does not matter if the field lines are bent because of another charge, or if they are bent because of acceleration.
This model would be a very simple description of the self-force, if true.
The centripetal self-force on a circling electron
On October 1, 2021 we calculated the self-force on an electron on a circular orbit, and got very roughly the correct result. We calculated the extra energy it requires to bend the field lines. From that we derived the centripetal force on the electron. Let us check if we get the same result by calculating the centripetal electric field which the bent field lines generate on the electron.
The deflection angle of the field lines is
2 r / s,
where r is the radius of the circling motion, and s is the distance where we assume that the field lines are bent.
We calculated a centripetal force
F = 4 r / s² * m_e c² * r_e / s,
where m_e is the mass of the electron and
r_e = 1 / (4 π ε₀) * e² / (m_e c²).
We get
F = 1 / (π ε₀) * e² r / s³.
The electric field strength at the distance s is
E = 1 / (4 π ε₀) * e / s².
Its centripetal component E_c is E * 2 r / s, and the centripetal force
F' = 1 / (2 π ε₀) * e² r / s³.
The order of magnitude is correct. If we adjust s to be a little smaller, then we get F' = F.
Our calculation suggests that the Conjecture might be true. The mysterious self-force on an electron is simply the force imposed on the electron by its own field, where the own field can be measured at some radius r from the electron.
Conclusions
It is time to forget the "rubber plate" model and the steel wire model of the static electric field of the electron, and move to the tense field line model.
We believe that the static, spherically symmetric electric field of an inertial electron has zero energy.
When the electric field lines bend, there is energy and momentum in the field of the electron. The force from bent lines can probably be calculated by looking at how much the field lines differ from spherically symmetric ones at some radius r from the electron. The difference gives us an electric field E which pulls on the electron. The self-force is
F = E e.
If there is an infinitesimal test charge close to the electron, it feels the same field E as the electron. The self-force field is not private to the electron, but visible to all charges.
The force on the electron from another charge can probably be calculated in the same way: determine how much the lines of force differ from spherically symmetric ones.
No comments:
Post a Comment