# The Fields of a Moving Point Charge and special relativity

David J. Griffiths Introduction to Electrodynamics page 456 shows the following figure:

... then page 461:

... and page 462:

The following thought experiment has three electrons, negative charges, in an intergalactic space observed from $$Q_1$$, $$Q_2$$ rest frame $$K$$:

The charges $$Q_1$$ and $$Q_2$$ are 'almost stationary' held by a device in $$K$$ reference frame, the charge $$q$$ accelerates straight down along the $$y$$ axis in $$-y$$ direction. This is an isolated system. The figure above shows the electric field as observed from electron accelerated momentarily comoving inertial frame (MCIF) in $$K$$ frame at the time $$t=7.54031\times10^-8s$$ and the position $$\{0,0.25,0\}$$ in meters. The isolated system is being observed from a moving inertial frame $$K'_1$$:

The figure above shows the electric field as observed from MCIF that is tangent to the accelerated electron trajectory in $$K'_1$$ inertial frame at the time $$t'=7.5783\times10^-8s$$ and the position $$\{-2.27192, 0.25, 0\}$$ in meters. Note: The flattened field is not correlated to the speed, it is bigger for the demonstration purpose.

Question: How do we transform the EM field from $$K$$ to $$K'_1$$? Is it safe to use equations (10.75), (10.76) in any inertial reference frame without any other considerations?

Edit 2024-08-01:

A new question arises, what does the frame dependent field mean for the Coulomb force?

David J. Griffiths Introduction to Electrodynamics page 460:

The charge $$q$$ repels $$Q_1$$ and $$Q_2$$ symmetrically in $$K$$ reference frame but the repulsion is not symmetrical in $$K'_1$$ reference frame. The Coulomb force has bigger magnitude between $$qQ_1$$ compared to $$qQ_2$$ in $$K'_1$$ frame. The isolated system body is not torqued in $$K$$ but it is torqued clockwise in $$K'_1$$ frame. The $$q$$ EM field 'rotates' counterclockwise in $$K'_1$$ and the conservation of angular momentum 'rotates' the isolated system body clockwise.

Angular momentum has two parts orbit and spin/rotation. Orbit is frame dependent but the direction of the spin/rotation is absolute.

Question: Is prediction of no spin/rotation of the isolated system body in $$K$$ and prediction of clockwise spin/rotation of the isolated system body in $$K'_1$$ a contradiction to special relativity?

• Commented Aug 1 at 18:11
• @Frobenius thank you, amazing work in the link you provided. Commented Aug 1 at 18:48

## 2 Answers

The answer is in the same book: David J. Griffiths Introduction to Electrodynamics pages 555, 556 the relativistic derivation:

Conclusion is the E field equations (10.75) and (12.93) are equal/invariant in any inertial reference frame. The same applies to B field equations (10.76) and (12.110). The text below of (12.93) explains the reason why the field flattens.

Observation: The above equations predict different, as frame dependent, field strength acting from $$q$$ charge towards $$Q_1$$ and $$Q_2$$ charges.

> Is it safe to use equations (10.75), (10.76) in any inertial reference frame without any other considerations?

No. Formula (10.75) only applies to uniformly moving point charges.

By the way, for accelerated point charges that move slower than approx. 30 percent of the speed of light, you can also consider using Weber-Maxwell electrodynamics (https://doi.org/10.1080/02726343.2024.2375328). Weber-Maxwell electrodynamics is based on the solution of Maxwell's equations for the force that is exerted by an arbitrarily moving point charge on a stationary test charge. In order to generalize the solution to arbitrarily moving test charges, a Galilean transformation is used instead of the Lorentz transformation. Although this is only an approximation for non-relativistic point charges, Weber-Maxwell electrodynamics provides excellent practical results and is much easier to use and understand. Essential properties, such as universal constancy of the propagation speed of electromagnetic waves with speed c for all test charges and the principle of relativity, apply even though the Lorentz transformation is bypassed. Weber-Maxwell electrodynamics can also be used to study effects such as bremsstrahlung. The only restriction is that the point charges must not move too rapidly. There is also a software framework for playing around and learning: https://github.com/StKuehn/OpenWME

• The equations (10.75) and (10.76) hold in momentarily comoving inertial frames. The equations support reasoning why there would be a 'rotation' of the field. The equation (10.74) includes acceleration and the delta in repulsion is there. The main question if the EM creates a special relativity contradiction in predicting the direction of the spin/rotation of the isolated system body stands. Commented Aug 2 at 13:26
• The problem has nothing to do with special relativity, but with an incorrect application of the Lorentz force law. The E and B fields calculated for point charges must not be used in the Lorentz force law, as the Lorentz force formula is only completely correct if a B field is generated by a direct current. Point charges are not direct currents. The topic is not trivial. See e.g. here: en.wikipedia.org/wiki/Weber_electrodynamics
– skn
Commented Aug 3 at 10:09
• From your link: "Experiments that do not support Weber electrodynamics". What are we talking about? @skn Commented Aug 3 at 13:42
• @Janooo: The sections "History" and "Lorentz force" are more interesting. Weber electrodynamics is just a stub that was not developed further for a long time. It only applies to very slow and minimally accelerated point charges. Consequently, there are experiments that "disprove" it.
– skn
Commented Aug 3 at 15:07