I am aware of this SE question, however, it does not solve my problem. Assume that a bulk uniformly charged nonconductive sphere is set in motion along the $x$-axis in the lab frame of reference. Since the electrical field complies with the inverse square law everywhere outside the sphere, some magnetic fields are anticipated to be induced, circulating the $x$-axis. However, if we consider specific lines of this magnetic field very close to the surface of the sphere, it seems that every infinitesimally small area element undergoes a Lorentz force $F$ (the red arrows) shown in the figure. If such forces exist, they exert tension on the sphere tending to change the shape of the sphere in the lab frame. However, how this tension is interpreted in the charge reference frame as there is no Lorentz force in the latter?! Recall that when $v$ approaches the speed of light, these forces can probably smash the sphere into pieces!

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1 Answer 1


Yes test charges on the surface of the moving charged sphere feel this force. Yes it affects the strain of the sphere of charge in the lab frame.

But don't forget that some external forces are needed to keep a charged sphere intact in the first place. Even in its rest frame there are electric fields pointing outward trying to make the charged sphere expand.

Note that $(1/4\pi\epsilon_0)q\hat{r}/r^2$ is not a correct expression for the electric field of a moving charged sphere. That is only the correct for the electric field of a motionless charged sphere. For a moving charged sphere, some of the electric field is transformed into magnetic field. I've seen this derived for a moving point charge by taking the electric field of a point charge and applying a Lorentz boost to the electromagnetic tensor $F_{\mu\nu}$.

The internal strains of the charged sphere have to be the same in both the lab frame and the co-moving frame (assuming $v\ll c$ so we can ignore time dilation). In the co-moving frame, all of the strain comes from the electric field. In the lab frame, some of the electric field is transformed into magnetic field, and some of the strain comes from the charged sphere's motion through its magnetic field, while some still comes from the electric field.


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