Electromagnetic rod shortening

[Pierre007080]

Hi Guys,
Your previous insights on the thread regarding gravitational rod shortening along the radial axis really helped me to have a naive but practical idea of what GR is about.
The next step in my thought process seems to be whether the EM force could induce rod shortening. The force formula (F = kq1q2/r^2) seems to have similar features to the gravitationalformula (except charge in stead of mass).

a)Are there simple formulae with regard to the radial rod shortening of a charged object (say negative) being attracted by another "STATIONARY" charge (say positive)?

b)Would it be possible to apply the same Lorentz type formulae as the case with gravity (GR)?

Perhaps you could use the theoretical example of a proton attracting an electron from infinity to explain?

MODERATORS EDIT: here is a link to the previous thread mentioned
https://www.physicsforums.com/showthread.php?t=441522

 

[WannabeNewton]

What do you mean by "radial rod shortening"?

Consider the simplest possible example one can think of in electromagnetism: we are in an inertial frame $S$ and a continuous line of charges are placed along the $x$ axis of frame $S$ (say from $z = 0$ to $z = L$) so as to represent a charged rod. We then switch on a uniform electric field $\vec{E} = E e_{\hat{x}}$. Each particle experiences a proper acceleration $a = \frac{q E}{mc^2}$ where $q$ is the charge of a given particle and $m$ its rest mass; let's keep the rest mass as a control variable.

Then if all the particles have the same charge $q$, they will all have the same proper acceleration $a$ which means that the proper length of the charged rod will increase until its tensile strength can no longer withstand the internal stresses arising from the stretching and the rod breaks.

If, on the other hand, you can arrange for the line charge density of the rod to vary such that each particle experiences a different proper acceleration $a$ in a manner that respects Born rigidity, then the proper length of the rod will be constant and its length as measured in $S$ will be continuously decreasing over time (as read by a clock in $S$). However the equation that relates the proper length of the rod to the (continuously decreasing) length of the rod as measured in $S$ will not be the usual Lorentz factor because each particle has a different proper acceleration and hence a different velocity relative to $S$.

 

[PAllen]

However, if the rod stops accelerating (if Born rigid, one end must stop accelerating earlier than the other end, I believe), settles into inertial motion, the length will be given by the Lorentz factor.

 

[WannabeNewton]

However, if the rod stops accelerating (if Born rigid, one end must stop accelerating earlier than the other end, I believe), settles into inertial motion, the length will be given by the Lorentz factor.

Good point PAllen.

@Pierre, I took a look through the thread that was recently edited into your post. Unfortunately it seems like you have misunderstood a few foundational aspects of GR. Finally, since your previous thread served as the impetus for this one, keep in mind the following: gravity in GR is intimately tied to space-time geometry, space-time measurements, global causal structure, and the dynamical evolution of all fields that propagate on space-time. Electromagnetism in GR is simply represented by one such field.

 

[Pierre007080]

Hi PAllen,
Thanks for taking the trouble to read the "Rod shortening of General Relativity" thread to understand my question better. My questions relate to two single charges of sufficiently different masses to ensure that one charge remains stationary. In the theoretical absence of any other fields or masses, the second charge would then accelerate toward the stationary charge along a radial of the stationary charge. The speed that this charge would attain at a certain radius (r) would relate to the rod shortening that I am enquiring about.