E Field and Field Transformations

[GRDixon]

Note: in the following, parentheses denote "subscript". "G" denotes "gamma".

A round, uncharged current loop is at rest in the xz plane of IRF K. The loop is centered on the Origin. Negative charge circulates around the loop, positive charge remains at rest. There is a nonzero B(y) field at points on the y axis.

Viewed from IRF K’, at time t'=0 dB(y)’/dt’=0 at points on the y’ axis. Yet E(z)’=GvB(y) (where G stands for “gamma”). Since the net charge is zero in K’ (as it is in K), and since dB(y)’/dt’=0 at points on the y’axis, what explains the nonzero E(z)’?

 

[bcrowell]

You haven't defined K'.

 

[GRDixon]

You haven't defined K'.

K' moves in the positive x direction of K at speed v. When the Origin clocks of K and K' coincide, they mutually read zero. The x/x' axes, etc., of the two frames are parallel.

 

[GRDixon]

K' moves in the positive x direction of K at speed v. When the Origin clocks of K and K' coincide, they mutually read zero. The x/x' axes, etc., of the two frames are parallel.

Here's a hint: the E' field in K' is the superposition of a conservative part (non-zero divergence) and a non-conservative part (non-zero curl). The non-zero E(z)' on the y' axis at time t'=0 is conservative. The question is, what engenders such a conservative field? The current loop has zero net charge.

 

[bcrowell]

Here's a hint: the E' field in K' is the superposition of a conservative part (non-zero divergence) and a non-conservative part (non-zero curl). The non-zero E(z)' on the y' axis at time t'=0 is conservative. The question is, what engenders such a conservative field? The current loop has zero net charge.

The loop's charge density doesn't vanish everywhere, as viewed in K'. For a simpler example, see subsection 4.2.4 of this: http://www.lightandmatter.com/html_books/genrel/ch04/ch04.html#Section4.2 (the situation described in figure b).

 

[GRDixon]

The loop's charge density doesn't vanish everywhere, as viewed in K'. For a simpler example, see subsection 4.2.4 of this: http://www.lightandmatter.com/html_books/genrel/ch04/ch04.html#Section4.2 (the situation described in figure b).

Good eye. In general, uncharged current loops that have an overall component of velocity in the plane of the loop are electrically polarized. It's interesting to model a ceramic, disc-shaped magnet as an array of microscopic uncharged current loops. When the magnet spins, the motion-associated tiny dipoles engender an electric field with a component toward/away from the parent magnet's rotation axis. I have read that even Einstein puzzled over some of the associated "homopolar" effects and their "seats of emf".