I've been reading Melvin Schwartz's mostly excellent book, Principles of Electrodynamics. He has a section on measuring (g-2) for the muon, and I'm trying to follow the argument. I've attached the figure for reference.(adsbygoogle = window.adsbygoogle || []).push({});

He takes the approach of considering an instantaneously comoving frame of reference (this is all within the context of special relativity) and measuring the various effects. Basically, for illustration sake, the problem we're considering is that of a particle in a uniform magnetic field [itex] B [/itex], moving at right angles to the magnetic field, so in the lab frame it should be in uniform circular motion. In the boosted frame, there is an electric field [itex] \bar{E} = \gamma \beta B [/itex], and the magnetic field there is [itex] \bar{B} = \gamma B [/itex] , and if we stayed in this frame (instead of moving to the next instantaneously comoving frame) the motion would be a cycloid.

Schwartz analyzes an infinitesimal portion of the orbit in each case, as follows. He says that in each frame, the path would appear to be a segment of arc, but in the comoving frame that segment is 'foreshortened'. In the attached figure, he's drawn both arc segments on top of each other.

Here's my problem: it seems to me that in the instantaneously comoving frame, at the instant when t=0, the particle has zero velocity. (We're measuring the velocity in a frame which has the same velocity as the particle, ergo it starts from rest in the primed frame.) Now the electric field is at right angles to the boost direction, so the initial motion should all be perpendicular to the boost. But in the lab frame, that infinitesimal portion of the arc should bealongthe boost direction, right? So it seems like the arc in one frame is perpendicular to the arc segment in the other frame, and so I don't understand why he draws them on top of each other.

It's not just a small point, either. Based on this diagram, he equates [itex] s =s^\prime[/itex], since dimensions perpendicular to the boost don't contract. Since [itex] d = \gamma d^\prime[/itex], he can relate [itex] d\theta [/itex] to [itex] d\theta^\prime [/itex], and thereby relate the (instantaneous) angular velocity in the primed frame, to the cyclotron frequency in the lab frame. It seems pretty central to his argument.

Can anyone help me out here? i.e. is my basic analysis -- that the arc segments are perpendicular, and that this invalidates his argument about [itex] s=s^\prime [/itex] -- essentially correct?

Best of all would be if someone who has access to the book (it's Dover and highly rated, so probably several people here have it) can explain to me what the heck he's talking about in section 4-11. It seems like really interesting physics and I'd love to learn it. :)

Thanks!

Chip

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# Following Schwartz's argument re: measuring (g-2)

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