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In Synchrotrons I think they use microwaves to accelerate the electrons bundles that fly through-how does putting a microwave through a cavity accelerate an electron? I know that the Electric and Magnetic fields could and do have an effect on the electron, but if the radiation is moving parallel to the electron, wouldn't the force be perpendicular? Also, how exactly does the microwave radiation look in the vacuum chamber? Are the two ends constant nodes?

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Electromagnetic waves aren't just magnetic which would justify your word "orthogonal". They also have electric fields and the electric force has the same direction as the electric field. –  Luboš Motl Jul 2 '13 at 7:54
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Here's my fairly clumsy attempt to draw the sort of RF cavities used in particle accelerators. The usual disclaimer applies, my knowledge of accelerator design is strictly at the popular science level so caveat emptor. I'm sure the particle physicists hereabouts will pick me up on any errors.

RF cavity

The RF cavity is just a box, within which is a standing wave. Typically the mode used would be 010, i.e. the electic field is directed along the length of the box (the particle direction of travel) and it's the lowest resonance. In my diagram the field gets more negative at upwards displacement i.e. a positive gradient of the red curve indicates a negative field gradient. So a positive particle like the protons in the LHC will be accelerated up a positive gradient.

The acceleration relies on synchronising the phase and frequency of the standing wave with the particle's transit through the cavity. At time zero the proton enters the resonator, and is accelerated forwards by the negative field gradient in the left half of the cavity. The frequency is adjusted so the period is twice the time the particle takes to cross the cavity, so after one quarter of the period, $\tau/4$, the particle has reached the middle of the box and the field has fallen to zero. At this point the particle is coasting. As the particle passes the centre of the resonator the field changes phase by $\pi$ and the particle is again accelerated by the field gradient in the right half of the cavity.

In big accelerators like the LHC the proton velocity is effectively constant (at 0.99 and a bit $c$) so the standing wave frequency is constant. It's just a matter of getting the phase right.

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"In [some] accelerators [] the [beam] velocity is effectively constant (at 0.99 and a bit c) so the standing wave frequency is constant." Indeed the CEBAF accelerator at JLAB takes advantage of this fact to recycle the beam on up to five passes through the same cavities continuously so that the cavities are occupied by five superimposed beams with different energies all being efficiently accelerated by the same RF at the same time in the same place. Cute, and a powerful demonstration of relativity. –  dmckee Jul 2 '13 at 13:30
this is a lecture on the physics of accelerators, including RF cavities. cas.web.cern.ch/cas/Baden/PDF/Physics-Accelerators-2.pdf . –  anna v Jul 2 '13 at 13:48
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