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For a hydrogen atom in the field B, torque on the electron is [itex]\tau =\mu Bsin\theta =\omega_pJsin\theta[/itex] => [itex]\omega_p=gqB/(2m)[/itex] If we only consider orbital motion (ignore spin), then g=1. So [itex]\omega_p=qB/(2m)[/itex]. (It follows the steps from Feynman's lectures VII section 34-3)

For a single electron in the field, [itex]qvB=v^2m/r[/itex] => [itex]\omega=qB/m[/itex]. I know these two cases are different, but why the results look the same (only a factor of 2 different). Is there any connection between them, or just coincidence?