We consider a class of electromagnetic fields that contains crossed fieldscombined with longitudinal electric and magnetic fields. We study the motion ofa classical particle (solutions of the Lorentz equations) in such fields. Then,we present an analysis that allows one to decide which fields from the classact as a beam guide for charged particles, and we find some time-independentand time-dependent configurations with beam guiding properties. We demonstratethat the Klein-Gordon and Dirac equations with all the fields from the classcan be solved exactly. We study these solutions, which were not known before,and prove that they form complete and orthogonal sets of functions.
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