The classical Coulomb and radiation fields from a relativistic charged particle or beam of particles passing out of a conductor into vacuum are calculated in detail, and certain features are pointed out and clarified. The similar case of pulsed acceleration is also studied. For the beam case, field structure is unusual in that Coulomb fields overlap radiation fields for an extended time. Radiation fields are proportional to beam current I (t) , not dI /dt . It is shown that for all practical times for beams of interest, the Coulomb fields drop off like 1/R , not 1/R2. The fields evolve through an "immersion" phase where the beam is still immersed in its own radiation fields, to a "separation" phase in which the radiation field has separated from the beam. We explain how the radiation fields break away from the beam, with field lines always remaining connected. Relation to more familiar field configurations (radiation by accelerated charges, dipole radiation, transition radiation, scattering from small conductors) is discussed, and energy radiated away is calculated. A radiation formula is presented for the general case in which the net current profile changes as the beam propagates (in air). We also discuss beams exiting an openended pipe. It is shown that the maximum radiation angle, = 1/γ, has a simple physical, geometric origin. Likewise the maximum angle of a longitudinally accelerated charge is shown to have a similar, readily understood, physical origin, making it intuitively clear why an accelerated electron radiates most at the angle ~ 1/2γ .