Electromagnetic radiation in a semi-compact space

In this note, we investigate the electromagnetic radiation emitted from a revolving point charge in a compact space. If the point charge is circulating with an angular frequency ω0 on the (x,y)-plane at z=0 with boundary conditions, x∼x+2πR and y∼y+2πR, it emits radiation into the z-direction of z∈[−∞,+∞]. We find that the radiation shows discontinuities as a function of ω0R at which a new propagating mode with a different Fourier component appears. For a small radius limit ω0R≪1, all the Fourier modes except the zero mode on (x,y)-plane are killed, but an effect of squeezing the electric field totally enhances the radiation. In the large volume limit ω0R→∞, the energy flux of the radiation reduces to the expected Larmor formula.