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 ${\omega }_0$ on the $(x,y)$-plane at $z=0$ with boundary conditions, $x\sim x+2\pi R$ and $y\sim y+2\pi R$, it emits radiation into the $z$-direction of $z$ in $[-\mathrm{\infty },+\mathrm{\infty }]$. We find that the radiation shows discontinuities as a function of $\omega_0 R$ at which a new propagating mode with a different Fourier component appears. For a small radius limit ${\omega }_{0}R\ll 1$, 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 $\omega_0 R \rightarrow \infty$, the energy flux of the radiation reduces to the expected Larmor formula.