In this paper, the dyadic Green's function technique has been employed to characterize electromagnetic radiation of an imposed current line source in the presence of an isotropic dielectric elliptic cylinder. The current density along the infinitely long wire has a constant amplitude but a varying phase. The elliptic cylinder is considered to be infinite in length. In order to analyze the problem, the dyadic Green's functions are expressed in terms of elliptic vector wave functions and the general equations needed to solve for the reflection and transmission coefficients are derived from the boundary conditions. These derived equations are transformed into, and solved using, a linear equation system. Numerically, the radiation patterns of the infinitely long wire are computed, plotted, and shown for various cases where the position and distance of the line source are varied. Both lossy and lossless dielectric media for the elliptic cylinder are considered. The results are believed to be very useful to many practical problems, and especially to characterize cable radiation or transmission line power leakage in tunnels.