The paper deals with the direct time domain calculation of a transient electric field in the upper half-space generated by a dipole antenna above a lossy ground. The transient current along the wire is governed by the Hallen integral equation which is numerically solved by means of the Galerkin-Bubnov scheme of the Indirect Boundary Element Method (GB- IBEM). Provided that the current distribution along the dipole is known, the related radiated field is computed from certain integral formulas. Some illustrative computational examples are presented in the paper.