Exact expressions for the electric and magnetic fields are derived for the case of a filamentary lightning current that flows in an extending channel. The treatment differs from those in the existing literature in that the expressions are found from fundamental principles applied to the appropriate bounded current rather than from the addition of an extra term in the radiation fields to account for the channel-tip discontinuity, the turn-on term. A different analytical form is obtained for the electrostatic field component, but this is found to be equivalent to existing relations. The radiation component of the magnetic field and a far-field expression for the electric field can each be expressed as a single expression in which the turn-on term from the channel tip is an integral part. However, the far-field expression for E can only be represented in this simple way if a term that arises from the induction field, the component that varies as 1/R2, is included with the remaining radiation terms, so that it is not a true radiation field. The Lorentz condition is shown to be equivalent to the continuity equation expressed in terms of the delayed time t-R/c. For a transmission line model with channel-tip speed v and current pulse propagation speed w, the same far-field expression is found for v=w as for v>w. The latter condition includes the case of a transmission-line current on a preexisting channel. ¿ 1999 American Geophysical Union