In this paper we study Maxwell's equations with a thermal effect. This system models an induction heating process where the electric conductivity σ strongly depends on the temperature u. We focus on a special one-dimensional case where the electromagnetic wave is assumed to be parallel to the y-axis. It is shown that the resulting hyperbolic-parabolic system has a global smooth solution if the electrical conductivity σ(u) grows like uq with 0 ≤q<8+4 √ 3. A fundamental element in this paper is the establishment of a maximum principle for wave equations with damping. This maximum principle provides an a priori bound for the first derivative with respect to both x and t of the solution without the imposition of any differentiability assumptions nor bounds on the coefficient of the damping term. The use of a nonlinear multiplier then permits (via a bootstrap procedure) the estimation of successively higher Lp-norms of the temperature function u.