The authors present a mathematical treatment of time-domain Maxwell's equations for scattering problems defined in an unbounded domain. The time-dependent scattering problem is first discretized in time by Newmark's time-stepping scheme. The resulting semidiscrete problem is proved to be well posed. Stability conditions are given for the time-marching scheme. Convergence proprieties of the finite element scheme for the fully discrete problem are also discussed. Numerical experiments are performed for two-dimensional cavity problems that demonstrate the accuracy and stability of the method.