The transient scattering of two-dimensional electromagnetic fields by an obstacle of finite extent is investigated with the aid of the time domain integral equation technique. In solving such equations with the marching-on-in-time method, numerical instabilities form a major problem. These instabilities can be attributed to errors in the discretization of the source type integrals that occur in the equations. In this paper, we formulate two so-called stability criteria for such a discretization that, if they are met, guarantee that the instability can be controlled by reducing the discretization step. With the aid of these criteria, we analyze the solution of two two-dimensional electromagnetic scattering problems, namely the scattering of a pulsed plane wave by a perfectly conducting and an inhomogeneous, lossy dielectric cylinder. Numerical results are presented and discussed.