  We prove that in contrast to the free wave equation in $\R^3$ there are no
incoming solutions of Maxwell's equations in the form of spherical or modulated
spherical waves. We construct solutions which are corrected by lower order
incoming waves. With their aid, we construct dissipative boundary conditions
and solutions to Maxwell's equations in the exterior of a sphere which decay
exponentially as $t \to +\infty$. They are asymptotically disappearing.
Disappearing solutions which are identically zero for $t \geq T > 0$ are
constructed which satisfy maximal dissipative boundary conditions which depend
on time $t$. Both types are invisible in scattering theory.
