The time integrated far field for Maxwell’s and D’Alembert’s equations

Abstract:

For $x$ large consider the electric field, $E(t,x)$, and its temporal Fourier Transform, $\hat E(\omega ,x)$. The D.C. component $\hat E(0,x)$ is equal to the time integral of the electric field. Experimentally, one observes that the D.C. component is negligible compared to the field. In this paper we show that this is true in the far field for all solutions of Maxwell’s equations. It is not true for typical solutions of the scalar wave equation. The difference is explained by the fact that though each component of the field satisfies the scalar wave equation, the spatial integral of $\partial _t E(t,x)$ vanishes identically. For the scalar wave equation the spatial integral of $\partial _t u(t,x)$ need not vanish. This conserved quantity gives the leading contribution to the time integrated far field. We also give explicit formulas for the far field behavior of the time integrals of the intensity.