Non-Stationary Electromagnetics, 2nd Edition

According to the causality principle, which is expressed by the Heaviside unit function in Eq. (1.41), the function G₀ is equal to zero if t′ > t. Therefore, the integration with respect to time in the convolution (1.59) is executed over the interval (−∞, t), where t is the observation moment. If the observation moment precedes the initial moment, t < t₀, then the integration is carried out over the whole interval (−∞, t] in the expression (1.59). If the observation occurs after the initial moment, t > t₀, then the integration time interval in Eq. (1.59) should be divided into two parts: before and after t₀. Consequently, the expression (1.59) is then rewritten as:

$\begin{array}{l}E=F-\frac{1}{v^2}\frac{{\partial }^2{G}_0}{\partial t^2}*\theta \left(t\right)\chi \frac{1}{{\epsilon }_0\epsilon }\left(P_1-P_{\text{ex}}\right)\\ -\frac{\partial G_0}{\partial t}*\theta \left(t\right)\{\text{curl}\chi {\mu }_0\mu \left(M_1-M_{\text{ex}}\right)+{\mu }_0\mu \end{array}$