For a given charge distribution and current distribution in , the electric and magnetic fields, and , satisfy Maxwell's equations, which in suitable units, read
The Poynting vector is defined as .
(a) For a closed surface around a volume , show that
(b) Suppose and consider an electromagnetic wave
where and are positive constants. Show that these fields satisfy Maxwell's equations for appropriate , and .
Confirm the wave satisfies the integral identity by considering its propagation through a box , defined by , and .