There is a situation, we have an electric field and a magnetic field, both are static. And we know the density of energy is u=E·D/2+B·H/2, so dU/dt=0, but Poynting vector S=ExH is not zero, which means energy is flowing. This confused me. Static field also has energy flux?
The static magnetic field is produced by a constant electric current. That means there is resistance, and energy is flowing into matter. "Hidden" momentum is not involved.
The relevant expression, from the conservation of energy, is: [tex]\frac{\partial u}{\partial t} + \nabla \bullet S = -J \bullet E[/tex]. Does this help?