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Conservation Laws. The local conservation of charge and the continuity equation. If the total charge in some volume changes, then exactly that amount of charge must have passed in or our through the surface. Energy conservation and Poynting’s theorem.
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Conservation Laws The local conservation of charge and the continuity equation If the total charge in some volume changes, then exactly that amount of charge must have passed in or our through the surface. Energy conservation and Poynting’s theorem Suppose we have some some charges and currents. The work done by electromagnetic forces acting on these charges in the interval dt is: We want to express the power in terms of of the fields alone. To this end, we use the Ampere-Maxwell law:
Poynting’s theorem (work-energy theorem of electrodynamics) total energy stored in the fields The rate at which the field energy energy is carried out of V across the boundary surface Poynting vector (the energy per unit time per unit area; energy flux density) Energy flux John Henry Poynting (1852-1914) For example, in a coaxial cable, the electric field is everywhere radial, while the magnetic field forms concentric circles around the inner conductor. The direction of propagation is along the axis of the cable.
Maxwell’s Stress Tensor (EM force, pressures, shears, and momentum) force density
Maxwell stress tensor The diagonal terms represent pressures, and off-diagonal terms are shears In the static case, the second term is zero. The first term is the integrated shear and pressure at the surface of the object.
Momentum conservation momentum stored in the electromagnetic fields momentum per unit time flowing in through the surface Thus represents the time rate of change of electromagnetic momentum out of a differential volume volume. It has two contributions. One is the change of mechanical momentum of the system and the second is the change of particle momentum in the system. momentum density momentum flux density Notice the very different role that S plays in conservation of energy (Poynting's theorem) and in conservation of momentum. In Poynting’s theorem, S is a flux that is integrated over an area to find the energy escaping a given volume. For conservation of momentum S is related to a momentum per unit volume.
Angular momentum of EM field angular momentum density What is really amazing is the fact that even a static field can contain nonzero angular momentum.
