- #1

fog37

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I am reviewing how classical EM problems are treated when dielectric materials are involved. Maxwell's equations relate the following vector fields: ##E(r,t)##, ##B(r,t)##, ##D(r,t)##, ##H(r,t)##, ##J(r,t)## and scalar field ##\rho(r,t)##. The two constitutive equations are also needed to relate ##E(r,t)##, ##B(r,t)##, ##D(r,t)##, ##H(r,t)## together via the material's permittivity and permeability.

**n the presence of dielectrics, the electric polarization ##P(r,t)## and magnetization ##M(r,t)## can be introduced instead of permittivity and permeability. ##P(r,t)## and ##M(r,t)## are vector fields generated by bound sources (bound charges and bound currents) .**

The ultimate goal is solving for the two fields ##E(r,t)##, ##B(r,t)##. IThe ultimate goal is solving for the two fields ##E(r,t)##, ##B(r,t)##. I

In theory, we could solve EM problems without using the auxiliary fields ##D(r,t)##, ##H(r,t)## fields that are controlled by free charges and conduction currents only. For a linear dielectric material only, the fields ##E(r,t)## and ##D(r,t)## are related in two different but equivalent ways: $$D(r,t) =\epsilon_0 E(r,t) +P(r,t)$$ $$E(r,t)=\epsilon(r,t) D(r,t)$$

The electric permittivity ##\epsilon## can be a constant, a function, a tensor function, etc.

Why is it so convenient to use the electric displacement vector ##D(r,t)## to find ##E(r,t)##? According to ##E(r,t)=\epsilon(r,t) D(r,t)##, to find E(r,t) we need both ##D(r,t)## and knowledge of the electric permittivity ##\epsilon##. Is it easier to gain knowledge of ##\epsilon## than gaining knowledge of the electric polarization ##P(r,t)## and solve for ##E(r,t)## using ##D(r,t) =\epsilon_0 E(r,t) +P(r,t)##?

Fundamentally, I see three possible scenarios:

Scenario 1: solving for ##E(r,t)## and ##B(r,t)## using Maxwell's equations, the total sources (##J(r,t)## and ##\rho(r,t)##) and ##P(r,t)## and ##M(r,t)## (no presence of the auxiliary fields).

Scenario 2: solving for ##E(r,t)## and ##B(r,t)## using Maxwell's equations, the auxiliary fields##D(r,t)##, ##H(r,t)##, the total sources ##J(r,t)## and ##\rho(r,t)##, ##P(r,t)## and ##M(r,t)##.

Scenario 3: solving for ##E(r,t)## and ##B(r,t)## using Maxwell's equations, the auxiliary fields##D(r,t)##, ##H(r,t)##, the total sources ##J(r,t)## and ##\rho(r,t)##, the electric permittivity ##\epsilon## and the magnetic permeability ##\mu##.

Are these approaches all equivalent to each other? It seems that scenario 3 is the most convenient for some reason but I am not sure why...

Thank you!