Electrodynamics MOC

Maxwell’s equations in materials

By introducing the auxiliary [[Electric displacement|⃗𝐃-field]] and H-field and separating free charge and current from those arising from electric polarization and magnetization, Maxwell’s equations may become

1. Gauß’s law for diëlectrics
2. Gauß’s law for magnetic flux
3. Faraday’s law of induction
4. Ampère’s law for magnets

Differential form

⃗∇⋅⃗𝐃=𝜌𝑓

⃗∇⋅⃗𝐁=0

⃗∇×⃗𝐄=−𝜕⃗𝐁𝜕𝑡

⃗∇×⃗𝐇=⃗𝐉𝑓+𝜕⃗𝐃𝜕𝑡

Integral form

⊂⊃∬𝜕Ω⃗𝐃⋅𝑑⃗𝐚=∭Ω𝜌𝑓𝑑𝜏′

⊂⊃∬𝜕Ω⃗𝐁⋅𝑑⃗𝐚=0

E=∮𝜕Σ⃗𝐄⋅𝑑⃗ℓ=−𝜕Φ𝐵𝜕𝑡=−𝜕𝜕𝑡∬Σ⃗𝐁⋅𝑑⃗𝐚

∮𝜕Σ⃗𝐇⋅𝑑⃗𝐫=𝜇0(𝐼𝑓,Σ+𝜕Φ𝐷,Σ𝜕𝑡)=(∬Σ⃗𝐉𝑓⋅𝑑⃗𝐚+𝑑𝑑𝑡∬Σ⃗𝐃⋅𝑑⃗𝐚)

Sources

Noting the expressions for bound charge density and bound current density, as well as current due to changes in electric polarization density, we have

𝜌=𝜌𝑓+𝜌𝑏=𝜌𝑓−⃗∇⋅⃗𝐏⃗𝐉=⃗𝐉𝑓+⃗𝐉𝑏+𝜕⃗𝐏𝜕𝑡=⃗𝐉𝑓+⃗∇×⃗𝐌+𝜕⃗𝐏𝜕𝑡

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