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Maxwell's equations in phasor form

Short answer

In phasor form for free space: ∇×E_s = −jωμH_s, ∇×H_s = jωεE_s, ∇·D_s = ρ_vs, ∇·B_s = 0.

Write Maxwell's equations in phasor form (for the time varying case in free space / a perfect dielectric).

For time-harmonic fields varying as e^(jωt), replace ∂/∂t by jω. In a perfect dielectric (σ = 0) or free space there is no conduction current (J = 0). Allowing a volume charge density ρ_v, and using D = εE and B = μH:

Point formName
∇×E_s = −jωμH_sFaraday's law
∇×H_s = jωεE_sAmpere's law (with displacement current)
∇·D_s = ρ_vs (∇·E_s = ρ_vs/ε)Gauss's law
∇·B_s = 0 (∇·H_s = 0)No magnetic monopoles

In a source-free region, ρ_vs = 0 so ∇·E_s = 0. For a conducting medium, the Ampere equation becomes ∇×H_s = (σ + jωε)E_s.

Here E_s and H_s are the phasors (functions of position only), related to the instantaneous fields by E = Re[E_s e^(jωt)].

✓ Answer checked by Claude Opus 5.5 (high effort)

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