Appendix: Maxwell’s Equations

A Appendix: Maxwell’s Equations

Maxwell’s Equations, differential form:

\begin{array}{rcl} \vec{\nabla} \times \vec{H} & = & σ \vec{E} + \frac{\partial \vec{D}}{\partial t} & (A.1) \\ \vec{\nabla} \times \vec{E} & = & - \frac{\partial \vec{B}}{\partial t} & (A.2) \\ \vec{\nabla} \cdot \vec{B} & = & 0 (no magnetic monopoles) & (A.3) \\ \vec{\nabla} \cdot \vec{D} & = & η_{q} (charge density) = 0 in conductive media & (A.4) \\ \vec{D} & = & ϵ \vec{E}, ϵ scalar in isotropic media & (A.5) \\ \vec{B} & = & μ \vec{H}, μ scalar in isotropic media & (A.6) \\ \vec{\nabla} \times \vec{H} & = & σ \vec{E} + ϵ \frac{\partial \vec{E}}{\partial t} & (A.7) \\ \vec{\nabla} \times \vec{E} & = & - μ \frac{\partial \vec{H}}{\partial t} & (A.8) \end{array}

Some identities from vector calculus ([1, frontispiece]):

\begin{array}{rcl} \vec{\nabla} \times (\vec{\nabla} \times \vec{A}) & = & \vec{\nabla} (\vec{\nabla} \cdot \vec{A}) - \nabla^{2} \vec{A} & (A.9) \\ \int_{S} (\vec{\nabla} \times \vec{A}) \cdot \hat{n} d s & = & \oint_{C} \vec{A} \cdot d \vec{l} Stokes’ Theorem & (A.10) \\ \int_{V} \vec{\nabla} \cdot \vec{A} d^{3} x & = & \int_{S} \vec{A} . \hat{n} d s Divergence Theorem & (A.11) \end{array}

Ampere’s Law is obtained by applying Stokes’ Theorem to (A.1)

\begin{array}{rcl} \int_{S} (\vec{\nabla} \times \vec{H}) \cdot \hat{n} d s & = & \int_{S} σ \vec{E} \cdot \hat{n} d s + \frac{\partial}{\partial t} \int_{S} \vec{D} \cdot \hat{n} d s & (A.12) \\ \oint_{C} \vec{H} \cdot d \vec{l} & = & I + \frac{\partial}{\partial t} \int_{S} \vec{D} \cdot \hat{n} d s (Ampere’s Law) & (A.13) \end{array}

and Faraday’s Law similarly to (A.2)

\begin{array}{rcl} \int_{S} (\vec{\nabla} \times \vec{E}) \cdot \hat{n} d s & = & - \frac{\partial}{\partial t} \int_{S} \vec{B} \cdot \hat{n} d s & (A.14) \\ \oint_{C} \vec{E} \cdot d \vec{l} & = & - \frac{\partial Φ_{B}}{\partial t} (Faraday’s Law) & (A.15) \end{array}

Obtain Maxwell’s equations in frequency domain by assuming $exp - i ω t$ time dependence ([2, convention]), and apply (A.9) to (A.1) and (A.2):

\begin{array}{rcl} \vec{\nabla} \times \vec{H} & = & (σ - i ω ϵ) \vec{E} & (A.16) \\ \vec{\nabla} \times (\vec{\nabla} \times \vec{H}) & = & - \nabla^{2} \vec{H} & (A.17) \\ = & (σ - i ω ϵ) \vec{\nabla} \times \vec{E} & (A.18) \\ = & (σ - i ω ϵ) i ω μ \vec{H} & (A.19) \\ \vec{\nabla} \times \vec{E} & = & i ω μ \vec{H} & (A.20) \\ \vec{\nabla} \times (\vec{\nabla} \times \vec{E}) & = & - \nabla^{2} \vec{E} & (A.21) \\ = & i ω μ (\vec{\nabla} \times \vec{H}) & (A.22) \\ = & i ω μ (σ - i ω ϵ) \vec{E} & (A.23) \end{array}

\begin{array}{rcl} \nabla^{2} \vec{H} + (σ - i ω ϵ) i ω μ \vec{H} & = & 0 & (A.24) \\ \nabla^{2} \vec{E} + (σ - i ω ϵ) i ω μ \vec{E} & = & 0 & (A.25) \\ (A.26) \end{array}

In one Cartesian dimension $\nabla^{2} = d^{2} ∕ d z^{2}$ and (A.24) admits the solutions

\begin{array}{rcl} H_{y} (ω, z) & = & A (ω) e^{i k z} + B (ω) e^{- i k z} & (A.27) \\ E_{x} (ω, z) & = & C (ω) e^{i k z} + D (ω) e^{- i k z} & (A.28) \\ k^{2} & = & i ω μ (σ - i ω ϵ) & (A.29) \\ = & ω^{2} μ ϵ - i ω μ σ & (A.30) \\ = & ω^{2} μ ϵ [1 - i σ ∕ (ω ϵ)] & (A.31) \\ = & ω^{2} μ ϵ (1 - i δ) & (A.32) \\ δ (ω) & \equiv & σ ∕ (ω ϵ) & (A.33) \end{array}

In free space $σ = 0$ , $μ_{o} ϵ_{0} = c^{- 2}$ and $k_{free} = ω ∕ c$ . In lossy media $σ > 0$ and things become more complex. But no where in Maxwell’s equations does $σ$ appear save through $(σ - i ω ϵ) \equiv ϵ (1 - i δ (ω))$ . If $σ$ is large enough and $ω$ small enough that $δ ≫ 1$ , as is usually (almost always) the case in sedimentary geology at audio frequencies and smaller, $k (ω)$ may safely be set to $k (ω) = \sqrt{- i μ σ ω}$ , Maxwell’s Equations change from wave-like to diffusive in character, and permittivity $ϵ$ need not apply.

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