2 Electromagnetic Waves
Maxwell’s equations in linear, isotropic, homogeneous media and the derivation of wave equations for vacuum, dielectric, and conducting media.
2.1 Medium Parameters
Parameters of a linear, isotropic, homogeneous electromagnetic medium: permittivity \(\varepsilon {\gt} 0\), permeability \(\mu {\gt} 0\), conductivity \(\sigma \ge 0\).
Wave speed in the medium: \(v = 1/\sqrt{\mu \varepsilon }\).
Squared wave speed: \(v^2 = 1/(\mu \varepsilon )\).
\(v^2 {\gt} 0\) since \(\mu , \varepsilon {\gt} 0\).
\(\mu \varepsilon {\gt} 0\).
\(\mu \ne 0\).
\(\varepsilon \ne 0\).
Vacuum medium: \(\varepsilon = \varepsilon _0\), \(\mu = \mu _0\), \(\sigma = 0\).
A lossless dielectric: general \(\varepsilon , \mu \) with \(\sigma = 0\).
A conducting medium: \(\sigma {\gt} 0\) introduces damping.
2.2 Maxwell’s Equations
Maxwell’s equations in a linear medium, packaging: Gauss’s law (\(\nabla \! \cdot \! \mathbf{E} = \rho /\varepsilon \)), no monopoles (\(\nabla \! \cdot \! \mathbf{B} = 0\)), Faraday (\(\nabla \! \times \! \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}\)), and Ampère-Maxwell (\(\nabla \! \times \! \mathbf{B} = \mu (\mathbf{J}_{\mathrm{free}} + \sigma \mathbf{E}) + \mu \varepsilon \frac{\partial \mathbf{E}}{\partial t}\)).
A source-free Maxwell system: \(\rho = 0\), \(\mathbf{J}_{\mathrm{free}} = 0\).
In source-free systems, \(\nabla \! \cdot \! \mathbf{E} = 0\).
In source-free systems, \(\nabla \! \times \! \mathbf{B} = \mu \sigma \mathbf{E} + \mu \varepsilon \frac{\partial \mathbf{E}}{\partial t}\).
2.3 Wave Equation Derivation
Smoothness hypotheses for the wave equation derivation: curl-time commutativity, \(C^2\) conditions for fields and their time derivatives, and differentiability of time derivatives.
Derived by taking curl of Faraday, applying curl-curl identity with \(\nabla \! \cdot \! \mathbf{E}=0\), and substituting Ampère’s law.
Derived by taking curl of Ampère, substituting Faraday, and using curl linearity.
\(\nabla ^2\mathbf{E} = \mu _0\varepsilon _0\, \frac{\partial ^2 \mathbf{E}}{\partial t^2}\) in vacuum (\(\sigma = 0\)).
\(\nabla ^2\mathbf{B} = \mu _0\varepsilon _0\, \frac{\partial ^2 \mathbf{B}}{\partial t^2}\) in vacuum.
\(c = 1/\sqrt{\mu _0\varepsilon _0}\).
\(\nabla ^2\mathbf{E} = \mu \varepsilon \, \frac{\partial ^2 \mathbf{E}}{\partial t^2}\) in a lossless dielectric.
\(\nabla ^2\mathbf{B} = \mu \varepsilon \, \frac{\partial ^2 \mathbf{B}}{\partial t^2}\) in a lossless dielectric.
\(v^2 = 1/(\mu \varepsilon )\) in a dielectric.
\(\nabla ^2\mathbf{E} = \mu \varepsilon \, \frac{\partial ^2 \mathbf{E}}{\partial t^2} + \mu \sigma \, \frac{\partial \mathbf{E}}{\partial t}\) (telegraph equation).
\(\nabla ^2\mathbf{B} = \mu \varepsilon \, \frac{\partial ^2 \mathbf{B}}{\partial t^2} + \mu \sigma \, \frac{\partial \mathbf{B}}{\partial t}\) (telegraph equation).
\(\nabla \! \times \! (\nabla \! \times \! \mathbf{E}) = -\frac{\partial }{\partial t}(\nabla \! \times \! \mathbf{B})\).