Electromagnetic media §

The idea, step by step.

1. Why two pairs of fields? Maxwell’s equations are most cleanly written with $\mathbf{E}$ and $\mathbf{H}$ (the fields that exert forces and do work), but the sources of the fields are charge and current. The auxiliary fields $\mathbf{D}$ and $\mathbf{B}$ are introduced so that Gauss’s law and the no-monopole law depend only on free charge and free current: $\nabla\cdot\mathbf{D}=\rho_{\text{free}}$ and $\nabla\cdot\mathbf{B}=0$. The constitutive relations are the material’s rule for converting between the two descriptions.

2. The relations are the material’s “personality.” A vacuum has $\varepsilon_0$, $\mu_0$, $\sigma=0$. A dielectric has a large $\varepsilon$ (it polarizes strongly). A ferromagnet has a large $\mu$ (its domains align). A metal has a large $\sigma$ (charges flow). Everything downstream — wave speed, reflection, loss — is a consequence of these three numbers.

3. They are not always simple scalars. The linear, isotropic, non-dispersive form above is the simplest case. Real media can make $\varepsilon$, $\mu$, $\sigma$ into tensors (anisotropic), frequency-dependent functions (dispersive), or field-dependent (nonlinear). Sections 5 and 8 develop these generalizations.

4. Authoritative source. The constitutive framework is standard; see Ellingson, Electromagnetic Properties of Materials and Jackson, Classical Electrodynamics.

The idea, step by step.

1. Polarization is charge separation. In a dielectric, the applied $\mathbf{E}$ field nudges the electrons slightly off the nuclei, creating tiny dipoles. The polarization $\mathbf{P}$ is the net dipole moment per unit volume. Bound charges pile up on the surfaces, producing an internal field that opposes the applied field.

2. Permittivity counts the polarization. The total displacement is $\mathbf{D} = \varepsilon_0 \mathbf{E} + \mathbf{P}$. For a linear dielectric $\mathbf{P} = \varepsilon_0 \chi_e \mathbf{E}$ (with $\chi_e$ the electric susceptibility), so $\mathbf{D} = \varepsilon_0(1+\chi_e)\mathbf{E} = \varepsilon \mathbf{E}$. The relative permittivity is $\varepsilon_r = 1 + \chi_e$.

3. Why it matters: capacitance and field reduction. A dielectric between capacitor plates raises the capacitance by $\varepsilon_r$ (it stores more charge at the same voltage) and reduces the field inside the material by $\varepsilon_r$ for a fixed free charge. Water has $\varepsilon_r \approx 80$, glass $\varepsilon_r \approx 4--10$, silicon $\varepsilon_r \approx 11.7$, air $\varepsilon_r \approx 1.0006$.

4. Authoritative source. See Ellingson, Electromagnetic Properties of Materials and Griffiths, Introduction to Electrodynamics.

The idea, step by step.

1. Magnetization is aligned magnetic dipoles. Atoms carry tiny magnetic moments (from electron spin and orbital motion). An applied $\mathbf{H}$ field tends to align them. The magnetization $\mathbf{M}$ is the net moment per unit volume, and $\mathbf{B} = \mu_0(\mathbf{H}+\mathbf{M})$.

2. Three regimes. In diamagnetic materials (copper, water, bismuth) the induced moments oppose the field, so $\mu_r$ is just below 1. In paramagnetic materials (aluminum, oxygen) the moments align weakly with the field, so $\mu_r$ is just above 1. In ferromagnetic materials (iron, nickel, cobalt) the moments lock into domains that align cooperatively, giving $\mu_r$ of hundreds to tens of thousands — but with hysteresis (the response depends on history) and saturation (it stops growing once all domains align).

3. Why it matters. High-$\mu$ materials concentrate magnetic flux, which is why transformer cores, inductor cores, and MRI magnets use them. Ferromagnetic response is nonlinear and history-dependent, so the simple $\mathbf{B}=\mu\mathbf{H}$ relation is only an approximation there; it is exact for the linear diamagnetic and paramagnetic cases.

4. Authoritative source. See Ellingson, Electromagnetic Properties of Materials and Griffiths, Introduction to Electrodynamics.

The idea, step by step.

1. Conduction is mobile charge. In a metal, a sea of free electrons drifts under $\mathbf{E}$. The current density is proportional to the field: $\mathbf{J} = \sigma \mathbf{E}$. This is the microscopic Ohm’s law; the circuit form $V = IR$ is its integral consequence.

2. Conductivity is the source of loss. Wherever $\sigma > 0$, the field does work on the charges at a rate $\mathbf{J}\cdot\mathbf{E} = \sigma E^2$ per unit volume, which is dissipated as heat (Joule heating). This is why a signal in a lossy medium attenuates — the energy is converted to thermal energy. See Poynting’s theorem for the energy bookkeeping.

3. The spectrum of materials. Conductors (copper, aluminum) have $\sigma \sim 10^7--10^8\,\text{S/m}$. Semiconductors (silicon) range over many orders of magnitude depending on doping. Electrolytes and biological tissue are weak conductors ($\sigma \sim 0.1--1\,\text{S/m}$). Good insulators (glass, dry air) have $\sigma \ll 10^{-12}\,\text{S/m}$.

4. Authoritative source. See Ellingson, Electromagnetic Properties of Materials and Griffiths, Introduction to Electrodynamics.

The idea, step by step.

1. Loss appears as an imaginary part. In a sinusoidal field of angular frequency $\omega$, the conduction current $\sigma \mathbf{E}$ is in phase with $\mathbf{E}$ while the displacement current $j\omega\varepsilon\mathbf{E}$ is $90^\circ$ out of phase. Combining them gives an effective complex permittivity $\varepsilon_c = \varepsilon - j\sigma/\omega$. The loss tangent $\tan\delta = \sigma/(\omega\varepsilon)$ compares the two: large when conduction dominates (good conductor), small when displacement dominates (dielectric).

2. The wave decays inside a conductor. Substituting $\varepsilon_c$ into the wave equation gives a complex propagation constant $\gamma = \alpha + j\beta$, where $\alpha$ is the attenuation (Np/m). In a good conductor ($\sigma \gg \omega\varepsilon$), $\alpha \approx \beta \approx \sqrt{\pi f \mu \sigma}$, and the field is $E(z) = E_0 e^{-z/\delta}$ with $\delta = 1/\alpha$.

3. Skin depth sets the penetration. The field (and current) are confined to a thin surface layer of thickness $\delta$. For copper at $f = 1\,\text{GHz}$, $\delta \approx 2.1\,\mu\text{m}$; at $60\,\text{Hz}$ it is $\approx 8.5\,\text{mm}$. This is why high-frequency currents flow on the surface (the skin effect), why RF shielding works, and why thick conductors are wasteful at high frequency.

4. Authoritative source. See Jackson, Classical Electrodynamics, and Ellingson, Electromagnetic Properties of Materials.

The idea, step by step.

1. The medium slows the wave. The same wave-equation derivation that gives $c = 1/\sqrt{\mu_0\varepsilon_0}$ in vacuum gives $v = 1/\sqrt{\mu\varepsilon}$ in a medium. Since $\mu \ge \mu_0$ and $\varepsilon \ge \varepsilon_0$, the wave is always slower than $c$. For a dielectric with $\mu_r \approx 1$, $v = c/\sqrt{\varepsilon_r}$.

2. The impedance sets the E/B ratio. In a plane wave the ratio of the electric to magnetic field is the intrinsic impedance $\eta = \sqrt{\mu/\varepsilon}$. In vacuum $\eta_0 \approx 377\,\Omega$. In a dielectric with $\varepsilon_r = 4$, $\eta = 377/2 \approx 189\,\Omega$. This impedance is what determines reflection at an interface (Section 7).

3. Why it matters. The speed sets the wavelength in the medium ($\lambda = v/f$), which governs the size of antennas, transmission lines, and optical paths. The impedance mismatch between two media is the root cause of reflection — the reason a lens needs an anti-reflection coating and a transmission line needs impedance matching.

4. Authoritative source. See Maxwell (1865) and Griffiths, Introduction to Electrodynamics.

The idea, step by step.

1. Fields must match at the boundary. Maxwell’s equations require the tangential $\mathbf{E}$ and $\mathbf{H}$ to be continuous across an interface. An incident wave arriving at the boundary therefore splits into a reflected wave (back into medium 1) and a transmitted wave (into medium 2), and the sum of incident and reflected fields must equal the transmitted field at the surface.

2. The coefficient is an impedance ratio. Enforcing the boundary conditions gives $\Gamma = (\eta_2-\eta_1)/(\eta_2+\eta_1)$. If the impedances match ($\eta_2 = \eta_1$), there is no reflection ($\Gamma = 0$). If medium 2 is a perfect conductor ($\eta_2 = 0$), the wave is fully reflected with a sign flip ($\Gamma = -1$).

3. Power reflection is $|\Gamma|^2$. The fraction of incident power reflected is $|\Gamma|^2$. For a wave going from air ($\eta_1 \approx 377\,\Omega$) into a dielectric with $\varepsilon_r = 4$ ($\eta_2 \approx 189\,\Omega$), $\Gamma \approx -0.33$ and $|\Gamma|^2 \approx 0.11$ — about 11% of the power is reflected. This is why glass reflects a little and why anti-reflection coatings (a thin quarter-wave layer) are used to cancel it.

4. Authoritative source. See Maxwell (1865) and Griffiths, Introduction to Electrodynamics.

The idea, step by step.

1. Anisotropy: direction matters. In a crystal the atomic arrangement is not the same in every direction, so $\mathbf{D}$ is not generally parallel to $\mathbf{E}$. The relation becomes $\mathbf{D} = \boldsymbol{\varepsilon}\,\mathbf{E}$ with $\boldsymbol{\varepsilon}$ a tensor. This is the origin of birefringence: an ordinary light beam splits into two polarized rays travelling at different speeds.

2. Dispersion: frequency matters. Because polarization and magnetization cannot respond instantaneously, $\varepsilon$ and $\mu$ depend on frequency. The result is dispersion: different colours travel at different speeds, which is why a prism spreads white light into a spectrum. The group velocity (the speed of a pulse) differs from the phase velocity, and near absorption resonances the dispersion is strong.

3. Nonlinearity: amplitude matters. At high field strengths (lasers), the polarization is no longer proportional to $\mathbf{E}$. The $\chi^{(2)}$ term mixes frequencies: two photons of frequency $\omega$ can combine into one of $2\omega$ (second-harmonic generation), the basis of green laser pointers.

4. Metamaterials: engineer the parameters. By arranging subwavelength metallic or dielectric elements, one can make an effective medium with negative $\varepsilon$ or $\mu$. A medium with both negative (a double-negative or left-handed medium) has a negative refractive index: light bends the “wrong” way, enabling negative refraction, cloaking, and superlenses that beat the diffraction limit.

5. Authoritative source. See Jackson, Classical Electrodynamics, and the review literature on metamaterials (e.g. Pendry, Negative Refraction, 2000).

Problem. A uniform plane wave at $f = 1\,\text{GHz}$ travels in air and strikes normally a large copper plate ($\sigma = 5.8\times10^7\,\text{S/m}$, $\mu \approx \mu_0$).

1. What is the skin depth of copper at this frequency?
2. What fraction of the incident power is reflected?
3. How far into the copper does the field penetrate before dropping to $1/e$ of its surface value?