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Key formulas
| Name | Formula | Variables / Notes |
|---|---|---|
| Gauss's Law (Electric) | ∮ D · dS = Q_enc | D: electric flux density (C/m²); dS: outward differential surface element (m²); Q_enc: total enclosed free charge (C) |
| Faraday's Law | ∮ E · dl = -dΦ_B / dt | E: electric field intensity (V/m); dl: differential path element (m); Φ_B: magnetic flux (Wb) = ∫ B · dS |
| Ampere's Law (with displacement current) | ∮ H · dl = I_enc + ∂D/∂t · ∫dS | H: magnetic field intensity (A/m); I_enc: enclosed conduction current (A); ∂D/∂t: displacement current density (A/m²) |
| Wave Equation in Free Space | ∇²E = μ_0 ε_0 * ∂²E/∂t² | E: electric field (V/m); μ_0: permeability of free space = 4π × 10⁻⁷ H/m; ε_0: permittivity of free space = 8.854 × 10⁻¹² F/m |
| Poynting Vector | P = E × H | P: power density vector (W/m²), direction gives direction of power flow; E: electric field (V/m); H: magnetic field (A/m) |
Key concepts
Maxwell's Equations
Four equations govern all classical electromagnetism: Gauss's law for E, Gauss's law for B (no magnetic monopoles), Faraday's law (changing B induces E), and Ampere-Maxwell law (current and changing E produce B). Together they predict electromagnetic wave propagation.
Boundary Conditions
At an interface between two media: tangential E is continuous (E_t1 = E_t2); normal D has a discontinuity equal to surface charge density (D_n1 - D_n2 = ρ_s); tangential H has a discontinuity equal to surface current density; normal B is continuous.
Skin Effect
In a conductor, high-frequency currents concentrate near the surface. Skin depth δ = sqrt(2 / (ωμσ)), where ω is angular frequency, μ is permeability, and σ is conductivity. At higher frequency, δ decreases and resistance increases.
Polarization of Waves
Linear polarization: E-field oscillates along one direction. Circular polarization: E-field rotates with constant magnitude. Elliptical polarization: E-field traces an ellipse. Polarization is defined by the locus of the E-field tip over one cycle.
Transmission Line Reflection
When a transmission line is terminated with load Z_L, the voltage reflection coefficient Γ = (Z_L - Z_0) / (Z_L + Z_0), where Z_0 is the characteristic impedance. Γ = 0 for matched load, Γ = 1 for open circuit, Γ = -1 for short circuit.
Tables
Maxwell's Equations: Integral and Differential Forms
| Law | Integral Form | Differential Form |
|---|---|---|
| Gauss E | ∮ D·dS = Q_enc | ∇·D = ρ_v |
| Gauss B | ∮ B·dS = 0 | ∇·B = 0 |
| Faraday | ∮ E·dl = -dΦ_B/dt | ∇×E = -∂B/∂t |
| Ampere-Maxwell | ∮ H·dl = I + ∂D/∂t·A | ∇×H = J + ∂D/∂t |
Material Properties
| Parameter | Symbol | Free Space Value |
|---|---|---|
| Permittivity | ε_0 | 8.854 × 10⁻¹² F/m |
| Permeability | μ_0 | 4π × 10⁻⁷ H/m |
| Wave velocity | c | 3 × 10⁸ m/s |
| Intrinsic impedance | η_0 | 377 Ω |
Quick facts
- Speed of light c = 1 / sqrt(μ_0 ε_0) = 3 × 10⁸ m/s.
- Intrinsic impedance of free space η_0 = sqrt(μ_0/ε_0) ≈ 377 Ω.
- Skin depth at 1 MHz in copper (σ = 5.8 × 10⁷ S/m): δ ≈ 66 μm.
- For a perfect electric conductor (PEC): E_tangential = 0 and B_normal = 0 at the surface.
- The divergence of B is always zero; there are no magnetic monopoles.
- Relative permittivity ε_r of water ≈ 80; of air ≈ 1.
- Power carried by a wave is proportional to |E|² and equals (1/2)|E|²/η for a plane wave.
Exam shortcuts
- Reflection coefficient shortcut: for a short circuit termination Z_L = 0, Γ = -1 immediately (no calculation). For open circuit Z_L = ∞, Γ = +1. These two cases appear frequently in standing wave problems.
- To apply Gauss's law, choose a Gaussian surface that matches the symmetry. Sphere for point/spherical charge, cylinder for line charge, pillbox for planar charge. The goal is to make |D| constant over the surface so ∮ D·dS = |D| * A.
- For phase velocity in a medium: v_p = c / sqrt(ε_r * μ_r). For most dielectrics μ_r = 1, so v_p = c / sqrt(ε_r). If ε_r = 4, then v_p = c/2 = 1.5 × 10⁸ m/s.