Maxwell Equations in Phasor Form

When electromagnetic fields vary sinusoidally with time at a single frequency, it is far more efficient to work in the frequency domain using phasor representation. The phasor form of Maxwell's equations replaces all time derivatives with a multiplication by jω, where j is the imaginary unit and ω is the angular frequency. This transformation converts the partial differential equations in time into algebraic equations in frequency, greatly simplifying analysis of antennas, waveguides, transmission lines, and any system excited by a single-frequency source.

Core Concept Explanation

Any time-varying field quantity that varies sinusoidally can be written as the real part of a complex exponential: E(x,y,z,t) = Re[Es(x,y,z) e^(jωt)], where Es is the phasor (complex amplitude). The phasor captures both the magnitude and phase of the field at each point in space, but contains no time information. When this form is substituted into the time-domain equations, the time derivative ∂/∂t acting on e^(jωt) simply brings down a factor of jω. This is the fundamental transformation that converts differential equations in time to algebraic equations in frequency.

This approach is identical to the phasor analysis used in AC circuits, where impedances replace resistors and V = IZ replaces Ohm's law. In field theory, the phasor form allows definition of the intrinsic impedance η = E/H, the propagation constant γ, and the concept of complex permittivity, all of which are frequency-domain quantities that simplify characterization of material and wave behavior.

It is important to note that phasor analysis is valid only when the system is driven by a single frequency (time-harmonic fields). For broadband signals, Fourier decomposition is used first, and phasor analysis is applied separately to each frequency component. In GATE problems, whenever the frequency is mentioned and fields involve sin or cos functions, the phasor form must be used.

Mathematical Expression

Applying the substitution ∂/∂t = jω to all time-domain Maxwell equations gives the phasor form. The subscript s denotes phasor quantities (complex, space-only functions):

∇ · Ds = ρvs

∇ · Bs = 0

∇ × Es = −jωBs = −jωμHs

∇ × Hs = Js + jωDs = (σ + jωε)Es = jωεc Es

The last equation introduces the complex permittivity εc = ε − j(σ/ω). The real part ε represents energy storage (polarization), and the imaginary part σ/ω represents energy dissipation (conduction losses). The loss tangent is defined as tan δ = σ/(ωε), which is the ratio of the imaginary to the real part of the complex permittivity. For a good dielectric, tan δ << 1. For a good conductor, tan δ >> 1.

For plane waves in a lossy medium, the propagation constant is γ = jω√(μεc) = α + jβ. Here α is the attenuation constant in Nepers per meter and β is the phase constant in radians per meter. The wave decays as e^(−αz) while oscillating as e^(−jβz), giving a decaying sinusoid in the time domain.

Practical Understanding

In practical antenna and RF design, all computations are performed in the phasor domain. Quantities like radiation resistance, input impedance, gain, and directivity are all frequency-domain concepts. The phasor form of Maxwell's equations is the foundation of the Helmholtz wave equation, which governs field distribution inside waveguides and resonant cavities:

∇²Es + k²Es = 0, where k² = ω²με = β²

This equation is solved with boundary conditions to find the allowed modes of propagation in a waveguide, such as TE and TM modes, each existing at frequencies above a cutoff frequency. The phasor form also naturally accommodates frequency-dependent permittivity ε(ω), which is a fundamental property of all real dielectric materials.

Given:
A lossy medium has ε = 4ε₀, μ = μ₀, σ = 0.02 S/m
Frequency f = 100 MHz (ω = 2π×10⁸ rad/s)
Find: loss tangent, complex permittivity, and whether this is a good dielectric or conductor.

Why this formula applies:
Complex permittivity εc = ε − j(σ/ω)
Loss tangent tan δ = σ/(ωε)

Formula:
ε = 4 × 8.854×10⁻¹² = 35.4×10⁻¹² F/m
ω = 2π×10⁸ rad/s

Substitution:
tan δ = σ/(ωε) = 0.02 / (2π×10⁸ × 35.4×10⁻¹²)

Calculation:
Denominator = 2π × 10⁸ × 35.4×10⁻¹² = 2.224×10⁻² = 0.02224
tan δ = 0.02 / 0.02224 = 0.899

εc = 35.4×10⁻¹² − j(0.02 / 2π×10⁸)
    = 35.4×10⁻¹² − j 31.83×10⁻¹² F/m
    = ε₀(4 − j3.59)

Final Answer with units:
tan δ ≈ 0.9 → this medium is neither a good dielectric nor a good conductor
(transition or quasi-conductor region at 100 MHz)
εc = ε₀(4 − j3.59)

Exam Tip: In GATE, a good dielectric has tan δ << 1 and a good conductor has tan δ >> 1. At a given σ, the classification changes with frequency: the same material may be a good conductor at low frequency and a good dielectric at very high frequency. Also note that jω in phasor domain replaces ∂/∂t, NOT 1/jω. Do not invert the substitution.

• Phasor transformation: ∂/∂t → jω, which converts time-domain PDEs to spatial-only equations.
• Complex permittivity: εc = ε − jσ/ω; the imaginary part accounts for ohmic loss in the medium.
• Loss tangent: tan δ = σ/(ωε); determines dielectric (<<1) vs conductor (>>1) classification.
• Propagation constant: γ = α + jβ = jω√(μεc); α gives attenuation and β gives phase variation.
• Helmholtz equation: ∇²Es + k²Es = 0, with k = β = ω√(με) for lossless media, governs waveguide and cavity field distributions.

Quick Revision

• Phasor rule: replace ∂/∂t with jω everywhere in Maxwell's equations.
• Phasor Maxwell equations: ∇·Ds = ρvs; ∇·Bs = 0; ∇×Es = −jωμHs; ∇×Hs = (σ+jωε)Es.
• Complex permittivity: εc = ε − jσ/ω = ε(1 − j tan δ).
• Loss tangent: tan δ = σ/(ωε); for copper at 1 GHz, tan δ >> 1 (good conductor).
• Helmholtz equation: ∇²Es + k²Es = 0 where k = ω√(με) for lossless medium.
• Intrinsic impedance: η = √(μ/εc); for free space η₀ = 120π ≈ 377 Ω.
• Exam trap: Classification as conductor or dielectric is frequency-dependent. Seawater is a conductor at low frequency but a leaky dielectric at optical frequencies.