--- title: "Derive E and H of plane wave" subject: "Electromagnetism" chapter: 4 key: "semester_3/electromagnetism:uniform-plane-wave-derive-e-h" source: https://www.bctengineer.com/q/semester_3/electromagnetism/uniform-plane-wave-derive-e-h site: BCT Engineering --- # Derive E and H of plane wave Electromagnetism · Chapter 4: Wave equation and Wave propagation - Asked in: 2075 Asoj, 6 marks; 2074 Asoj, 5+3 marks; 2072 Chaitra, 5+3 marks; 2069 Chaitra, 7 marks; 2068 Chaitra, 8 marks (asked 5 times) - Video: [EMF-Electromagnetic Fields- Relation between E and H - By Dr Namarta Kad](https://www.youtube.com/watch?v=WBFopudplsI) — Dr. Namarta Kad - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_3/electromagnetism/uniform-plane-wave-derive-e-h - Whole chapter: https://www.bctengineer.com/md/semester_3/electromagnetism/chapter-4 ## Question Derive the expressions for the electric and magnetic fields for a uniform plane wave propagating in a perfect dielectric medium (free space). ## Short answer In a perfect dielectric with zero conductivity, the wave equation yields E = E₀ cos(ωt − βz) a_x and H = (E₀/η) cos(ωt − βz) a_y, where β = ω√(με) and η = √(μ/ε). The fields are perpendicular to each other and to the propagation direction (+z). ## Answer A uniform plane wave is one whose E and H are constant over every plane perpendicular to the direction of travel (the fields have no variation in the transverse plane), and the fields are perpendicular to each other and to the direction of propagation (transverse electromagnetic, TEM). #### Wave equation for a perfect dielectric (σ = 0, ρv = 0, J = 0) Maxwell's equations: ```text ∇×E = −μ ∂H/∂t ∇×H = ε ∂E/∂t ∇·E = 0, ∇·H = 0 ``` Take the curl of the first and use the second: ```text ∇×(∇×E) = −μ ∂(∇×H)/∂t = −με ∂²E/∂t² ∇×(∇×E) = ∇(∇·E) − ∇²E = −∇²E ⇒ ∇²E = με ∂²E/∂t² ``` and likewise ∇²H = με ∂²H/∂t². For time-harmonic fields e^{jωt}: ∇²E_s = −ω²με E_s = −β² E_s, with β = ω√(με). #### Solution for a wave travelling in +z Let E = E_x(z, t) a_x only (E_x does not vary with x or y). The equation reduces to ```text ∂²E_x/∂z² = με ∂²E_x/∂t² ``` Its solution is a forward wave plus a backward wave. For the forward wave: **E = E₀ cos(ωt − βz) a_x V/m** with ```text β = ω √(με) (rad/m), v = ω/β = 1/√(με), λ = 2π/β ``` In free space v = 1/√(μ₀ε₀) = c = 3×10⁸ m/s and β = ω/c. #### Magnetic field Use ∇×E = −μ ∂H/∂t. Only E_x(z, t) exists, so ∇×E = (∂E_x/∂z) a_y, with ∂E_x/∂z = E₀ β sin(ωt − βz). Hence H has only a y-component: ```text −μ ∂H_y/∂t = E₀ β sin(ωt − βz) H_y = (E₀ β / μω) cos(ωt − βz) ``` Since β/(ωμ) = √(με)/μ = √(ε/μ) = 1/η: **H = (E₀/η) cos(ωt − βz) a_y A/m** where **η = √(μ/ε) Ω** is the intrinsic impedance of the medium. For free space η₀ = √(μ₀/ε₀) = 120π ≈ 377 Ω. #### Properties - E and H are in time phase, mutually perpendicular, and both perpendicular to the z-direction (a_x × a_y = a_z, the direction of E × H). - Ratio E/H = η. - Phase velocity v = 1/√(με); the wave moves without attenuation in a lossless dielectric. - In phasor form: E_s = E₀ e^{−jβz} a_x, H_s = (E₀/η) e^{−jβz} a_y. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. A uniform plane wave is one in which: A. E and H are constant over each plane perpendicular to propagation and are transverse (TEM) B. E varies in the transverse plane C. E is parallel to the direction of travel D. H is zero Answer: A. Fields are uniform over transverse planes, perpendicular to each other and to the direction of propagation. (Key point: Plane wave definition) 2. What wave equation follows for a perfect dielectric? A. Laplacian E = sigma mu dE/dt B. Laplacian E = mu epsilon d^2E/dt^2 C. Laplacian E = 0 D. Laplacian E = -rho_v/epsilon Answer: B. Taking the curl of curl E = -mu dH/dt and using curl H = epsilon dE/dt with div E = 0 gives Laplacian E = mu epsilon d^2E/dt^2. (Key point: Wave equation) 3. For E = E0 cos(wt - beta z) a_x, beta and v are: A. beta = w / sqrt(mu epsilon), v = sqrt(mu epsilon) B. beta = w sqrt(mu epsilon), v = 1/sqrt(mu epsilon) C. beta = sqrt(mu epsilon)/w, v = w beta D. beta = 2 pi w, v = c always Answer: B. beta = w sqrt(mu epsilon) rad/m and v = w/beta = 1/sqrt(mu epsilon). (Key point: Phase constant and velocity) 4. What is H for this wave? A. (E0/eta) cos(wt - beta z) a_y, with eta = sqrt(mu/epsilon) B. (E0 eta) cos(wt - beta z) a_y C. (E0/eta) cos(wt - beta z) a_x D. (E0/eta) sin(wt - beta z) a_z Answer: A. From curl E = -mu dH/dt, H_y = (E0 beta/mu w) cos(...) = (E0/eta) cos(...), in time phase with E and in the a_y direction. 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