Plane Waves and Properties
What is Plane Waves and Properties?
An electromagnetic wave in which the electric and magnetic fields are uniform in planes perpendicular to the direction of propagation.
Key formula / rule: Phase Velocity
Key points
- Understand the characteristics of plane electromagnetic waves.
- Derive the wave equation from Maxwell's equations for a source-free, lossless medium.
- Calculate propagation velocity and intrinsic impedance for different media.
- Relate electric and magnetic field strengths.
Common exam trap
Assuming E, H, and k are parallel.
Definitions
- Term
Plane Wave
- Meaning
An electromagnetic wave in which the electric and magnetic fields are uniform in planes perpendicular to the direction of propagation.
- Term
Wavefront
- Meaning
A surface over which an optical disturbance has a constant phase.
- Term
Intrinsic Impedance
- Meaning
The ratio of the transverse electric field strength to the magnetic field strength in a uniform plane wave.
- Term
Poynting Vector
- Meaning
A vector whose direction is the direction of energy flow of an electromagnetic field and whose magnitude is the rate of energy flow per unit area.
Learning objectives
Understand the characteristics of plane electromagnetic waves.
Derive the wave equation from Maxwell's equations for a source-free, lossless medium.
Calculate propagation velocity and intrinsic impedance for different media.
Relate electric and magnetic field strengths.
Analyze power flow using the Poynting vector.
Formulae
- Name
Phase Velocity
- Note
Velocity of propagation of a point of constant phase.
- Expression
v = 1/√(μϵ)
- Name
Intrinsic Impedance
- Note
Ratio of the electric field magnitude to the magnetic field magnitude.
- Expression
η = √(μ/ϵ)
- Name
Wave Number
- Note
Spatial frequency of the wave.
- Expression
k = ω/v = 2π/λ
- Name
Poynting Vector
- Note
Instantaneous power flow per unit area.
- Expression
S = E × H
- Name
Average Poynting Vector
- Note
Time-averaged power flow per unit area.
- Expression
<S> = ½ Re(E × H*)
- Name
Wavelength
- Note
Distance over which the wave's shape repeats.
- Expression
λ = v/f
- Name
Free Space Constants
- Note
Values for vacuum.
- Expression
c = 1/√(μ₀ϵ₀) ≈ 3 × 10⁸ m/s, η₀ = √(μ₀/ϵ₀) ≈ 377 Ω
Prerequisites
Maxwell's Equations
Vector Calculus
Wave Equation
Basic understanding of electric and magnetic fields.
Common mistakes
Assuming E, H, and k are parallel.
Confusing phase velocity with group velocity (in dispersive media).
Incorrectly calculating intrinsic impedance or wave number.
Ignoring the medium's properties (μ, ϵ) and assuming free space values.
Keywords
Plane wave
Electromagnetic wave
Phase velocity
Intrinsic impedance
Poynting vector
Transverse wave
Wave number
Wavelength
Frequency
Frequency
Lossless medium
Free space
Practice preview
What is the relationship between the electric field (E), magnetic field (B), and the speed of light (c) in a vacuum for a plane electromagnetic wave?…
easy
For a plane electromagnetic wave propagating in the positive z-direction in a vacuum, if the electric field is given by E = E₀ cos(kz - ωt) x, what is the direction of the magnetic field?…
medium
Which of the following statements about the Poynting vector (S) for a plane electromagnetic wave is INCORRECT?…
medium