- Explain uniformity within a plane and variation along the propagation axis.
- Connect Maxwell’s equations to coupled electric and magnetic fields in a source-free medium.
- Describe the geometry, motion, impedance, and power flow of a uniform plane wave.
- Represent reflection with forward and backward waves and extend sinusoidal analysis through Fourier decomposition.
- Basic familiarity with vectors, electric fields, and magnetic fields.
- Basic understanding of sinusoidal variation and spatial coordinates.
Fields That Are Uniform Across a Plane
A plane field is unchanged within a transverse plane but may vary with distance, time, or both.
Electromagnetic waves carry energy and information through space in radio systems, microwave links, light, antennas, and waveguides. Maxwell’s equations supply their mathematical description. Their solutions depend on the medium, the sources, and how the fields change; the uniform plane wave is the simplest important case considered here.
Take an electric field directed along x. Uniformity in the xy plane means that changing x or y leaves both its magnitude and orientation unchanged. Its magnitude can still depend on z and time.
- The unit vector x-hat gives the field orientation.
- gives its magnitude.
- The magnitude may change with z and with time t.
- Neither x nor y appears as a dependence.
At one z coordinate, every point of the corresponding xy plane shares the field’s magnitude, direction, and phase. Different z planes may nevertheless carry positive, zero, or negative values.
Here, β indicates the rate of spatial change. With time included, a sinusoidal pattern moving toward positive z can be written as follows.
- Hold time constant and scan along z: the value changes sinusoidally between planes.
- Remain at one position: the value oscillates sinusoidally as time passes.
The magnetic field may be uniform across a plane in the same way while changing with position, time, or both.
A field is unchanged when you move along x or y, but changes as you move along z. Which axis is the natural propagation direction?
The z axis, because field variation occurs perpendicular to the uniform xy planes.
The Four Maxwell Relationships
Maxwell’s equations relate fields to charges, currents, and time-varying companion fields.
Maxwell’s four equations govern electric and magnetic fields, their sources, and their interaction. They also lead to the electromagnetic wave equation.
Gauss’s electric law identifies charge as the source of electric flux. Divergence describes outward spreading or inward convergence: positive charge behaves as a source and negative charge as a sink. In integral terms, enclosed charge determines the electric flux through a closed surface.
Gauss’s magnetic law says that the model contains no isolated magnetic charge. Magnetic-field lines therefore close on themselves instead of starting or stopping at a lone source or sink.
Faraday’s law connects a magnetic field changing in time with a circulating, spatially varying electric field. Curl represents this rotational field behavior.
Ampère’s law with Maxwell’s addition gives two causes of circulating magnetic behavior: conduction-current density J and a time-changing electric displacement. Thus, both a wire current and a changing electric field can produce a magnetic field.
Ideal Conditions for the Simplest Wave
An unbounded, homogeneous, source-free region removes local charges, currents, and boundaries while retaining coupled fields.
- Unbound: no physical boundary interrupts the indefinitely extended medium.
- Source-free: the region contains neither electric charge nor electric current.
- Homogeneous: electromagnetic properties are the same throughout the region.
Free space or another material can serve as the medium when these ideal assumptions apply.
Zero divergence means no local static electric or magnetic charges originate the fields in the region. It does not eliminate the fields: the curl equations still link a spatially varying electric field to a changing magnetic field and a spatially varying magnetic field to a changing electric field.
The two components must coexist. Each field supplies through Maxwell’s curl relationships the variation required by the other, so a passing wave needs no charge or current at every point.
Without local charge or current, what keeps an electromagnetic wave propagating through a source-free region?
Its time-changing electric and magnetic fields are coupled: each produces the spatial variation of the other through Maxwell’s curl equations.
Forming the Electric–Magnetic Wave Pair
For travel along z, x-directed electric and y-directed magnetic fields are coupled and mutually perpendicular.
Begin with an x-directed electric field that is uniform over xy and varies only with z and time. Maxwell’s equations require its companion magnetic field to point along y for propagation along z.
- E is aligned with x.
- H is aligned with y.
- Propagation is aligned with z.
All three directions are perpendicular. The two fields are not separate solutions; their changes in space and time are tied together by Maxwell’s equations.
Freezing time reveals the same spatial sinusoid for E and H, although their amplitudes are different. Their x and y orientations remain fixed while their magnitudes vary between z planes.
At a selected location, both fields oscillate with time; a nonreference position also carries its spatial phase. Every point on the same xy plane undergoes identical temporal change.
- Field vectors are evaluated at stationary observation locations.
- Their magnitudes, and possibly their signs, change there with time.
- The coordinated pattern advances because corresponding values occur at different positions at different times.
Geometry and Visualization of the Uniform Wave
The full wave is a moving sequence of parallel, internally uniform field planes.
- For z-directed travel, xy is the transverse field plane.
- The electric field is transverse to propagation.
- The magnetic field is also transverse to propagation.
- The two fields are perpendicular to one another.
- Both magnitudes depend on z and time.
- All points of one xy plane share the same field values.
A plot at one point on the z axis shows only one line through the wave. The same field pair is repeated over the complete xy plane, producing a stack of parallel planes. Each plane is internally uniform, adjacent planes may differ, and the sequence shifts along z with time.
The magnitude on each plane can be encoded by a color, with distinct colors for positive, zero, and negative values. As the sinusoidal values change, the colored sequence advances along z. This displays field-pattern propagation; it does not show material planes moving.
Actual media are not perfectly infinite and homogeneous; they contain boundaries, sources, objects, losses, or material changes. Even so, when a region’s dimensions greatly exceed the wavelength, the boundaries may be treated as very distant and the model can describe local propagation well.
- communication systems
- antennas
- waveguides
- free-space propagation
- electromagnetic power transport
- reflection at surfaces and between materials
Why is the uniform plane-wave approximation useful when every real medium has finite dimensions and imperfections?
In a region much larger than the wavelength, boundaries can appear effectively remote, so the ideal wave gives a simple local account of propagation.
Two Fields Acting Together
Electric and magnetic components jointly transport power and resemble voltage and current waves on a transmission line.
A complete electromagnetic wave requires both components. On a transmission line, voltage and current change along the line, impedance fixes their relationship, and together they convey electrical power. In an unbound medium, E and H vary through space, characteristic impedance relates them, and together they convey electromagnetic power.
How the Medium Sets the Field Ratio
Characteristic impedance fixes the proportional relationship between electric- and magnetic-field magnitudes.
The field relation parallels Ohm’s law. Just as circuit impedance determines current for a specified voltage, η determines the H magnitude paired with a specified E magnitude. Its value is governed by the medium’s permeability and permittivity, so different materials give different field ratios.
For a free-space plane wave, H remains perpendicular to E and scales directly with it through the fixed ratio of about 377 ohms.
Direction and Rate of Energy Transport
The electric–magnetic cross product points along propagation and describes average electromagnetic power flow.
A wave transfers energy. With sinusoidal fields expressed as complex quantities, average power flow is obtained from the real part of the following cross product.
The conjugate H* appears because the sinusoidal fields use complex-exponential representations, whereas physical fields and average power are real quantities.
A cross product is normal to both input vectors. Consequently, perpendicular E and H fields produce a power-flow vector along z, the travel direction.
- Circuit voltage and current establish circuit power.
- Electric and magnetic fields establish electromagnetic power flow.
- Impedance connects voltage with current.
- Characteristic impedance connects E with H.
Thus free space or another insulating medium can carry power without conduction current in a wire: the coupled fields carry it. Treating free space as approximately 377 ohms for a plane wave is especially useful when studying how antennas exchange energy between circuits and fields.
Why does E crossed with H point along propagation instead of along either field?
A cross product points perpendicular to both factors. Because E and H span the transverse plane, their cross product points normal to that plane, along propagation.
Waves Traveling in Opposite Directions
An interface can add a backward reflected component to the original forward wave.
Uninterrupted propagation is the simplest case. A wave may instead meet a reflecting surface or enter a material whose permeability or permittivity differs. Depending on the surface or medium, one portion may be dissipated or transmitted while another returns toward the original region.
- The forward-traveling component continues in the initial direction.
- The backward-traveling component represents reflection in the opposite direction.
The magnetic field has an analogous sum. Superposing components that travel oppositely can create interference, so the total field need not resemble either component alone. This incident-and-reflected description matches the corresponding transmission-line picture and follows from the Maxwell-derived wave equation.
Using Sinusoids to Study Any Waveform
Fourier analysis converts an arbitrary signal into sinusoidal components that can be analyzed and recombined.
Sinusoidal waves are a convenient starting point because their temporal and spatial changes have a simple, predictable connection. Electromagnetic signals, however, may instead be pulses, pulse sequences, or other time-dependent shapes.
- Rewrite the original waveform as a sum of sinusoids.
- Determine how every sinusoidal component travels through the medium.
- Add the propagated components to recover the original signal’s behavior.
Because the Maxwell and wave equations permit systematic treatment of each sinusoid, the separate responses can be added to obtain the response of a complicated waveform. Sinusoidal uniform plane waves therefore serve as the elementary pieces for broader signal analysis.
Common misconceptions
Watch out: Setting ρ and J to zero forces every electric and magnetic field in the region to vanish.
Instead: Those conditions remove local charge and current sources. A traveling wave may still exist because its changing electric and magnetic components generate the variations required of each other.
Watch out: Propagation means that the observation points or field vectors physically move down the z axis.
Instead: The locations stay fixed. Their field values change with time, and the organized spatial pattern is what advances.
Watch out: The 377-ohm comparison means free space is literally a conventional resistor or conducting wire.
Instead: It is a plane-wave analogy: characteristic impedance relates E to H in the same mathematical role that circuit impedance relates voltage to current.
Lesson recap
- A plane wave has constant field behavior across every plane transverse to travel.
- With propagation along z, uniformity lies in each xy plane.
- E and H are perpendicular to one another.
- Each field is also normal to the propagation direction.
- Faraday’s law links a changing magnetic field to electric-field variation.
- Maxwell’s addition to Ampère’s law links a changing electric field to magnetic-field variation.
- Dependence on position and time makes the pattern travel while measurement locations remain stationary.
- The medium’s characteristic impedance is E divided by H.
- For free space, this ratio is approximately 377 ohms.
- The E–H cross product identifies electromagnetic power-flow direction.
- Interfaces and material changes can produce reflected components and possible dissipation.
- Fourier decomposition extends sinusoidal wave analysis to more general signals.
Although idealized, the uniform plane wave is the foundation for explaining electromagnetic propagation, power transfer without a wire, antenna radiation, boundary interactions, and the movement of energy and information through space.