Wave Polarization & the Poynting Vector
How the E-field orients in space and how much power a wave actually carries
You now know that a plane wave has E and H locked in a fixed ratio η. But as the wave travels, which way does the E-field vector point? And how much energy does the wave actually deliver? Polarization describes the orientation; the Poynting vector quantifies the flow.
What Is Wave Polarization?
In the previous lesson we wrote the electric field of a plane wave as E=E₀cos(ωt−βz)x̂, pointing along a fixed direction x̂. As the wave travels in the +z direction, the E-field vector oscillates back and forth along that fixed axis. This is called linear polarization — the simplest and most common case.
But linear polarization is not the only possibility. If two orthogonal field components (along x̂ and ŷ) have different amplitudes or a phase difference, the tip of the E-field vector traces out a more complex path as the wave propagates. When the two components have equal amplitudes and a 90° phase difference, the tip traces a circle — circular polarization. When the amplitudes differ or the phase shift is not exactly 90°, the trace becomes an ellipse — elliptical polarization.
Polarization is exploited heavily in engineering. Satellite communications, radar, and antenna design all rely on controlling wave polarization. A circularly polarized wave can be received by an antenna regardless of its rotational orientation — which is why satellite antennas often use circular polarization. 3D movie glasses use orthogonal polarizations to separate left-eye and right-eye images. While the full mathematical treatment of circular and elliptical polarization involves superposing two phase-shifted linear components, the key physical idea is simple: the E-field vector can rotate, not just oscillate along a line.
The Poynting Vector: Power Flow per Unit Area
Polarization tells us the orientation of the fields; the Poynting vector tells us how much power the wave carries and in which direction:
Direction and Magnitude
The direction of S=E×H is determined by the right-hand rule. For the plane wave from the previous lesson — E along x̂, H along ŷ — the cross product x̂×ŷ=ẑ, exactly the direction of propagation. The Poynting vector always points in the direction of wave propagation, confirming that the wave carries energy forward.
The magnitude |S|=E·H gives the instantaneous power flowing through a unit area perpendicular to the propagation direction, in watts per square meter. Substituting H=E/η:
Time-Average Power Density
The instantaneous Poynting vector oscillates because E itself oscillates as cos(ωt−βz). In practice, what we measure is the time-average power density. For a sinusoidal signal, the average of cos² over a period is 1/2, so:
In circuit theory, the average power dissipated in a resistor R by a sinusoidal voltage with peak amplitude V₀ is P=V₀²/(2R). The Poynting formula S_avg=E₀²/(2η) has exactly the same structure — E₀ plays the role of voltage, η plays the role of resistance. We can also write S_avg=E_rms²/η, the direct analog of P=V_rms²/R. The intrinsic impedance η truly is the electromagnetic counterpart of a circuit impedance.
- Use the time-average Poynting formula: S_avg = E₀²/(2η₀).
- Substitute: S_avg = (100)²/(2 × 377) = 10000/754.
- Compute: 10000/754 = 13.26 W/m².
Putting It Together
A 100 V/m field in free space — a moderate field strength comparable to what you might find near a radio transmitter — carries about 13 W per square meter. Double the field amplitude and the power density quadruples, because power goes as the square of the field. This square-law relationship is why radar systems need enormous peak powers to detect distant targets, and why even modest reductions in path loss can dramatically improve signal quality.
Combined with the intrinsic impedance from the previous lesson, the Poynting vector completes the basic toolkit for analyzing plane waves. In the lessons that follow, we apply these tools to reflection and transmission at material boundaries, and then to transmission lines — where these same concepts reappear in circuit-theory form.
- Use S_avg = E₀²/(2η₀) = (50)²/(2×377) = 2500/754.
- Compute: 2500/754 = 3.315 W/m².
Check your understanding
- Wave polarization describes the orientation of the E-field vector in the transverse plane. Linear polarization keeps E along a fixed axis; circular and elliptical polarization involve rotation of the E-field vector.
- Circular polarization arises when two orthogonal E-field components have equal amplitudes and a 90° phase difference.
- The Poynting vector S = E × H gives the instantaneous power density (W/m²) and always points in the direction of wave propagation.
- The time-average power density for a sinusoidal plane wave is S_avg = E₀²/(2η) — the electromagnetic analog of P = V₀²/(2R) from circuit theory.