EMG-05

Poynting vector of a plane wave

S = E B / μ0 = E² / Z0 for a vacuum plane wave.

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Electromagnetic wavesPoynting 1884

Governing equation

S=\dfrac{E B}{\mu_0}=\dfrac{E^2}{Z_0},\quad Z_0=\sqrt{\mu_0/\varepsilon_0}

$E \in \left[0.1, 1\times 10^{5}\right]$

where

E: Electric field (peak) (V/m)
S: Peak Poynting (W/m²)
B: Peak B (µT)

The figure, part by part

Each numbered marker wears the color of its card. Hover a card to light the marker. Tap a marker to jump to that card. The slider walks a red point along the curve; Animate drawing traces it like a pen.

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①

Governing law — red

S=\dfrac{E B}{\mu_0}=\dfrac{E^2}{Z_0},\quad Z_0=\sqrt{\mu_0/\varepsilon_0}

$E \in \left[0.1, 1\times 10^{5}\right]$

$E = 0.1$ $S = 2.65\times 10^{-5}$

Along the curve

②

Domain of the variable — blue

E_{\min} = 0.1,\quad E_{\max} = 1\times 10^{5}

$E \in \left[0.1, 1\times 10^{5}\right]$

$E = 0.1$ $E = 0.1$

Along the curve

③

Response curve — green

S=\dfrac{E B}{\mu_0}=\dfrac{E^2}{Z_0},\quad Z_0=\sqrt{\mu_0/\varepsilon_0}

$E \in \left[0.1, 1\times 10^{5}\right]$

$E = 0.1$ $S = 2.65\times 10^{-5}$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}S}{\mathrm{d}E}

$E \in \left[0.1, 1\times 10^{5}\right]$

$E = 629$ $S^{\prime} = 3.339$

Along the curve

Lecture brief

Historical brief

Coulomb, Gauss, Ampère, Faraday and Maxwell (1861–65) unified charge, current and light. The lab computes fields, induction, Poynting flux and the electromagnetic wave in SI. This sheet (EMG-05 — Poynting vector of a plane wave) is the form associated with Poynting 1884. Working symbols: $E$ $\rightarrow$ $S$ , $B$ . Poynting identified S = E×H as the energy-flux density. In a TEM plane wave |H| = |E|/Z0 so |S| = E²/Z0.

Purpose

Purpose: compute

S

B

from

E

in Electromagnetism via

S=\dfrac{E B}{\mu_0}=\dfrac{E^2}{Z_0},\quad Z_0=\sqrt{\mu_0/\varepsilon_0}

S = E B / μ0 = E² / Z0 for a vacuum plane wave. Use it when a real electromagnetism question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

E = 100.000\,\mathrm{V/m}

, the governing relation

S=\dfrac{E B}{\mu_0}=\dfrac{E^2}{Z_0},\quad Z_0=\sqrt{\mu_0/\varepsilon_0}

yields

S = 26.544\,\mathrm{W/m^{2}}

B = 0.333564\,\mathrm{\mu T}

. RMS or peak consistently: here E is peak, S is instantaneous peak. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Electric field (peak) E100 V/m

Outputs

Peak Poynting S26.544 W/m²
Peak B B0.333564 µT

Reading speed

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Narration of this film

RMS or peak consistently: here E is peak, S is instantaneous peak.

Poynting identified S = E×H as the energy-flux density. In a TEM plane wave |H| = |E|/Z0 so |S| = E²/Z0.

Reading speed

Watch on YouTube