Foundations — Poynting vector — energy flux in EM waves

Before you can read the parent note Poynting vector, you must own every symbol it fires at you. Below, each piece is built from nothing: plain words → a picture → why the topic needs it. They are ordered so each rests on the one before.

0. The picture everything sits inside

Everything happens in ordinary 3D space. A vector is an arrow: it has a length (how big) and a direction (which way). We write vectors with a little hat-arrow, like $\vec{E}$. A plain letter like $E$ (no arrow) means just the length of that arrow — a single number with no direction.

1. The electric field $\vec{E}$

Why the topic needs it: half the wave's energy lives in $\vec{E}$. The energy density term $\frac{1}{2}{\epsilon }_0{E}^2$ and the whole Poynting vector both start here.

2. The magnetic field $\vec{B}$

Why the topic needs it: the other half of the wave's energy lives in $\vec{B}$, and $\vec{S}$ is literally $\vec{E}$ combined with $\vec{B}$. In a light wave $\vec{E}$ and $\vec{B}$ are always at right angles and locked together (you'll see $E=cB$).

3. The cross product $\vec{E}\times \vec{B}$

This is the single most important operation in the topic, so we build it carefully.

Look at the figure. Why the $\sin \theta$? It measures how "spread apart" the two arrows are:

• If they point the same way ($\theta =0$), $\sin 0=0$ — the result is zero. Two parallel fields carry no propagating wave, so no energy flow.
• If they are perpendicular ($\theta =90^{\circ }$), $\sin 90^{\circ }=1$ — the result is as big as possible.

In a light wave $\vec{E}\perp \vec{B}$, so $\theta =90^{\circ }$ and the length simplifies beautifully: $|\vec{E}\times \vec{B}|=EB\sin 90^{\circ }=EB.$ That is exactly why the parent note can write $S=EB/{\mu }_0$.

4. The dot product and the surface element $d\vec{A}$

Why the topic needs both: to get the total power crossing a surface, the parent writes $P=\oint \vec{S}\cdot d\vec{A}$. The dot product picks out only the part of the energy flow that actually pierces each patch (energy skimming sideways along the surface crosses nothing). The $\oint$ symbol just means "add up over the whole closed surface."

5. Constants ${\epsilon }_0$, ${\mu }_0$, and the speed $c$

Why the topic needs them: $\vec{S}=\frac{1}{{\mu }_0}\vec{E}\times \vec{B}$ carries a ${\mu }_0$. The identity $\frac{1}{{\mu }_{0}c}={\epsilon }_{0}c$ (which you get by multiplying top and bottom of $\frac{1}{{\mu }_{0}c}$ by ${\epsilon }_{0}c$ and using $c^2=1/({\mu }_0{\epsilon }_0)$) is what turns $S=E^2/({\mu }_{0}c)$ into $S={\epsilon }_{0}c{E}^2$. See EM wave equation for where $c$ comes from.

6. Energy density $u$

Why the topic needs it: the derivation starts by tracking how $u$ changes in time and asks "where did the energy go?" The answer is $\vec{S}$. Prerequisite: Energy density of electric and magnetic fields.

7. Rate of change $\frac{\partial u}{\partial t}$ and divergence $\nabla \cdot$

Why the topic needs them: Poynting's theorem is a bookkeeping line, $\frac{\partial u}{\partial t}+\nabla \cdot \vec{S}=-\vec{J}\cdot \vec{E},$ read as "(energy piling up) + (energy flowing out) = (minus the work done on charges)." $\nabla \cdot$ is the exact tool for "flowing out."

8. Curl $\nabla \times$, current density $\vec{J}$, and Maxwell's laws

Why the topic needs them: the derivation swaps time-derivatives of the fields for curls using Maxwell's equations — the only laws that say how $\vec{E}$ and $\vec{B}$ evolve. Then one line of Vector calculus identities, $\nabla \cdot (\vec{E}\times \vec{B})=\vec{B}\cdot (\nabla \times \vec{E})-\vec{E}\cdot (\nabla \times \vec{B}),$ packages two curl terms into the single divergence of $\vec{E}\times \vec{B}$ — and out pops $\vec{S}$.

9. Time-average $⟨⟩$ and intensity $I$

Why the topic needs it: $E_0$ is the peak field amplitude; measurements give $I$. The factor $\frac{1}{2}$ (the famous forgotten hero) comes entirely from $⟨{\cos }^2⟩=\frac{1}{2}$. More on this in Intensity and amplitude of waves.

Prerequisite map

Equipment checklist

Test yourself — you should be able to answer each before opening the parent note.