Visual walkthrough — Poynting vector — energy flux in EM waves

We are chasing one plain-English sentence and turning it into an equation:

"The energy stored in a little box of space can only change if energy flows across its walls, or if the fields do work on charges inside."

That's it. That sentence, drawn carefully, is Poynting's theorem.

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Step 0 — The three things we must already picture

Before symbols, three pictures. If any feels shaky, that's what the arrows below fix.

We build everything from Maxwell's equations (the laws that say how $\vec{E}$ and $\vec{B}$ change) and from the energy density $u$.

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Step 1 — Draw the little box and the bookkeeping

WHAT. We pick a tiny cubic box of space, fixed in place, and count energy going in and out — exactly like counting money in a till.

WHY. Energy is conserved. If the joules inside the box drop, they didn't vanish — they either leaked out through a wall or were spent doing work on charges inside. Writing that balance is the whole game; the Poynting vector is going to be the name we give to "leak through a wall."

PICTURE. Look at the box below. Green arrows = energy crossing walls (flux). The lightning-bolt symbol inside = fields pushing charges (work). The stack of coins = stored energy $u$.

In words, our target equation is:

$$\underset{\begin{array}{c}\text{how fast the}\\ \text{stored energy}\\ {\text{changes, per m}}^3\end{array}}{\underbrace{\frac{\partial u}{\partial t}}}=-\underset{\text{we will call this }\nabla \cdot \vec{S}}{\underbrace{({\text{energy leaking out per m}}^3)}}-\underset{\vec{J}\cdot \vec{E}}{\underbrace{({\text{work done on charges per m}}^3)}}$$

Every term here is per unit volume per unit time — that's what $\partial u/\partial t$ forces on us, so the other two terms must match. We don't yet know what $\vec{S}$ is. We are going to discover it.

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Step 2 — Write the stored energy, then watch it change

WHAT. Write $u$ from the two field energies, then take its time derivative $\partial u/\partial t$.

WHY. We want the left-hand side of the bookkeeping equation — the rate the coins change. The symbol $\partial /\partial t$ means "how fast this changes as time ticks, holding position fixed." We use it (not an ordinary $d/dt$) because $u$ depends on both place and time, and here we sit still and only let time flow.

Now differentiate. The rule $\frac{d}{dt}(E^2)=2E\frac{dE}{dt}$ (a vector version) gives:

$$\frac{\partial u}{\partial t}={\epsilon }_0\underset{\begin{array}{c}\text{how fast the}\\ \text{E-arrow grows,}\\ \text{measured along E}\end{array}}{\underbrace{\vec{E}\cdot \frac{\partial \vec{E}}{\partial t}}}+\frac{1}{{\mu }_0}\underset{\begin{array}{c}\text{same for}\\ \text{the B-arrow}\end{array}}{\underbrace{\vec{B}\cdot \frac{\partial \vec{B}}{\partial t}}}$$

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Step 3 — Ask Maxwell how the fields actually change

WHAT. Replace $\partial \vec{E}/\partial t$ and $\partial \vec{B}/\partial t$ using two of Maxwell's equations.

WHY. So far $\partial u/\partial t$ is written in terms of how the fields change in time — but we have no independent handle on that. Maxwell's laws are the only laws that tell us how fields evolve, and they express time-changes in terms of the fields' spatial twisting (the curl). Swapping them in is what secretly introduces geometry — and geometry is where a flow direction can hide.

We need the symbol $\nabla \times$ (read "curl"). Picture it as a tiny paddlewheel: $\nabla \times \vec{B}$ measures how much $\vec{B}$ circulates around a point — how hard it would spin the wheel.

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Step 4 — Substitute and stare at what appears

WHAT. Put the two Maxwell replacements into the $\partial u/\partial t$ expression from Step 2.

WHY. This is pure substitution — but it converts "rate of stored energy" into an expression built from $\vec{E}$, $\vec{B}$, their curls, and $\vec{J}$. The $\vec{J}$ term will become the work-on-charges term; the curl terms are the seed of the flow term.

$$\frac{\partial u}{\partial t}=\vec{E}\cdot \left(\underset{={\epsilon }_0\partial \vec{E}/\partial t}{\underbrace{\frac{1}{{\mu }_0}\nabla \times \vec{B}-\vec{J}}}\right)+\frac{1}{{\mu }_0}\vec{B}\cdot \left(\underset{=\partial \vec{B}/\partial t}{\underbrace{-\nabla \times \vec{E}}}\right)$$

Expand and group:

$$\frac{\partial u}{\partial t}=\underset{\text{two curl terms — the mystery}}{\underbrace{\frac{1}{{\mu }_0}[\vec{E}\cdot (\nabla \times \vec{B})-\vec{B}\cdot (\nabla \times \vec{E})]}}-\underset{\begin{array}{c}\text{power delivered to}\\ {\text{charges, per m}}^3\end{array}}{\underbrace{\vec{J}\cdot \vec{E}}}$$

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Step 5 — The identity that packages two curls into one flow

WHAT. Use one line of vector calculus: $\nabla \cdot (\vec{E}\times \vec{B})=\vec{B}\cdot (\nabla \times \vec{E})-\vec{E}\cdot (\nabla \times \vec{B})$

WHY. Notice the right side is the negative of our bracket in Step 4. This identity is the one tool that folds two curl terms into the divergence of a single vector. And "divergence" — the symbol $\nabla \cdot$ — literally means net outflow per unit volume: exactly the leak we wanted.

Applying the identity to our bracket:

$$\frac{1}{{\mu }_0}[\vec{E}\cdot (\nabla \times \vec{B})-\vec{B}\cdot (\nabla \times \vec{E})]=-\frac{1}{{\mu }_0}\nabla \cdot (\vec{E}\times \vec{B})$$

Define the vector whose divergence just appeared:

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Step 6 — Read off Poynting's theorem

WHAT. Substitute the packaged term back. The bracket became $-\nabla \cdot \vec{S}$.

WHY. Now every piece matches Step 1's bookkeeping sentence, symbol for symbol.

Rearranged into plain English: the stored energy drops ($-\partial u/\partial t$) by exactly the amount that flows away ($\nabla \cdot \vec{S}$) plus the amount handed to charges ($\vec{J}\cdot \vec{E}$). Nothing is created or destroyed.

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Step 7 — Edge case A: empty space, no charges

WHAT. Set $\vec{J}=0$ (vacuum — no charges to push).

WHY. A light wave travels through empty space. We must show the theorem still makes sense there, or the whole picture is suspect.

$$\frac{\partial u}{\partial t}+\nabla \cdot \vec{S}=0$$

This is a pure continuity equation: energy is neither made nor spent, only moved. Whatever leaves one box enters the next — energy surfs forward. This degenerate case is the one that governs sunlight reaching Earth (see Intensity and amplitude of waves).

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Step 8 — Edge case B: a plane wave, and the sign/geometry check

WHAT. Specialize to a plane wave where $\vec{E}\perp \vec{B}$ and both point across the direction of travel, with $E=cB$.

WHY. We must check all cases of the geometry: what is $|\vec{S}|$ when the fields are perpendicular (the wave case), and what happens when they are parallel or zero (degenerate cases)? A formula you can't stress-test isn't trustworthy.

Because $|\vec{E}\times \vec{B}|=EB\sin \theta$ and here $\theta =90^{\circ }$ so $\sin \theta =1$:

$$S=\frac{EB}{{\mu }_0}=\frac{E(E/c)}{{\mu }_0}=\frac{E^2}{{\mu }_{0}c}={\epsilon }_{0}c{E}^2$$

using $c^2=1/({\mu }_0{\epsilon }_0)$, hence $\frac{1}{{\mu }_{0}c}={\epsilon }_{0}c$.

The time-average of $S$ (over one oscillation) is the intensity, using $⟨{\cos }^2⟩=\frac{1}{2}$:

$$I=⟨S⟩=\frac{1}{2}{\epsilon }_{0}c{E}_0^2$$

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The one-picture summary

Everything above, compressed: start with a box of stored energy, ask Maxwell how the fields change, use one vector identity, and out falls a conserved flow $\vec{S}$ that — for a wave — points where the light goes.

Recall Feynman retelling — the whole walkthrough in plain words

Picture a tiny glass box floating in space, with electric and magnetic field-arrows threading through it. Inside the box is stored energy, like coins in a till. If the number of coins drops, the money went somewhere — it either leaked out through a wall, or it got spent buying work on some charges inside. That single honest sentence is the whole theorem.

To turn it into math we asked: how fast does the stored energy change? That's the derivative of $u$. But we can't answer that alone, so we asked Maxwell's laws how the fields themselves change — and Maxwell answers in terms of how the fields curl through space. We plugged that in. Out came three pieces: two twisty curl terms, and one clean term that is obviously "work done on charges." Then one magic line of vector calculus folded the two twisty terms into the outflow of a single new arrow. We named that arrow $\vec{S}$ — the Poynting vector. It was never assumed; it fell out because energy has to be conserved.

Finally we checked the corners: empty space (energy just surfs forward), parallel fields (no flow at all, and rightly so), and a real light wave (fields at right angles, flow maximal and pointing exactly where the wave travels). Point your fingers along $\vec{E}$, curl them to $\vec{B}$, and your thumb points where the light — and its energy — is going.