Speed of light c = 1 - √(ε₀ μ₀): Foundations

Before you can derive $c = 1/ ε_{0} μ_{0}$ you have to be fluent in a small pile of symbols: fields, arrows-on-letters, upside-down triangles, dots, crosses, and two Greek constants. The parent note assumes you already read those fluently. Here we earn each one, in an order where every symbol only uses symbols defined before it.

1. A field — what does $E$ even mean?

Plain words: imagine standing anywhere near a charged ball and asking "if I put a speck of positive charge right here, which way does it get shoved, and how strongly?" The answer is one arrow. Do that for every point and you have painted space with arrows — that carpet of arrows is the field.

The picture: look at Figure 1. Around a positive charge the arrows point outward like a hedgehog; around a negative charge they point inward. The arrow's length = strength, the arrow's direction = push direction.

Figure — Speed of light c = 1 - √(ε₀ μ₀)

Why the topic needs it: an electromagnetic wave is a pattern of these arrows that marches through space. You cannot talk about a wave of "field" until you know a field is just "an arrow at every point."

$E$ = electric field (arrow at every point).
$B$ = magnetic field (arrow at every point, but its arrow tells which way a compass needle would turn, not where a charge is pushed).

2. Rate of change — the symbol $\frac{\partial}{\partial t}$

Plain words: stand at one fixed point in space and just watch the field arrow there. Is it growing? Shrinking? Flipping? The number that says "the arrow here is getting longer at 5 units per second" is $\partial E / \partial t$ .

Why "partial" and not the ordinary $d / d t$ ? Because the field depends on both place and time. The curly $\partial$ is a promise: "I am asking about change in this one variable (time), keeping the others (position) still." That is exactly the question a wave forces on us — "at this fixed spot, how is the field wobbling?"

Why the topic needs it: the whole engine of light is "a changing electric field", and "changing in time at a fixed point" is precisely $\partial / \partial t$ . No $\partial / \partial t$ , no change, no wave.

3. Change across space — $\nabla$ (nabla / del)

Think of $\nabla$ as three partial-derivative speedometers pointing along the three directions of a room. Attaching $\nabla$ to a field lets us ask geometric questions about how the arrow-carpet varies from point to point. There are four such questions — gradient, divergence, curl, Laplacian — and each has its own symbol.

3a. Gradient $\nabla f$ — "which way is uphill?"

Plain words: if $f$ is the height of a hill at each spot, $\nabla f$ is the little arrow a marble would roll away from — straight uphill, and longer where the slope is steeper. Divergence, curl and Laplacian are all built by feeding $\nabla$ into the field in different ways, so gradient is the raw ingredient behind all of them.

Why the topic needs it: we need it specifically to define the Laplacian in §3d, because $\nabla^{2} f = \nabla \cdot (\nabla f)$ — "take the uphill arrows, then ask whether they net-flow out." Everything downstream leans on this one operation.

3b. Divergence $\nabla \cdot E$ — "is this point a source?"

The picture: Figure 2, left panel. Draw a tiny box. If more arrow flows out than in, $\nabla \cdot E > 0$ and there must be positive charge inside. In vacuum there are no charges, so nothing flows net-out anywhere: $\nabla \cdot E = 0$ . That single fact is used in the parent's Step 2.

Why a dot? The dot is the same "dot product" idea — you line up $\nabla$ 's direction-slopes with the field's own directions and add them, producing one plain number (a scalar). Dot ⇒ scalar answer.

3c. Curl $\nabla \times E$ — "does this point swirl?"

The picture: Figure 2, right panel. If the arrows curl around a point like water spiralling down a drain, a paddle-wheel spins and curl is nonzero, pointing along the spin axis (right-hand rule).

Why a cross? The cross is the "cross product" idea — it takes two directions and gives a new perpendicular direction (an arrow), unlike the dot which gave a plain number. Cross ⇒ vector answer, and we need a vector because a swirl has an axis.

Why the topic needs both: the two "engine" equations of light are curl equations. They hold in vacuum — no charges ( $ρ = 0$ ) and no currents ( $J = 0$ ), which is what makes them so clean: $\nabla \times E = - \frac{\partial B}{\partial t}, \nabla \times B = μ_{0} ε_{0} \frac{\partial E}{\partial t} .$ Read in plain words: a swirl in the electric field is caused by a magnetic field changing in time, and a swirl in the magnetic field is caused by an electric field changing in time. That mutual swirling is the "self-sustaining handshake" of the parent note. (Full statements in Faraday's Law of Induction and the Ampère–Maxwell Law.)

3d. Laplacian $\nabla^{2}$ — "how curved is the field in space?"

Why we care: a wave equation is nothing but "spatial curvature = (slowness)² × time curvature." When the parent writes $\nabla^{2} E = μ_{0} ε_{0} \partial^{2} E / \partial t^{2}$ , it is literally saying "how each component of the field bends in space equals a constant times how it bends in time" — and that constant is the slowness squared.

4. The two constants of empty space

The picture: Figure 3 draws them as two "stiffnesses" of space, like two spring constants. The stiffer space is to each kind of field, the harder — and here, counter-intuitively, the slower — the handshake propagates.

Why they multiply and why the reciprocal. These are not slopes or derivatives — they are fixed material numbers of vacuum. When the derivation couples the two curl equations, both constants land side by side as the product $ε_{0} μ_{0}$ .

5. Reading the whole formula out loud

Now every symbol is earned, so the parent's headline sentence becomes readable:

Prerequisite map

This map shows the flow: vectors and fields feed nabla; nabla's four flavours (gradient → Laplacian, plus divergence and curl) plus the time-derivative build the Maxwell equations; those combine (using displacement current) into the wave equation; and the two space-constants set the speed. From there the parent topic continues into Electromagnetic Waves and Refractive Index and n = c/v.

Equipment checklist

Recall Self-test: can you say each in one sentence?

What does the arrow over $E$ tell you that plain $E$ does not? ::: That $E$ has a direction, not just a size. In plain words, what is a field? ::: A rule attaching an arrow (or number) to every point in space. What does $\partial E / \partial t$ measure? ::: How fast the field is changing in time at one fixed point. Why is $\partial$ curly instead of a straight $d$ ? ::: Because we change only time while holding position frozen. What does the gradient $\nabla f$ give you? ::: An arrow at every point pointing the way $f$ increases fastest, as long as it is steep. What does $\nabla \cdot E$ (divergence) tell you, and its value in vacuum? ::: Whether a point is a source/sink of arrows; it is $0$ in vacuum (no charge). What does $\nabla \times E$ (curl) tell you, and what kind of object is it? ::: How the field swirls; it is itself a vector (an arrow along the spin axis). Why does curl use a cross while divergence uses a dot? ::: Cross gives a new perpendicular vector (an axis); dot gives a single scalar. What does $\nabla^{2}$ (Laplacian) measure, and how is it built from gradient? ::: The spatial curvature; $\nabla^{2} f = \nabla \cdot (\nabla f)$ . When $\nabla^{2}$ acts on the vector $E$ , what does it do? ::: Acts component by component: $\nabla^{2} E = (\nabla^{2} E_{x}, \nabla^{2} E_{y}, \nabla^{2} E_{z})$ . Under what conditions do the curl equations we use hold? ::: In vacuum, with $ρ = 0$ and $J = 0$ . Why is there a minus sign in Faraday's law? ::: Lenz's law — the induced electric swirl opposes the change that made it, keeping the wave stable. What is $ε_{0}$ physically? ::: The permittivity — how readily vacuum permits an electric field. What is $μ_{0}$ physically, and what are equivalent units? ::: The permeability; units $T m A^{- 1} = H m^{- 1} = N A^{- 2}$ . Why must you take a reciprocal to get $c$ from $ε_{0} μ_{0}$ ? ::: Because $ε_{0} μ_{0}$ has units of slowness-squared ( $s^{2} / m^{2}$ ), so $1/ ε_{0} μ_{0}$ is a speed.