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

This is a rapid-fire trap-spotting deck for the speed-of-light note. Each line is a claim, an error, a "why", or an edge case. Read the left side, commit to an answer out loud with a reason, then reveal. Bare "true/false" earns nothing — the reasoning is the whole point.

Before you start, two words you must already own from the parent note:

permittivity $ε_{0}$ — how readily empty space permits (supports) an electric field.
permeability $μ_{0}$ — how readily empty space permits a magnetic field.

Everything below tests whether you can tell a real fact from a plausible-sounding lie. See also Maxwell's Equations, Displacement Current, Electromagnetic Waves, Refractive Index and n = c/v, and Wave Equation.

True or false — justify

A red herring here is that many false statements sound like textbook lines. Say the reason before revealing.

Blue light travels faster than red light in vacuum.

False —

c = 1/ ε_{0} μ_{0}

has no frequency in it, so every colour travels at the same

c

in vacuum. Dispersion (colours separating) only happens in a medium.

A brighter (higher-intensity) laser beam moves faster than a dim one.

False — amplitude/intensity never appears in

c = 1/ ε_{0} μ_{0}

; the speed is set by vacuum properties alone, not by how strong the fields are.

In vacuum the magnetic field of light is negligibly weak and can be ignored.

False —

E = c B

means

B

is only numerically small (because

c

is huge), but the wave literally cannot propagate without

B

; drop it and Faraday's law has nothing to feed the relay.

Light slows down inside glass because the glass "absorbs and re-emits" it more slowly.

Broadly true in mechanism, but the clean statement is

v = c / ε_{r} μ_{r}

— the medium's larger permittivity (

ε = ε_{r} ε_{0}

) changes the "give" of the material, lowering the wave speed.

The formula

c = 1/ ε_{0} μ_{0}

was derived by measuring the speed of light first.

False — the beauty is the reverse:

ε_{0}

came from electrostatics and

μ_{0}

from magnetism, both measured with no optics, and their combination predicted the known speed of light.

An EM wave can consist of a pure electric field with no magnetic field.

False — a changing

E

generates

B

(Ampère–Maxwell Law) and vice versa (Faraday's Law of Induction); neither can travel alone, they re-create each other.

In a medium with

μ_{r} = 1

and

ε_{r} = 4

, the refractive index is

n = 2

True —

n = ε_{r} μ_{r} = 4 \cdot 1 = 2

, so

v = c /2

; the derivation is identical to the vacuum one with material constants substituted.

ε_{0} μ_{0}

is a speed.

False — it has units s/m, which is a slowness (seconds per metre). You must take the reciprocal:

c = 1/ ε_{0} μ_{0}

has units m/s.

Spot the error

Each line contains one planted mistake. Name it and correct it.

"In vacuum

J = 0

, so

\nabla \times B = μ_{0} J = 0

, hence no EM wave can exist."

The error is dropping the Displacement Current. The full law is

\nabla \times B = μ_{0} J + μ_{0} ε_{0} \partial E / \partial t

; even with

J = 0

the second term survives and is what launches the wave.

"Taking the curl of Faraday's law gives

\nabla^{2} E

directly."

You first get the double curl

\nabla \times (\nabla \times E)

; only after using the identity

\nabla \times (\nabla \times E) = \nabla (\nabla \cdot E) - \nabla^{2} E

and

\nabla \cdot E = 0

does

- \nabla^{2} E

appear.

"Comparing to the wave equation,

\frac{1}{v ^{2}} = \frac{1}{μ _{0} ε _{0}}

, so

v = μ_{0} ε_{0}

Sign of the exponent is flipped. The wave equation gives

\frac{1}{v ^{2}} = μ_{0} ε_{0}

, hence

v = 1/ μ_{0} ε_{0}

"Light slows in a medium because the frequency

f

decreases."

Frequency is fixed by the source and does not change on entering a medium; it is the wavelength

λ

(and hence

v = f λ

) that shrinks.

"Since

E = c B

and

c

is large, the electric field carries almost all the energy and the magnetic field carries none."

In an EM wave the electric and magnetic energy densities are equal; the factor

c

is just a unit-conversion between the numerical values of

E

and

B

, not an energy split.

ε_{0} μ_{0}

has units of m²/s², like a speed squared."

It has units s²/m² (the inverse), which is why

1/ ε_{0} μ_{0}

— not

ε_{0} μ_{0}

— comes out as m/s.

Why questions

Force yourself to give the mechanism, not just the label.

Why does the term

\nabla (\nabla \cdot E)

vanish in the derivation?

Because in vacuum there are no charges, so Gauss's law gives

\nabla \cdot E = 0

; the whole gradient term is then zero, leaving the clean Laplacian.

Why must the displacement current be nonzero for light to exist?

It is the only source of

\nabla \times B

in charge-free vacuum; without it a changing

E

could not create a

B

, breaking the self-sustaining

E \leftrightarrow B

relay (Electromagnetic Waves).

Why do

E

and

B

travel at exactly the same speed?

Taking the curl of Ampère–Maxwell Law instead of Faraday's law yields the identical wave equation for

B

with the same

μ_{0} ε_{0}

, so both fields propagate at one shared

c

Why is it significant that

ε_{0}

came from Coulomb's law and

μ_{0}

from forces between wires?

Because both were measured with zero reference to optics, yet their combination reproduces the measured speed of light — that "coincidence" is the proof that light is electromagnetism.

Why does the same derivation give

v = c / n

in a material?

You simply swap

ε_{0} \to ε_{r} ε_{0}

and

μ_{0} \to μ_{r} μ_{0}

; the algebra is unchanged and out drops

v = c / ε_{r} μ_{r} = c / n

(see Refractive Index and n = c/v).

Why is the speed of light called a property of space rather than of the light source?

The formula contains only

ε_{0}

and

μ_{0}

, which describe the electric and magnetic "stiffness" of the vacuum itself — nothing about who emitted the wave or how.

Edge cases

Push each quantity to a limit and check the formula still behaves.

What would

c

be in a hypothetical vacuum with double the permittivity,

2 ε_{0}

c^{'} = 1/ 2 ε_{0} μ_{0} = c / 2

; more electric "give" makes the relay slower, so light would travel about

30%

slower.

ε_{r} \to \infty

in a medium (a perfect dielectric), what happens to

v

v = c / ε_{r} μ_{r} \to 0

; infinite permittivity would freeze the wave, an idealised limit no real material reaches.

For a non-magnetic material (

μ_{r} = 1

), does the refractive index depend on

μ_{0}

at all?

The speed still depends on

μ_{0}

through

ε_{0} μ_{0}

, but the index

n = ε_{r} μ_{r} = ε_{r}

reduces to permittivity alone, which is why most optics tables list only

ε_{r}

In the zero-frequency (static) limit, is there any EM wave?

No — a truly static field has

\partial E / \partial t = 0

, so the displacement current vanishes and the relay never starts; propagation requires changing fields.

Can

n < 1

occur, meaning

v > c

The phase velocity can exceed

c

for some frequencies (e.g. near resonances), but no energy or signal travels faster than

c

, so causality is safe — the trap is confusing phase velocity with signal speed.

μ_{r}

and

ε_{r}

were both slightly less than

1

, could light go faster than in vacuum?

For that you'd need

ε_{r} μ_{r} < 1

; ordinary matter has

ε_{r} \geq 1

, so this doesn't happen for real broadband media, though engineered metamaterials can mimic it over narrow bands.

Recall One-line summary of the whole deck

Every trap here is one of four lies: (1) $c$ depends on the wave (frequency/intensity) — no, only on vacuum; (2) you can drop $B$ or the displacement current — no, they are the wave; (3) the formula is $ε_{0} μ_{0}$ instead of its reciprocal — no, check units; (4) frequency changes in a medium — no, wavelength does.