Electromagnetic waves — derivation from Maxwell's equations: Question bank

Definition Notation you must have before line one

Three pieces of shorthand appear all over this page. Read these first.

The hat $^$ means "unit vector." A little hat on a letter, like $\hat{E}$ , $\hat{B}$ , $\hat{k}$ , means "an arrow of length exactly $1$ pointing in that thing's direction." So $\hat{E}$ is just the direction the electric field points, with its size stripped off; $\hat{k}$ is the pure direction the wave travels. Picture a full-length field arrow, then shrink it to length one keeping its heading.
The Laplacian $\nabla^{2}$ means "how much a field bulges away from its neighbourhood average." For each component of a field it adds up the second space-derivatives, $\nabla^{2} f = \partial_{x}^{2} f + \partial_{y}^{2} f + \partial_{z}^{2} f$ . Intuitively: is the value here higher or lower than the average of the points immediately around it? A wave equation is exactly a rule linking this spatial bulge to how the field accelerates in time.
The imaginary unit $i$ means the number with $i^{2} = - 1$ . It does not correspond to any measurement — see the box below. It rides along in $e^{i (k x - ω t)}$ purely as bookkeeping.

Definition Every symbol on this page, in plain words

Look up any symbol here before reading a trap.

$E$ — the electric field (units V/m), the arrow of electric push at a point; $\hat{E}$ is its unit-vector direction.
$B$ — the magnetic field (units tesla, T), the arrow of magnetic influence at a point; $\hat{B}$ is its direction.
$ρ$ (rho) — the charge density: how much electric charge sits in a tiny volume. In vacuum $ρ = 0$ .
$J$ — the current density: how much charge flows through a tiny area each second. In vacuum $J = 0$ .
$ε_{0} \approx 8.854 \times 1 0^{- 12} F/m$ — the permittivity of free space, measured from electric (electrostatic) force experiments; it sets how strong an electric field a charge makes.
$μ_{0} = 4 π \times 1 0^{- 7} T\cdotm/A$ — the permeability of free space, measured from magnetic force experiments; it sets how strong a magnetic field a current makes.
$c = 1/ μ_{0} ε_{0} \approx 3.00 \times 1 0^{8}$ m/s — the speed of light in vacuum.
$k$ — the wave number, $k = 2 π / λ$ : how many radians of wiggle fit per metre of space. $k$ is the arrow pointing along the wave's travel; $\hat{k}$ is that direction as a unit vector.
$ω$ (omega) — the angular frequency, $ω = 2 π f$ : how many radians of wiggle happen per second.
$E_{0}, B_{0}$ — the amplitudes (peak heights) of the electric and magnetic wiggles.
$λ$ (lambda) — the wavelength; $f$ — the ordinary frequency in Hz; related by $c = f λ$ .
$\nabla^{2}$ — the Laplacian, defined in the box above.

Keep the divergence and curl questions in mind — almost every trap below is really about which of them is switched on.

True or false — justify

TF1. In vacuum with no charges,

E

field lines can still begin and end on nothing.

False.

\nabla \cdot E = 0

says there are no sources or sinks, so lines never start or stop — they either close on themselves or run off to infinity.

TF2. Removing the displacement current term still leaves a wave equation, just a slower wave.

False. Without

μ_{0} ε_{0} \partial_{t} E

, Ampère's law in vacuum gives

\nabla \times B = 0

; Step 3 then yields

0

and the second time-derivative disappears entirely — no wave of any speed.

TF3. The relation

c = 1/ μ_{0} ε_{0}

was measured by timing a light pulse.

False. It falls out of the wave equation's coefficient;

μ_{0}

(from magnetic force experiments) and

ε_{0}

(from electrostatic experiments) are measured with no light involved, yet their combination predicted light's speed.

TF4. Because

E_{0} = c B_{0}

and

c

is huge, the electric field carries far more energy than the magnetic field.

False. The energy densities are equal,

\frac{1}{2} ε_{0} E_{0}^{2} = \frac{1}{2 μ _{0}} B_{0}^{2}

; the large numerical ratio is a purely SI-unit artefact, not a physical imbalance.

TF5. An EM wave in vacuum can have

E

pointing along its direction of travel.

False.

\nabla \cdot E = 0

forces

ik E_{0 x} = 0

, so the component along

k

(the travel direction) vanishes — the wave is strictly transverse.

TF6.

E

and

B

in a plane wave are

9 0^{\circ}

out of phase, like current and voltage in a reactive circuit.

False. Faraday's law with

e^{i (k x - ω t)}

(real part

cos (k x - ω t)

) gives

k E_{0} = ω B_{0}

with no relative phase, so

E

and

B

peak and vanish together, in phase.

TF7. The curl-of-curl identity always leaves

\nabla^{2} E = μ_{0} ε_{0} \partial_{t}^{2} E

with no extra term.

False. In general the

\nabla (\nabla \cdot E)

term survives; only the charge-free condition

\nabla \cdot E = 0

kills it. Inside a charged medium it stays and complicates things.

TF8. Both

E

and

B

satisfy the same wave equation with the same speed

c

True. Taking the curl of Ampère–Maxwell and using

\nabla \cdot B = 0

mirrors the

E

derivation exactly, giving

\nabla^{2} B = μ_{0} ε_{0} \partial_{t}^{2} B

TF9. Polarization refers to which of

E

B

is larger.

False. Polarization is the direction

E

oscillates in the plane transverse to

k

; see Polarization. Amplitude size is a separate matter and

E_{0} = c B_{0}

always.

TF10. Linear polarization (E oscillating along one fixed line) is the only kind an EM wave can have.

False. If two perpendicular transverse components are

9 0^{\circ}

out of phase and equal, the tip of

E

traces a circle — circular polarization; unequal amplitudes or other phases trace an ellipse — elliptical polarization. Linear is just the special case of zero phase offset.

Spot the error

SE1. "From

\nabla \times E = - \partial_{t} B

, we substitute

B

from Gauss's law

\nabla \cdot B = 0

You can't extract

B

from a divergence equation — it only says

B

has no sources. The substitution that closes the loop uses Ampère–Maxwell (4), which supplies

\nabla \times B

SE2. "Since

E_{0} = c B_{0}

, in SI units the

E

wave and

B

wave are different physical waves."

They are one wave.

E

and

B

are locked together in phase, orthogonal, moving as a single ripple; the factor

c

is just the numerical bridge between their units.

SE3. "

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

by the identity."

The identity is

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

; you may only drop the first term after invoking

\nabla \cdot E = 0

. Writing it as

- \nabla^{2} E

without stating that assumption skips the key physics step.

SE4. "A plane wave

e^{i (k x - ω t)}

solves the wave equation for any

ω

and

k

Only if the dispersion relation

ω = c k

holds. Plugging in gives

- k^{2} = - μ_{0} ε_{0} ω^{2}

, which pins

ω / k = c

; other pairs are not solutions.

SE5. "

\hat{E} \times \hat{B}

points backward, opposite to the wave's travel."

For a right-handed triad

\hat{E} \times \hat{B} = \hat{k}

, which points along the direction of travel — the same direction as the Poynting vector

S = \frac{1}{μ _{0}} E \times B

carrying the energy.

SE6. "In vacuum

J = 0

, so Ampère–Maxwell reduces to

\nabla \times B = 0

The Displacement Current term

μ_{0} ε_{0} \partial_{t} E

remains even when the current density

J = 0

. Dropping it is exactly the mistake that erases light from the equations.

SE7. "Swapping

\partial_{t}

and

\nabla \times

in Step 3 needs an extra physical assumption."

It only needs the fields to be smooth (mixed partials commute), which is a mathematical regularity condition, not new physics.

Why questions

WHY1. Why do we take the curl of Faraday's law a second time rather than just solving the coupled pair directly?

The second curl brings in

\nabla \times B

, which Ampère–Maxwell lets us replace by

\partial_{t} E

, eliminating

B

and collapsing two coupled first-order equations into one closed second-order equation in

E

alone.

WHY2. Why does the charge-free condition

ρ = 0

matter for getting a clean wave equation?

It sets

\nabla \cdot E = 0

, which annihilates the

\nabla (\nabla \cdot E)

term from the curl-of-curl identity, leaving the pure Laplacian

\nabla^{2} E

that the standard Wave Equation template needs.

WHY3. Why is it "stunning" that

c

comes from

μ_{0}

and

ε_{0}

Neither constant was measured using light —

μ_{0}

from magnetic forces,

ε_{0}

from electric forces — yet their combination equals the speed of light, revealing that light is an electromagnetic phenomenon.

WHY4. Why must EM waves be transverse while sound can be longitudinal?

Sound is a pressure oscillation with no divergence-free constraint; EM waves obey

\nabla \cdot E = 0

, which forbids any oscillation component along

k

, so the wiggle must be sideways.

WHY5. Why do all colours of light — radio to gamma — share the same speed

c

in vacuum?

c = 1/ μ_{0} ε_{0}

contains no frequency, so every frequency across the Electromagnetic Spectrum travels at the identical vacuum speed; only

λ

changes via

c = f λ

WHY6. Why does Faraday's law force

B

to be perpendicular to

E

With

E

along

\overset{y}{^}

and travel along

\overset{x}{^}

, the curl

\nabla \times E

points along

\overset{z}{^}

; since it equals

- \partial_{t} B

, the field

B

must also lie along

\overset{z}{^}

— perpendicular to

E

WHY7. Why is the displacement current called the ingredient "that makes light possible"?

It is the only term coupling a changing

E

back into a curling

B

in vacuum; without it the feedback loop (E wiggles B, B wiggles E) breaks and the self-sustaining ripple cannot exist.

WHY8. Why does the Poynting vector

S = \frac{1}{μ _{0}} E \times B

point along the travel direction?

Because

\hat{E} \times \hat{B} = \hat{k}

for the right-handed triad, so the cross product inherits the direction

\hat{k}

— the energy streams the same way the wave moves.

Edge cases

EC1. What does the wave equation say for a static (time-independent) field,

\partial_{t} = 0

It reduces to

\nabla^{2} E = 0

, Laplace's equation — a valid electrostatic field, but no propagating wave, since a wave requires the time-derivative term to be nonzero.

EC2. If

B_{0} = 0

everywhere, can a pure

E

wave travel in vacuum?

No. A changing

E

necessarily curls up a

B

via Ampère–Maxwell, and that

B

is what regenerates

E

; killing

B

kills the feedback, so no wave survives.

EC3. What happens to

E_{0} / B_{0} = c

if the wave enters a medium where the speed is

v < c

The ratio becomes

E_{0} / B_{0} = v

, not

c

, because the amplitude relation comes from

ω / k

, which equals the actual phase speed in that medium.

EC4. In the limit

ω \to 0

(a wave that barely wiggles), what happens to the wavelength?

From

λ = 2 π c / ω

, the wavelength diverges to infinity; a truly non-oscillating field is the static limit — the "wave" spreads out until it is no longer a wave.

EC5. Does the derivation change if the wave travels along

- x

instead of

+ x

No physics changes; use

e^{i (- k x - ω t)}

or let

k < 0

. The triad flips so that

\hat{E} \times \hat{B}

still points along the new travel direction, and

E_{0} = c B_{0}

is unchanged.

EC6. Is

E_{0} = c B_{0}

a vector equation?

No — it relates the scalar amplitudes. As vectors

E

and

B

point in different (perpendicular) directions, so you cannot write

E_{0} = c B_{0}

EC7. What if we chose

E

along

\overset{z}{^}

instead of

\overset{y}{^}

for a wave going in

+ x

Perfectly allowed — that is simply a different linear polarization. The derivation is identical with axes relabelled, and

B

ends up along whichever remaining transverse axis completes the right-handed triad.

EC8. What distinguishes circular from elliptical polarization at the boundary between them?

Both come from two perpendicular transverse components with a phase offset; when amplitudes are equal and the offset is exactly

9 0^{\circ}

the tip of

E

traces a perfect circle (circular), and any other amplitude/phase combination traces an ellipse (elliptical) — circular is the symmetric special case.

Divergence-zero ⇒ transverse and sourceless; displacement current ⇒ the wave exists; curl-of-curl ⇒ the wave equation; in-phase orthogonal triad with $E_{0} = c B_{0}$ , equal energy, and energy flow $S = \frac{1}{μ _{0}} E \times B$ along $\hat{k}$ ; polarization (linear/circular/elliptical) is the path traced by the tip of $E$ ; $c$ frequency-independent from two static constants.

Question bank — Electromagnetic waves — derivation from Maxwell's equations