Maxwell's equations

8 min readBuilds on Light

All of classical optics, and every circuit in this series, follows from four equations that James Clerk Maxwell assembled from what was known about electricity and magnetism. They relate the electric and the magnetic field to each other and to the charges and currents that produce them [1]. This article states the four laws in words and shows how a wave, and its speed, follow from them. We will not solve Maxwell’s equations; for this series it is enough to know what they say and what follows from them.

Four laws for two fields

Each of the four laws describes one way in which a field is produced.

Gauss’s law for the electric field. Electric charges are the sources of the electric field: field lines begin on positive charges and end on negative ones. The total field flowing out through any closed surface is proportional to the charge inside it [1].
Gauss’s law for the magnetic field. There are no magnetic charges, so magnetic field lines have no beginning and no end: they close on themselves [1].
Faraday’s law of induction. A magnetic field that changes in time produces an electric field that circles around the direction of the change. This is how a generator or a transformer works [1].
Ampère’s law, with Maxwell’s addition. An electric current produces a magnetic field that circles around it. Maxwell added that an electric field changing in time does the same, even where no charge moves. He called this term the displacement current, and it is the piece that makes light possible [1].

Figure 1 sketches the four laws. The last two describe fields that circle around a changing field, in opposite senses. The difference is the minus sign in Faraday’s law; this matters in the derivation under How do we model it, where the sign is what turns the two laws into a wave equation.

a Gauss's law for E: the lines of E begin on charges. The field through a closed surface (dashed) is set by the charge inside it.

b Gauss's law for B: there are no magnetic charges, so the lines of B have no ends. Around a magnet they close on themselves, through the magnet and around it.

c Faraday's law: a growing magnetic field is circled by an electric field.

d Ampère–Maxwell: a growing electric field, or a current, is circled by a magnetic field, in the opposite sense.

Figure Maxwell’s four laws as sketches,

\mathbf{E}

in the colour of the text,

\mathbf{B}

in grey. The growing fields in c and d point upwards; the circling fields are seen slightly from above.

In empty space, far from any charges and currents, the four laws take the form [1]

\begin{aligned} \nabla \cdot \mathbf{E} &= 0, & \nabla \times \mathbf{E} &= -\frac{\partial \mathbf{B}}{\partial t}, \\ \nabla \cdot \mathbf{B} &= 0, & \nabla \times \mathbf{B} &= \mu_0 \varepsilon_0 \frac{\partial \mathbf{E}}{\partial t}. \end{aligned} \tag{1}

Here $\nabla \cdot$ is the divergence, which measures how much a field flows out of a point, and $\nabla \times$ is the curl, which measures how much it circulates around a point [1]. The constants $\varepsilon_0 \approx 8.85 \times 10^{-12}\,\mathrm{F/m}$ and $\mu_0 = 4\pi \times 10^{-7}\,\mathrm{H/m}$ are the permittivity and the permeability of vacuum; they set how strongly charges and currents produce fields [1]. Read in words, Eq. (1) says: in empty space no field lines begin or end, a changing magnetic field makes an electric field circulate, and a changing electric field makes a magnetic field circulate.

Fields that sustain each other

The two curl equations, on the right of Eq. (1), are the ones that matter for light. A changing magnetic field produces an electric field; if that electric field changes too, it produces a magnetic field, which changes and produces an electric field again. Once a disturbance of the fields has been started, for example by an accelerated charge in an antenna, it keeps itself going and moves away from its source, independently of it [1]. The two fields pass the disturbance to each other, step by step, through empty space.

This picture can be made exact. Combining Faraday’s and Ampère’s laws gives an equation for the electric field alone [1]:

\nabla^2 \mathbf{E} = \mu_0 \varepsilon_0 \, \frac{\partial^2 \mathbf{E}}{\partial t^2}, \tag{2}

and the same equation holds for the magnetic field. Equations of this form were known long before Maxwell: they are wave equations, and they describe waves that travel with the speed $v$ given by $1/v^2 = \mu_0 \varepsilon_0$ [1]. The derivation is short and is given in How do we model it below.

The speed of light

The speed that follows from Eq. (2) contains nothing but the two constants of electricity and magnetism:

v = \frac{1}{\sqrt{\mu_0 \varepsilon_0}} \approx 3.00 \times 10^8 \ \text{m/s}. \tag{3}

Maxwell evaluated this with the results of electrical experiments by Weber and Kohlrausch and found agreement with the speed of light that Fizeau had measured with a rotating toothed wheel, 315 300 km/s. He concluded that light itself is an electromagnetic wave [1]. The argument is worth appreciating: two constants measured with charges, magnets and wires predict how fast light travels. Maxwell did not see the experimental confirmation; he died eight years before Hertz produced electromagnetic waves in his laboratory [1].

In vacuum the speed does not depend on the wavelength, since Eq. (2) contains none. The same equations therefore describe radio waves, light and X-rays, which differ only in wavelength (Light).

What the equations say about the wave

Three consequences of Eq. (1) are used throughout this series.

The wave is transverse. For a wave travelling along $x$ , the equation $\nabla \cdot \mathbf{E} = 0$ allows no electric field along $x$ ; the fields oscillate only across the direction of travel [1]. The magnetic field is perpendicular to the electric field, oscillates in step with it, and its size follows from the electric field: $E = cB$ [1]. Figure 2, which the article Light introduced, shows these relations.

The motion is slowed down about 5 × 10¹⁴ times. B is drawn c times larger than its value, so that the two fields appear the same size.

A light wave travels along x. Its electric field oscillates along y, its magnetic field along z, at right angles to each other and to the direction of travel, and both reach their crests at the same places.

Figure A plane light wave as Maxwell’s equations describe it, drawn after Hecht [1]:

\mathbf{E}

(violet) along

y

\mathbf{B}

(grey) along

z

, both across the direction of travel

x

and in step.

The equations are linear: the fields appear only to the first power. The sum of two solutions is therefore again a solution. Two light waves can overlap without disturbing each other, and the field where they overlap is the sum of their fields [2]. This is the principle of superposition, on which the article Interference builds.

One field component is enough. Each component of the electric field obeys the same scalar wave equation [1]. Wave optics describes light by one such scalar function, an approximation to the full electromagnetic theory that holds whenever the direction of the field does not matter [2]. In this series the direction of the field is fixed: each waveguide carries light of one polarisation only (APX-006). From here on, the field $E$ is a single number at each point and instant.

Key idea

A changing electric field produces a magnetic field and a changing magnetic field produces an electric field, so a disturbance of the two fields sustains itself and travels as a wave, in vacuum at the speed

1/\sqrt{\mu_0\varepsilon_0}

Light in a material

Glass, silicon and every other transparent material contain charges. The electric field of the light pushes the electrons of the atoms back and forth, and the material responds with a polarisation that adds to the field [2]. For a uniform, non-absorbing material, the net effect on Maxwell’s equations is simple: the vacuum permittivity $\varepsilon_0$ is replaced by the material’s permittivity $\varepsilon$ , and $\mu_0$ by its permeability $\mu$ [1]. The wave equation keeps its form, and light travels with the speed $v = 1/\sqrt{\mu\varepsilon}$ , slower than in vacuum.

How the atoms slow the wave down can be followed one layer at a time. Each electron is bound to its atom like a mass on a spring, with a resonance frequency of its own, and the light drives it at the light’s frequency. Below the resonance, the electron moves in step with the force on it [1]. An oscillating electron radiates a wave of its own, the secondary wave, which lags behind the light. The light itself is not held up by the atoms: apart from being weakened, it travels on as in vacuum [1]. The wave in the material is the sum of the light and all the secondary waves, and after every layer of atoms it lags a little further behind the same wave in vacuum. A crest therefore arrives later than it would in vacuum, which means that the wave is slower [1]. Figure 3 shows this for a slab of atoms. The closer the frequency of the light comes to the resonance, the more strongly the electrons respond, and the larger the lag and the refractive index become [1].

the wave through the atoms
the same wave in vacuum

at the exit: vacuum wave (dashed) + secondary wave = the wave

Light frequency / resonance, ω/ω₀0.80

Density of atoms80 %

refractive index n: 1.173
wavelength inside, λ/n: 1322 nm
behind the vacuum wave at the exit: 0.17 cycle
delay at the exit: 0.86 fs
electron response, 1/(1 − (ω/ω₀)²): 2.78

Light of 1550 nm, slowed down about 5 × 10¹⁴ times; the slider moves the atoms' resonance, not the light. The electrons' motion is enlarged enormously. One resonance, no absorption, no reflection at the faces (APX-018).

A wave of 1550 nm passes through a slab of atoms 1.5 µm thick, at 0.80 of the atoms' resonance frequency and 80 % density. The electrons oscillate with the field, and the wave inside has the refractive index 1.173: its wavelength is shorter, and it leaves the slab 0.17 cycle behind the same wave in vacuum.

Figure A wave of 1550 nm passing through a slab of atoms, each drawn as an electron bound to its nucleus by a spring, after the account of Hecht [1]. The dashed line is the same wave in vacuum. The clock shows the wave at the exit as the sum of the vacuum wave and the secondary waves of all atoms. A model material with one resonance, without absorption or reflection (APX-018); the motion of the electrons is enlarged enormously.

The ratio of the two speeds is the refractive index $n$ :

n = \frac{c}{v} = \sqrt{\frac{\varepsilon\mu}{\varepsilon_0\mu_0}} \approx \sqrt{\frac{\varepsilon}{\varepsilon_0}}, \tag{4}

where the last step holds for materials that are not magnetic, $\mu \approx \mu_0$ , such as glass and silicon [1]. At a wavelength of 1550 nm, silicon has $n \approx 3.47$ [3] and silicon dioxide, the glass that surrounds silicon waveguides on a chip, $n \approx 1.44$ [3]. In silicon, light therefore covers about 8.6 cm per nanosecond instead of 30 cm.

Equation (4) comes with a caveat. The permittivity that belongs in it is the one at the frequency of the light, and it differs from the permittivity measured with static fields. For water the static value gives $\sqrt{\varepsilon/\varepsilon_0} = 8.96$ , while the refractive index of water for visible light is 1.333 [1]. The electrons in a material follow the field differently at different frequencies, so the refractive index depends on the wavelength. This effect, dispersion, is why a prism separates colours [1]. For the circuits in this series, it means that every number that depends on the refractive index holds for one wavelength.

The refractive index is the link between Maxwell’s equations and the rest of the series. It decides how much the phase of a wave advances over a given length (Phase), and the difference in index between silicon and glass is what holds light inside a waveguide (Waveguides).

How do we model itOptional · from Maxwell's equations to the wave equation

Take the curl of Faraday’s law and use Ampère’s law in vacuum from Eq. (1):

\nabla \times (\nabla \times \mathbf{E}) = -\frac{\partial}{\partial t} (\nabla \times \mathbf{B}) = -\mu_0 \varepsilon_0 \frac{\partial^2 \mathbf{E}}{\partial t^2}.

For any vector field, $\nabla \times (\nabla \times \mathbf{E}) = \nabla(\nabla \cdot \mathbf{E}) - \nabla^2 \mathbf{E}$ , and in empty space $\nabla \cdot \mathbf{E} = 0$ . What remains is Eq. (2), $\nabla^2 \mathbf{E} = \mu_0 \varepsilon_0 \, \partial^2 \mathbf{E} / \partial t^2$ . The same steps, starting from the curl of Ampère’s law, give the equation for $\mathbf{B}$ [2].

Written for one component that depends only on $x$ and $t$ , Eq. (2) reads

\frac{\partial^2 E_y}{\partial x^2} = \frac{1}{v^2} \frac{\partial^2 E_y}{\partial t^2}, \qquad v = \frac{1}{\sqrt{\mu_0 \varepsilon_0}}.

Any function of the form $E_y(x, t) = f(x - vt)$ solves it: both sides equal $f''(x - vt)$ . Such a function is a fixed shape that moves along $x$ with the speed $v$ . The harmonic wave $E_y = E_0 \cos(kx - \omega t)$ is the special case that the article Describing a wave uses throughout, with $\omega / k = v$ [1].

In short

Maxwell’s four equations say that charges produce electric fields, that there are no magnetic charges, and that a changing magnetic field produces an electric field and a changing electric field a magnetic one.
Together the last two allow a self-sustaining wave, whose speed in vacuum, $1/\sqrt{\mu_0\varepsilon_0} \approx 3.00 \times 10^8\,\mathrm{m/s}$ , equals the measured speed of light; the wave is transverse, and its equations are linear, so waves add.
In a material light travels slower by the refractive index $n = c/v \approx \sqrt{\varepsilon/\varepsilon_0}$ , about 3.47 for silicon and 1.44 for silicon dioxide at 1550 nm; $n$ depends on the wavelength.

Next · FoundationsDescribing a waveHow a light wave is described by its wavelength, frequency and amplitude, written as a cosine or as a complex exponential, and measured by its power.Read next

References

1Hecht, E. (2017). Optics (5th edition, global edition). Pearson Education Limited. link
2Saleh, B. E. A., Teich, M. C. (2019). Fundamentals of Photonics (3rd edition). Wiley. link
3Chrostowski, L., Hochberg, M. (2015). Silicon Photonics Design. Cambridge University Press. doi:10.1017/CBO9781316084168

Approximations used on this page: APX-006 single mode, single polarisation, APX-018 atoms as driven oscillators with one resonance.