Maxwell's Equations

In the early nineteenth century, electricity and magnetism looked like two separate curiosities. Rubbed amber attracted scraps of paper. Lodestones pointed north. Batteries could make wires warm. Nobody had a reason to think these things had anything to do with sunlight.

Then the connections started to appear, one experiment at a time. In 1820 Hans Christian Ørsted noticed that a wire carrying current deflected a compass needle. In 1831 Michael Faraday showed the reverse: moving a magnet near a coil of wire made a current flow in it ^[1]. Electricity could make magnetism, and magnetism could make electricity. What was still missing was a single description that explained why.

That description arrived in the 1860s, when James Clerk Maxwell gathered the known laws into one mathematical system, found that one of them was inconsistent, repaired it, and then noticed something unexpected. His equations predicted waves of electric and magnetic fields travelling through empty space, and the speed of those waves came out equal to the measured speed of light ^[2,3].

That is the core of why Maxwell's equations matter. They are not only a summary of electricity and magnetism. They showed that light itself is an electromagnetic phenomenon, and they carried a hidden conflict with Newtonian ideas about motion that, forty years later, produced special relativity ^[4].

The Basic Idea: Fields That Fill Space

Faraday did not think of electric and magnetic forces as reaching instantly across empty space. He pictured "lines of force" filling the region around charges and magnets. Maxwell took this picture seriously and made it quantitative. The modern word for it is a field.

A field assigns a quantity to every point in space. Picture a weather map with a small arrow at every location showing wind speed and direction. That map is a vector field: each point gets an arrow with a length and a direction.

Electromagnetism uses two such maps:

the electric field $\mathbf{E}$ , which tells a charged particle placed at a point which way it will be pushed and how hard;
the magnetic field $\mathbf{B}$ , which pushes on charges only when they move, and at right angles to their motion.

The combined push on a particle with charge $q$ moving at velocity $\mathbf{v}$ is the Lorentz force:

\mathbf{F} = q\left(\mathbf{E} + \mathbf{v}\times\mathbf{B}\right).

Maxwell's equations do not describe the force on particles. They describe the fields themselves: where they come from, what shapes they can take, and how a change in one produces the other.

Two Questions You Can Ask of Any Arrow Map

A useful trick, emphasised in Grant Sanderson's visual introduction to the topic ^[5], is to imagine that the arrows in a field describe the flow of some imaginary fluid, even when the field is really electric or magnetic. There is no literal electric fluid. But the picture gives an intuitive reading of the two mathematical operations Maxwell's equations are written in.

Divergence: Is Something Coming Out of Here?

Draw a tiny imaginary bubble around a point. If more of the imaginary fluid flows out of the bubble than into it, the point behaves like a source, like a tap. If more flows in than out, it behaves like a sink, like a drain. If the flow in and out balance, the fluid behaves like water in a pipe: it moves, but it is not created or destroyed there.

The number that measures this is the divergence, written $\nabla\cdot\mathbf{F}$ . Formally, it is the net outward flow through a small closed surface, divided by the volume enclosed, as that volume shrinks to zero:

\nabla\cdot\mathbf{F} = \lim_{V\to 0}\frac{1}{V}\oint_{S}\mathbf{F}\cdot d\mathbf{A}.

Here $d\mathbf{A}$ is a small patch of the bubble's surface, pointing outward, and the circle on the integral sign means the surface is closed. Positive divergence means source, negative means sink, zero means "flows through without piling up."

Curl: Would a Paddle Wheel Spin Here?

Now imagine dropping a tiny paddle wheel into the flow, pinned at its centre so it can rotate but not drift. If the flow pushes harder on one side of the wheel than the other, it spins. The flow does not have to be a whirlpool for this to happen: a current that is fast on top and slow underneath will also turn the wheel.

The measure of this tendency to rotate is the curl, written $\nabla\times\mathbf{F}$ . It is a vector: its direction is the axis the paddle wheel would spin around (by the right-hand rule), and its length is how strongly it would spin. Formally, the component of the curl along a direction $\hat{\mathbf{n}}$ is the circulation around a small loop perpendicular to $\hat{\mathbf{n}}$ , divided by the loop's area:

\hat{\mathbf{n}}\cdot(\nabla\times\mathbf{F}) = \lim_{A\to 0}\frac{1}{A}\oint_{C}\mathbf{F}\cdot d\boldsymbol{\ell}.

Here $d\boldsymbol{\ell}$ is a small step along the loop $C$ . The symbol $\nabla$ ("nabla" or "del") behaves like a vector of derivatives, $\nabla = (\partial_x, \partial_y, \partial_z)$ , which is why divergence looks like a dot product and curl like a cross product. As Sanderson points out, this is more than a notational convenience: divergence really does measure how much a small step tends to change the field along the step, and curl how much it tends to change the field across it ^[5].

With just these two ideas, the four equations can be read almost in plain English.

The Four Equations

The equations below are in their modern vector form, in SI units, for fields in a vacuum that may contain charges and currents. Maxwell's own 1865 paper wrote the theory as around twenty equations in component form ^[2]. The compact four-equation version was produced in the following decades, notably by Oliver Heaviside, using the vector notation above.

The symbols are:

$\rho$ : electric charge density (charge per unit volume);
$\mathbf{J}$ : current density (current per unit area);
$\varepsilon_0$ : the vacuum permittivity, a constant setting the strength of electric effects;
$\mu_0$ : the vacuum permeability, a constant setting the strength of magnetic effects;
$t$ : time.

\begin{aligned} \nabla\cdot\mathbf{E} &= \frac{\rho}{\varepsilon_0} \\[4pt] \nabla\cdot\mathbf{B} &= 0 \\[4pt] \nabla\times\mathbf{E} &= -\frac{\partial\mathbf{B}}{\partial t} \\[4pt] \nabla\times\mathbf{B} &= \mu_0\mathbf{J} + \mu_0\varepsilon_0\frac{\partial\mathbf{E}}{\partial t} \end{aligned}

Maxwell's Equations (differential form)

1. Gauss's Law: Charges Are Sources

\nabla\cdot\mathbf{E} = \frac{\rho}{\varepsilon_0}

Intuitive reading: electric field lines spring out of positive charge and disappear into negative charge. Where there is no charge, the field neither starts nor stops; it only passes through.

This single statement contains Coulomb's inverse-square law. Surround a point charge with a sphere: the total outward "flow" through the sphere is fixed by the charge inside, and the sphere's area grows as $r^2$ , so the field strength must fall as $1/r^2$ .

2. Gauss's Law for Magnetism: No Magnetic Charges

\nabla\cdot\mathbf{B} = 0

Intuitive reading: magnetic field lines never begin or end. They always form closed loops. Cut a bar magnet in half and you get two smaller magnets, each with a north and south pole, never an isolated north pole.

An isolated magnetic pole, a magnetic monopole, would show up as a nonzero divergence of $\mathbf{B}$ . Some theories beyond the Standard Model allow monopoles, and physicists have searched for them for decades. None has been confirmed. Within Maxwell's theory as it stands, this equation is a statement of that absence.

3. Faraday's Law: A Changing Magnetic Field Makes a Swirling Electric Field

\nabla\times\mathbf{E} = -\frac{\partial\mathbf{B}}{\partial t}

Intuitive reading: wherever the magnetic field is changing in time, the electric field curls around it. That circulating electric field pushes charges around a loop of wire, which is exactly the induced current Faraday observed ^[1].

The minus sign (Lenz's law) says the induced effects oppose the change that caused them. This is the principle behind every electric generator and transformer on the power grid.

4. The Ampère–Maxwell Law: Currents and Changing Electric Fields Make Swirling Magnetic Fields

\nabla\times\mathbf{B} = \mu_0\mathbf{J} + \mu_0\varepsilon_0\frac{\partial\mathbf{E}}{\partial t}

Intuitive reading: magnetic field curls around electric currents, as Ørsted's compass showed. It also curls around places where the electric field is changing, even if no charge is moving there.

That second term is Maxwell's own contribution, and it is the key to everything that follows.

The Same Laws in Integral Form

The differential form describes what happens at each point. The integral form describes what happens over a whole surface or loop, which is often closer to what one actually measures. The two are mathematically equivalent, connected by the divergence theorem and Stokes' theorem.

\begin{aligned} \oint_{S}\mathbf{E}\cdot d\mathbf{A} &= \frac{Q_{\text{enc}}}{\varepsilon_0} \\[4pt] \oint_{S}\mathbf{B}\cdot d\mathbf{A} &= 0 \\[4pt] \oint_{C}\mathbf{E}\cdot d\boldsymbol{\ell} &= -\frac{d}{dt}\int_{\Sigma}\mathbf{B}\cdot d\mathbf{A} \\[4pt] \oint_{C}\mathbf{B}\cdot d\boldsymbol{\ell} &= \mu_0 I_{\text{enc}} + \mu_0\varepsilon_0\frac{d}{dt}\int_{\Sigma}\mathbf{E}\cdot d\mathbf{A} \end{aligned}

Maxwell's Equations (integral form)

Here $S$ is any closed surface, $Q_{\text{enc}}$ the charge inside it, $C$ any closed loop, $\Sigma$ any surface bounded by that loop, and $I_{\text{enc}}$ the current passing through $\Sigma$ . The quantity $\int\mathbf{B}\cdot d\mathbf{A}$ is the magnetic flux: roughly, how many field lines pierce the surface.

Read in words:

The electric flux out of a closed surface counts the charge inside.
The magnetic flux out of a closed surface is always zero.
The voltage around a loop equals the rate at which magnetic flux through it is decreasing.
The magnetic circulation around a loop counts the current through it, plus the rate of change of electric flux through it.

The Missing Piece: Displacement Current

Before Maxwell, the fourth law was Ampère's law, without the $\partial\mathbf{E}/\partial t$ term:

\nabla\times\mathbf{B} = \mu_0\mathbf{J}.

It worked well for steady currents. But it fails in a simple, everyday situation.

A Scenario: Charging a Capacitor

Connect a battery to a capacitor: two metal plates separated by a gap. Current flows along the wire, and charge piles up on the plates. No charge crosses the gap.

Now draw a loop around the wire and apply the integral form of Ampère's law. It asks for the current through any surface bounded by that loop. Choose a flat disc that the wire pierces, and the answer is the current $I$ . Choose instead a bag-shaped surface with the same rim that bulges out to pass between the plates, and no current crosses it at all. The law gives two different answers for the same loop. Something is wrong.

The Formal Problem

The contradiction can be stated more generally. The divergence of any curl is always zero, $\nabla\cdot(\nabla\times\mathbf{B}) = 0$ . Taking the divergence of Ampère's original law therefore forces $\nabla\cdot\mathbf{J} = 0$ : current could never pile up anywhere. But conservation of charge says

\frac{\partial\rho}{\partial t} + \nabla\cdot\mathbf{J} = 0,

so $\nabla\cdot\mathbf{J}$ must be nonzero whenever charge density is changing, as it is on the capacitor plates.

Maxwell's Repair

Maxwell added a term proportional to the changing electric field ^[3]. Using Gauss's law, $\rho = \varepsilon_0\nabla\cdot\mathbf{E}$ , the divergence of the corrected law becomes

\nabla\cdot\mathbf{J} + \varepsilon_0\frac{\partial}{\partial t}(\nabla\cdot\mathbf{E}) = \nabla\cdot\mathbf{J} + \frac{\partial\rho}{\partial t} = 0,

which is exactly charge conservation. In the capacitor, the growing electric field in the gap plays the role of a current, so both surfaces now give the same answer. Maxwell called the new term the displacement current, $\varepsilon_0\,\partial\mathbf{E}/\partial t$ .

Maxwell originally motivated it with a mechanical model of the "ether," the medium then assumed to fill space ^[3]. The model has not survived. The term has. What matters for the history is this: the correction was introduced to make the equations consistent, and its most dramatic consequence was not yet known when it was proposed.

What the Equations Predict: Light

Consider empty space: no charges, no currents, $\rho = 0$ and $\mathbf{J} = 0$ . The equations become beautifully symmetric. A changing $\mathbf{B}$ produces a curling $\mathbf{E}$ ; a changing $\mathbf{E}$ produces a curling $\mathbf{B}$ . Each field can sustain the other.

Intuitively: a disturbance in the electric field creates a magnetic disturbance next to it, which creates an electric disturbance next to that, and so on. The disturbance walks through space on its own, needing no wire and no charge to carry it. As Sanderson puts it in passing, this back-and-forth between the last two equations is what gives rise to light waves ^[5].

Formally: take the curl of Faraday's law and use a standard vector identity:

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

since $\nabla\cdot\mathbf{E} = 0$ in empty space. The left side is also

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

Setting them equal gives:

\nabla^2\mathbf{E} = \mu_0\varepsilon_0\,\frac{\partial^2\mathbf{E}}{\partial t^2}, \qquad c = \frac{1}{\sqrt{\mu_0\varepsilon_0}}

Electromagnetic wave equation

This is the standard wave equation, the same form that describes waves on a string or sound in air. The operator $\nabla^2$ (the Laplacian) measures how a quantity at a point differs from its average nearby; the equation says that this curvature in space drives acceleration in time. The magnetic field obeys the identical equation. The constant in front fixes the wave speed $c$ .

Without the displacement current, the $\partial^2\mathbf{E}/\partial t^2$ term would not appear, and there would be no wave. The consistency fix was also the prediction of light.

The Numbers

$\varepsilon_0$ and $\mu_0$ can be measured with tabletop experiments on capacitors, currents and magnets. Nothing about light goes into them. In Maxwell's time, the relevant ratio of electrical units had been measured by Wilhelm Weber and Rudolf Kohlrausch, and Maxwell found it matched the optical measurements of the speed of light to within experimental error ^[3]. In the 1862 paper he concluded that one can "scarcely avoid the inference" that light consists of undulations in the same medium responsible for electric and magnetic phenomena ^[3].

With modern CODATA values, $\varepsilon_0 \approx 8.854\,187\,8188\times10^{-12}\ \text{F/m}$ and $\mu_0 \approx 1.256\,637\,061\,27\times10^{-6}\ \text{N/A}^2$ ^[6,7], and

\frac{1}{\sqrt{\mu_0\varepsilon_0}} \approx 2.997\,924\,58\times10^{8}\ \text{m/s},

which is the speed of light ^[8]. A historical caution is worth noting here: today the relation runs partly the other way. Since 1983 the metre has been defined so that $c$ is exactly $299\,792\,458$ m/s, and since the 2019 SI redefinition $\mu_0$ and $\varepsilon_0$ are measured quantities tied together by $\mu_0\varepsilon_0 c^2 = 1$ ^[6–8]. In 1862, however, the agreement was a genuine, independent match between electrical measurements and optics. It was a prediction, not a fit.

The equations also said that light should be a transverse wave, with $\mathbf{E}$ and $\mathbf{B}$ perpendicular to each other and to the direction of travel, which matched the long-known fact that light can be polarised. And they said nothing restricts the wavelength. Visible light should be only a narrow band in a much wider spectrum.

Evidence: Hertz Makes the Waves

For two decades the electromagnetic theory of light remained a theory. It explained optics elegantly, but nobody had produced an electromagnetic wave from electrical equipment and watched it behave like light.

Heinrich Hertz did so between 1886 and 1888. He built circuits in which a spark jumped rapidly back and forth across a gap, producing oscillating currents at frequencies far higher than anything used before ^[9]. A separate loop of wire with its own tiny gap, placed across the room, showed faint sparks when the transmitter fired, even though nothing connected the two.

Hertz then showed that these invisible disturbances behaved like waves. By reflecting them off a metal wall he set up standing waves, located their nodes, and measured their wavelength. Combining wavelength with the known oscillation frequency gave a propagation speed comparable to that of light ^[10]. He also showed that the waves reflect, refract and can be polarised. Within a generation, the same physics became radio.

Established fact: electromagnetic waves exist, travel at $c$ in vacuum, and span a continuous spectrum from radio waves through microwaves, infrared, visible light, ultraviolet, X-rays and gamma rays. Every antenna, fibre-optic cable, MRI scanner and mobile phone is a daily test of Maxwell's equations.

The Crack: Which Frame Is the Speed of Light Measured In?

Maxwell's equations give a single number, $c$ , for the speed of light. They do not say relative to what.

In Newtonian mechanics, velocities simply add. If you walk at 1 m/s along a train moving at 30 m/s, you move at 31 m/s relative to the ground. The mathematical rule that converts between the two frames is the Galilean transformation, $x' = x - vt$ , $t' = t$ . Newton's laws look the same in every frame connected this way.

Maxwell's equations do not. Apply a Galilean transformation and the wave equation changes form: light would move at $c - v$ in one direction and $c + v$ in the other. So either:

Maxwell's equations are true only in one special frame, presumably at rest relative to the ether; or
The Galilean rule for converting between frames is wrong.

Nineteenth-century physicists mostly assumed the first option. If so, the Earth, moving around the Sun at about 30 km/s, should feel an "ether wind," and light should travel at slightly different speeds in different directions. In 1887 Albert Michelson and Edward Morley built an interferometer sensitive enough to detect this difference. They found no effect of the expected size ^[11].

Hendrik Lorentz, George FitzGerald and Henri Poincaré developed a set of transformations under which Maxwell's equations do keep their form. These Lorentz transformations mix space and time:

x' = \gamma(x - vt), \qquad t' = \gamma\left(t - \frac{vx}{c^2}\right), \qquad \gamma = \frac{1}{\sqrt{1 - v^2/c^2}}.

For everyday speeds, $\gamma \approx 1$ and they reduce to the Galilean rule.

In 1905 Albert Einstein took the second option seriously and made it a principle. His paper "On the Electrodynamics of Moving Bodies" opens not with the ether experiments but with an asymmetry in electromagnetism itself: a magnet moving past a wire and a wire moving past a magnet produce the same current, yet the theory of the time described the two cases differently ^[4]. Einstein proposed that the laws of physics, Maxwell's included, are the same in every inertial frame, and that the speed of light in vacuum is the same for all observers. The Lorentz transformations then follow, and with them time dilation and length contraction (see Time Dilation).

In a sense, Maxwell's equations were relativistic before relativity existed. It was Newtonian mechanics, not electromagnetism, that had to be modified.

In the language of relativity, the four equations collapse into two, using the electromagnetic field tensor $F^{\mu\nu}$ , which packages $\mathbf{E}$ and $\mathbf{B}$ into a single object:

\partial_\mu F^{\mu\nu} = \mu_0 J^{\nu}, \qquad \partial_{[\lambda}F_{\mu\nu]} = 0.

What one observer calls a purely electric field, another observer moving relative to the first will see as partly magnetic. Electricity and magnetism are two views of one thing.

Limitations and What the Equations Do Not Say

Maxwell's equations are among the most thoroughly tested laws in physics, but they have a well-defined domain.

They are classical. They treat fields as smooth and continuous. At the level of individual photons, electromagnetism is described by quantum electrodynamics (QED). Maxwell's equations remain the correct large-scale limit, but they cannot explain the photoelectric effect or the discreteness of light on their own.
They are linear in vacuum. Two light beams pass through each other without interacting. QED predicts tiny departures from this at extreme field strengths; the classical equations do not include them.
Matter needs extra modelling. Inside materials, the response of atoms is summarised by permittivity, permeability and conductivity, often introduced through auxiliary fields $\mathbf{D}$ and $\mathbf{H}$ . Those material properties come from experiment or from quantum theory, not from Maxwell's equations themselves.
Magnetic monopoles remain open. The equations can be extended symmetrically to include magnetic charge. Whether nature contains any is unknown; none has been observed.
The ether did not survive. Maxwell's mechanical picture of a medium was historically important in producing the displacement current, but modern physics treats the electromagnetic field as a fundamental entity that needs no medium.

Four Equations, Three Revolutions

Maxwell's equations did three things at once.

They unified. Electricity, magnetism and optics, once three separate subjects, became aspects of one field. This became the model for later unification attempts, from the electroweak theory (see Higgs Field) to the Kaluza–Klein idea that electromagnetism might arise from gravity in a higher dimension (see Kaluza–Klein Reduction).

They predicted. A term added for mathematical consistency implied invisible waves that no one had seen, at a speed fixed by tabletop constants. Hertz found them. Few theoretical predictions have had larger practical consequences.

They exposed a contradiction. The fixed speed $c$ could not coexist with Newton's rule for combining velocities. Resolving that contradiction produced special relativity and a new picture of space and time.

At everyday and astronomical scales, the equations describe electromagnetic phenomena with extraordinary accuracy. Possibilities such as magnetic monopoles remain open to observation, while the quantum nature of light required a later theory. Four short lines, written with two operations that measure sources and swirls, turned out to contain radio, optics and the structure of spacetime.

References

[1] Faraday, M. (1832). "Experimental Researches in Electricity." Philosophical Transactions of the Royal Society of London, 122, 125–162.
https://doi.org/10.1098/rstl.1832.0006

[2] Maxwell, J. C. (1865). "A Dynamical Theory of the Electromagnetic Field." Philosophical Transactions of the Royal Society of London, 155, 459–512.
https://doi.org/10.1098/rstl.1865.0008

[3] Maxwell, J. C. (1862). "On Physical Lines of Force. Part III: The Theory of Molecular Vortices Applied to Statical Electricity." The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 23(151), 12–24.
https://doi.org/10.1080/14786446208643207

[4] Einstein, A. (1905). "Zur Elektrodynamik bewegter Körper" (On the Electrodynamics of Moving Bodies). Annalen der Physik, 322(10), 891–921.
https://doi.org/10.1002/andp.19053221004

[5] Sanderson, G. (3Blue1Brown). (2018). "Divergence and curl: The language of Maxwell's equations, fluid flow, and more." Video, YouTube. Further watching.
https://www.youtube.com/watch?v=rB83DpBJQsE

[6] NIST. (2022 CODATA). "Vacuum electric permittivity, $\varepsilon_0$ ." The NIST Reference on Constants, Units, and Uncertainty.
https://physics.nist.gov/cgi-bin/cuu/Value?ep0

[7] NIST. (2022 CODATA). "Vacuum magnetic permeability, $\mu_0$ ." The NIST Reference on Constants, Units, and Uncertainty.
https://physics.nist.gov/cgi-bin/cuu/Value?mu0

[8] NIST. (2022 CODATA). "Speed of light in vacuum, $c$ ." The NIST Reference on Constants, Units, and Uncertainty.
https://physics.nist.gov/cgi-bin/cuu/Value?c

[9] Hertz, H. (1887). "Ueber sehr schnelle electrische Schwingungen" (On Very Rapid Electric Oscillations). Annalen der Physik, 267(7), 421–448.
https://doi.org/10.1002/andp.18872670707

[10] Hertz, H. (1888). "Ueber die Ausbreitungsgeschwindigkeit der electrodynamischen Wirkungen" (On the Propagation Velocity of Electrodynamic Effects). Annalen der Physik, 270(7), 551–569.
https://doi.org/10.1002/andp.18882700708

[11] Michelson, A. A., & Morley, E. W. (1887). "On the Relative Motion of the Earth and the Luminiferous Ether." American Journal of Science, s3-34(203), 333–345.
https://doi.org/10.2475/ajs.s3-34.203.333