Electromagnetism · Physics

Maxwell's Equations Explained: All Four Laws, Integral and Differential Forms, and How They Predict Light

Four equations describe every classical electric and magnetic phenomenon, from a compass needle to Wi-Fi to sunlight. This guide states each law in integral and differential form, explains what it means physically with an example, shows why Maxwell's displacement current was essential, derives the electromagnetic wave equation step by step, and lets you explore electromagnetic waves across the spectrum with an interactive calculator.

Integral + differential formsWave equation derivedInteractive EM waveSpectrum calculator

By Bimal Ghimire · Published October 30, 2026 · 18 min read · Constants: CODATA 2022

Key facts

Maxwell's equations are four laws: Gauss's law, Gauss's law for magnetism, Faraday's law of induction, and the Ampère–Maxwell law.
Together with the Lorentz force F = q(E + v × B), they form the complete theory of classical electromagnetism.
Maxwell added the displacement current ε₀∂E/∂t to Ampère's law; this makes the equations consistent with charge conservation and allows electromagnetic waves.
In vacuum they predict waves travelling at c = 1/√(μ₀ε₀) = 299,792,458 m/s, the speed of light.
In an electromagnetic wave, E and B are perpendicular to each other and to the direction of travel, in phase, with |E| = c|B|.
James Clerk Maxwell published the theory in 1865; Heinrich Hertz generated and detected the predicted waves in 1887–1888.

At a Glance

The Four Maxwell's Equations (SI Units, Vacuum)

Law	Integral form	Differential form	Physical meaning
Gauss's law	$\oint_S \mathbf E\cdot d\mathbf A=\dfrac{Q_{enc}}{\varepsilon_0}$	$\nabla\cdot\mathbf E=\dfrac{\rho}{\varepsilon_0}$	Electric field lines start and end on charges.
Gauss's law for magnetism	$\oint_S \mathbf B\cdot d\mathbf A=0$	$\nabla\cdot\mathbf B=0$	No magnetic monopoles; B field lines form closed loops.
Faraday's law	$\oint_C \mathbf E\cdot d\boldsymbol\ell=-\dfrac{d\Phi_B}{dt}$	$\nabla\times\mathbf E=-\dfrac{\partial\mathbf B}{\partial t}$	A changing magnetic field creates a circulating electric field.
Ampère–Maxwell law	$\oint_C \mathbf B\cdot d\boldsymbol\ell=\mu_0 I_{enc}+\mu_0\varepsilon_0\dfrac{d\Phi_E}{dt}$	$\nabla\times\mathbf B=\mu_0\mathbf J+\mu_0\varepsilon_0\dfrac{\partial\mathbf E}{\partial t}$	Currents and changing electric fields create circulating magnetic fields.

The integral and differential forms are equivalent: the divergence theorem converts surface integrals to volume integrals, and Stokes' theorem converts loop integrals to surface integrals. Faraday's integral form is written for a stationary loop.

Equation 1

Gauss's Law: Charges Are Sources of Electric Field

$$\oint_S\mathbf E\cdot d\mathbf A=\frac{Q_{enc}}{\varepsilon_0}\qquad\Longleftrightarrow\qquad\nabla\cdot\mathbf E=\frac{\rho}{\varepsilon_0}$$

The total electric flux out of any closed surface equals the enclosed charge divided by ε₀ = 8.8541878188 × 10⁻¹² F/m. For a point charge q and a sphere of radius r around it, symmetry gives $E\cdot4\pi r^2=q/\varepsilon_0$, so $E=q/(4\pi\varepsilon_0r^2)$: Coulomb's law follows directly. Gauss's law also explains why the excess charge on a conductor sits on its surface and why the field inside a closed conducting shell is zero. Explore point-charge fields in our electric field simulator.

Equation 2

Gauss's Law for Magnetism: No Magnetic Monopoles

$$\oint_S\mathbf B\cdot d\mathbf A=0\qquad\Longleftrightarrow\qquad\nabla\cdot\mathbf B=0$$

As much magnetic flux enters any closed surface as leaves it. Cut a bar magnet in half and you get two smaller magnets, each with a north and a south pole, never an isolated pole. Magnetic field lines therefore have no beginning or end; they form closed loops, as you can see in the field map of our magnetic field calculator. Searches for magnetic monopoles continue, but none has been observed.

Equation 3

Faraday's Law: Changing Magnetic Fields Make Electric Fields

$$\mathcal E=\oint_C\mathbf E\cdot d\boldsymbol\ell=-\frac{d\Phi_B}{dt},\quad\Phi_B=\int_S\mathbf B\cdot d\mathbf A\qquad\Longleftrightarrow\qquad\nabla\times\mathbf E=-\frac{\partial\mathbf B}{\partial t}$$

A time-varying magnetic flux through a loop induces an electromotive force around it. The minus sign is Lenz's law: the induced current opposes the change that produced it, which is required by energy conservation. Unlike an electrostatic field, the induced electric field forms closed loops and is not conservative. This law is the basis of generators, transformers, induction cooktops and wireless chargers. See the full treatment in Faraday's law and Lenz's law explained.

Equation 4

Ampère–Maxwell Law and the Displacement Current

$$\oint_C\mathbf B\cdot d\boldsymbol\ell=\mu_0I_{enc}+\mu_0\varepsilon_0\frac{d\Phi_E}{dt}\qquad\Longleftrightarrow\qquad\nabla\times\mathbf B=\mu_0\mathbf J+\mu_0\varepsilon_0\frac{\partial\mathbf E}{\partial t}$$

Why Maxwell had to add the second term

Taking the divergence of Ampère's original law, $\nabla\times\mathbf B=\mu_0\mathbf J$, gives $\nabla\cdot\mathbf J=0$, because the divergence of a curl is always zero. But charge conservation requires $\nabla\cdot\mathbf J=-\partial\rho/\partial t$, which is not zero while charge accumulates, for example on a charging capacitor plate. Adding $\mu_0\varepsilon_0\,\partial\mathbf E/\partial t$ and using Gauss's law, $\nabla\cdot(\varepsilon_0\partial\mathbf E/\partial t)=\partial\rho/\partial t$, restores consistency exactly.

Capacitor picture: a current I flows in the wire to a charging capacitor, but no charge crosses the gap. A loop around the wire must give the same $\oint\mathbf B\cdot d\boldsymbol\ell$ whether the surface it bounds cuts the wire or passes between the plates. Between the plates, the changing electric flux supplies exactly $\varepsilon_0\,d\Phi_E/dt=I$, the displacement current.

Materials

Maxwell's Equations in Matter: D and H

$$\nabla\cdot\mathbf D=\rho_f,\quad\nabla\cdot\mathbf B=0,\quad\nabla\times\mathbf E=-\frac{\partial\mathbf B}{\partial t},\quad\nabla\times\mathbf H=\mathbf J_f+\frac{\partial\mathbf D}{\partial t}$$

$\mathbf D=\varepsilon_0\mathbf E+\mathbf P$ (P = polarisation), $\mathbf H=\mathbf B/\mu_0-\mathbf M$ (M = magnetisation). Only free charge ρ_f and free current J_f appear. For linear isotropic media $\mathbf D=\varepsilon\mathbf E$, $\mathbf B=\mu\mathbf H$, and waves travel at $v=1/\sqrt{\mu\varepsilon}=c/n$.

Derivation

From Maxwell's Equations to Electromagnetic Waves

In empty space there are no charges or currents (ρ = 0, J = 0). Take the curl of Faraday's law and use the identity $\nabla\times(\nabla\times\mathbf E)=\nabla(\nabla\cdot\mathbf E)-\nabla^2\mathbf E$:

Curl of Faraday's law: $\nabla\times(\nabla\times\mathbf E)=-\dfrac{\partial}{\partial t}(\nabla\times\mathbf B)$
Use Gauss's law ($\nabla\cdot\mathbf E=0$): the left side becomes $-\nabla^2\mathbf E$.
Use Ampère–Maxwell with J = 0: $\nabla\times\mathbf B=\mu_0\varepsilon_0\,\partial\mathbf E/\partial t$.
Result, the wave equation: $$\nabla^2\mathbf E=\mu_0\varepsilon_0\frac{\partial^2\mathbf E}{\partial t^2},\qquad\nabla^2\mathbf B=\mu_0\varepsilon_0\frac{\partial^2\mathbf B}{\partial t^2}$$
Read off the speed: comparing with $\nabla^2f=\frac{1}{v^2}\frac{\partial^2f}{\partial t^2}$ gives $$c=\frac{1}{\sqrt{\mu_0\varepsilon_0}}=\frac{1}{\sqrt{(1.25663706127\times10^{-6})(8.8541878188\times10^{-12})}}=2.99792458\times10^{8}\ \text{m/s}$$

Notice that the displacement current is what made step 3 possible: without it, there would be no wave. For a plane wave $\mathbf E=E_0\cos(kx-\omega t)\,\hat{\mathbf y}$, Faraday's law gives $\mathbf B=(E_0/c)\cos(kx-\omega t)\,\hat{\mathbf z}$. The energy flux is the Poynting vector $\mathbf S=\mathbf E\times\mathbf B/\mu_0$, whose time average is the intensity $I=\tfrac12c\varepsilon_0E_0^2$.

Example, sunlight: at the top of Earth's atmosphere the solar irradiance is about 1361 W/m². Treating it as a single plane wave, $E_0=\sqrt{2I/(c\varepsilon_0)}\approx1013$ V/m and $B_0=E_0/c\approx3.38$ μT. The ratio of the fields equals the impedance of free space, $\mu_0c\approx376.73\ \Omega$.

Interactive Electromagnetic Wave and Spectrum Calculator

E field B field

Enter a value

Unit

Refractive index n1 = vacuum; wavelength in medium = λ/n

Field amplitude E₀ (V/m)

← long wavelengthshort wavelength →

Band boundaries are approximate and vary between conventions (for example, visible light is often quoted as 400–700 nm or 380–750 nm). Constants: c and h are exact in the SI.

History

How the Equations Came Together

Year	Discovery
1785	Coulomb measures the inverse-square law of electric force with a torsion balance.
1820	Ørsted observes a compass needle deflected by an electric current; Biot, Savart and Ampère quantify the magnetic effects of currents.
1831	Faraday discovers electromagnetic induction.
1861–1865	Maxwell introduces the displacement current and, in "A Dynamical Theory of the Electromagnetic Field" (1865), concludes that light is an electromagnetic wave.
1880s	Heaviside and Gibbs rewrite the theory as four vector equations.
1887–1888	Hertz generates and detects electromagnetic waves, confirming Maxwell's prediction.

Keep Learning

Related Tools and Guides

FAQ

Frequently Asked Questions

What are Maxwell's equations in simple words?

They are four laws that describe all classical electric and magnetic phenomena. Electric charges create electric fields (Gauss's law). There are no magnetic charges, so magnetic field lines never begin or end (Gauss's law for magnetism). A changing magnetic field creates a circulating electric field (Faraday's law). Electric currents and changing electric fields create circulating magnetic fields (Ampère–Maxwell law).

What is displacement current?

Displacement current is the term ε₀ ∂E/∂t that Maxwell added to Ampère's law. It is not a flow of charge; it is a changing electric field that produces a magnetic field just as a current does. Without it Ampère's law would contradict conservation of charge (for example in a charging capacitor), and electromagnetic waves could not exist.

How do Maxwell's equations predict the speed of light?

In empty space the equations combine into a wave equation for E and B whose speed is c = 1/√(μ₀ε₀). With μ₀ = 1.25663706127 × 10⁻⁶ N/A² and ε₀ = 8.8541878188 × 10⁻¹² F/m this gives 299,792,458 m/s, the speed of light, showing that light is an electromagnetic wave.

What is the difference between the integral and differential forms?

The integral forms relate fields on a closed surface or loop to the charge, current or flux inside it, which is convenient for symmetric problems and for intuition. The differential forms state the same laws at a single point using divergence and curl. They are equivalent through the divergence theorem and Stokes' theorem.

Why is there no magnetic charge in Maxwell's equations?

Because no isolated magnetic monopole has ever been observed, the divergence of B is zero: every magnetic field line closes on itself. If monopoles were discovered, Gauss's law for magnetism and Faraday's law would gain magnetic charge and current terms, making the equations more symmetric.

How many equations did Maxwell originally write?

Maxwell's 1865 paper "A Dynamical Theory of the Electromagnetic Field" listed twenty equations in twenty variables, written component by component. Oliver Heaviside and Josiah Willard Gibbs later recast them in vector notation as the four equations used today.

Do Maxwell's equations work in materials?

Yes. In materials it is convenient to use the displacement field D = ε₀E + P and the magnetic field strength H = B/μ₀ − M, so that only free charges and free currents appear: ∇·D = ρ_f and ∇×H = J_f + ∂D/∂t. For linear materials D = εE and B = μH.

Sources

References

Maxwell, J. C. (1865). A dynamical theory of the electromagnetic field. Philosophical Transactions of the Royal Society of London, 155, 459–512.

Griffiths, D. J. (2017). Introduction to Electrodynamics (4th ed.), Chapters 7 and 9. Cambridge University Press.

Feynman, R. P., Leighton, R. B., & Sands, M. (1964). The Feynman Lectures on Physics, Vol. II, Chapter 18: The Maxwell Equations. feynmanlectures.caltech.edu

Ling, S. J., Sanny, J., & Moebs, W. (2016). University Physics Volume 2, Chapter 16: Electromagnetic Waves. OpenStax. openstax.org

NIST (2024). CODATA Recommended Values of the Fundamental Physical Constants: 2022. physics.nist.gov/cuu/Constants

Kopp, G., & Lean, J. L. (2011). A new, lower value of total solar irradiance: Evidence and climate significance. Geophysical Research Letters, 38, L01706.

Cite this article: Ghimire, B. (2026). Maxwell's Equations Explained: Integral & Differential Forms. Engineers Views. https://engineersviews.com/blog/maxwells-equations-explained/