Maxwell's Equations
Maxwell's Equations are a set of four partial differential equations formulated by James Clerk Maxwell (1861–1865) that completely describe the behaviour of electric and magnetic fields and their interactions with matter and charge. They unify electricity, magnetism, and optics into a single coherent theory and predicted the existence of electromagnetic waves travelling at the speed of light. Maxwell's Equations are among the greatest achievements in theoretical physics and form the foundation of classical electrodynamics, modern optical theory, and electrical engineering.
Key Formula
div E = ρ/ε₀ | div B = 0 | curl E = −∂B/∂t | curl B = μ₀J + μ₀ε₀(∂E/∂t)
LaTeX: \nabla \cdot \vec{E} = \frac{\rho}{\epsilon_0}, \quad \nabla \cdot \vec{B} = 0, \quad \nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}, \quad \nabla \times \vec{B} = \mu_0\vec{J} + \mu_0\epsilon_0\frac{\partial \vec{E}}{\partial t}
| Symbol | Meaning | Unit |
|---|---|---|
| ∇·E | Divergence of electric field | V/m² |
| ρ | Electric charge density | C/m³ |
| ε₀ | Permittivity of free space | F/m |
| ∇·B | Divergence of magnetic field (always zero — no magnetic monopoles) | T/m |
| ∇×E | Curl of electric field | V/m² |
| ∂B/∂t | Time rate of change of magnetic field (Faraday's Law) | T/s |
| ∇×B | Curl of magnetic field | T/m |
| μ₀ | Permeability of free space | T·m/A |
| J | Current density | A/m² |
Worked Example
Problem
Using Maxwell's third equation (Faraday's Law in differential form), explain qualitatively why a time-varying magnetic field creates a circulating electric field even in empty space with no charges present.
Solution
Step 1: Write Maxwell's third equation. ∇ × E = −∂B/∂t Step 2: Interpret the equation. The curl of the electric field (∇ × E) equals the negative time derivative of the magnetic field. A non-zero curl means the electric field circulates in closed loops. Step 3: Apply to the scenario. If ∂B/∂t ≠ 0 (magnetic field is changing), then ∇ × E ≠ 0 even when no charges or currents are present (ρ = 0, J = 0). Step 4: Physical implication. The changing magnetic field acts as a source of circulating electric field lines — this is exactly what Faraday observed experimentally: changing B induces an EMF in a loop, which is the integral form of this same equation. Step 5: Consequence for EM waves. This and Maxwell's fourth equation create a self-sustaining feedback: changing E induces B; changing B induces E — producing a propagating electromagnetic wave.
Answer
A changing magnetic field (∂B/∂t ≠ 0) creates a non-zero curl in E, meaning E circulates in closed loops even without charges — this is the differential form of Faraday's Law and the mechanism behind electromagnetic wave propagation.
Maxwell's Four Equations: Names, Forms and Physical Meanings
| Equation | Name | Differential Form | Physical Meaning |
|---|---|---|---|
| 1st | Gauss's Law for Electricity | ∇·E = ρ/ε₀ | Electric charges are sources/sinks of electric field lines |
| 2nd | Gauss's Law for Magnetism | ∇·B = 0 | No magnetic monopoles; B-field lines always close on themselves |
| 3rd | Faraday's Law of Induction | ∇×E = −∂B/∂t | Changing B creates circulating E (basis of generators, transformers) |
| 4th | Ampère–Maxwell Law | ∇×B = μ₀J + μ₀ε₀(∂E/∂t) | Currents and changing E create circulating B; predicts EM waves |
Interactive Tools
Wolfram Alpha — Maxwell's Equations
Explore and compute results from Maxwell's Equations symbolically
Open ToolKhan Academy — Maxwell's Equations Overview
Conceptual introduction to all four of Maxwell's Equations
Open ToolBrilliant — Maxwell's Equations
In-depth interactive derivations and physical interpretation of each equation
Open Tool
Wikimedia Commons, CC BY-SA
Related Terms
Electromagnetic Wave
An electromagnetic wave is a self-propagating transverse wave consisting of oscillating electric and magnetic fields that are perpendicular to each other and to the direction of propagation. Predicted theoretically by James Clerk Maxwell in 1865 and confirmed experimentally by Heinrich Hertz in 1887, electromagnetic waves require no medium and travel at the speed of light (3 × 10⁸ m/s) in vacuum. The electromagnetic spectrum spans radio waves, microwaves, infrared, visible light, ultraviolet, X-rays, and gamma rays — all governed by the same wave equations.
Faraday's Law of Induction
Faraday's Law of Induction states that the electromotive force (EMF) induced in a closed loop is equal to the negative rate of change of magnetic flux through the loop. This fundamental law explains how changing magnetic fields produce electric currents, forming the basis of electric generators, transformers, and induction motors. It was discovered experimentally by Michael Faraday in 1831 and independently by Joseph Henry around the same time.
Magnetic Flux
Magnetic flux is the total quantity of magnetic field lines passing perpendicularly through a given surface area, measuring how much of the magnetic field is captured by that surface. It is a scalar quantity defined as the dot product of the magnetic field vector and the area vector of the surface. Magnetic flux is fundamental to understanding electromagnetic induction, transformer operation, and the behaviour of inductors in circuits.
Named after James Clerk Maxwell (1831–1879), a Scottish physicist. "Equation" from Latin "aequatio" meaning "making equal". Maxwell compiled and extended the work of Gauss, Faraday, and Ampère into this unified set, publishing them in 1861 and the complete theory in 1865.