Maxwell's Equations
The four equations that describe how electric and magnetic fields are generated by charges and currents, and how they interact — the complete classical theory of electromagnetism.
Formula
Variables
Example
Together they predict electromagnetic waves travelling at c = 1/√(ε₀μ₀) ≈ 3×10⁸ m/s
How it works
Maxwell's equations are the compact mathematical statement of how electric charge and current create electric and magnetic fields, and how changing one field induces the other. The first and third equations are source laws: charge acts like a source of electric field, while magnetic field lines have no beginning or end, so there are no isolated magnetic charges in classical physics. The curl equations describe field circulation: a changing magnetic field creates an electric field, and currents plus changing electric fields create magnetic fields, which is why the laws are consistent with conservation of charge and with self-propagating electromagnetic disturbances.
When to use it
Use Maxwell's equations when analyzing electric and magnetic fields in space and time, especially for conductors, capacitors, inductors, waves, antennas, or waveguides. They are the starting point for deriving field behavior from charge/current distributions and for checking whether a proposed electromagnetic solution is physically consistent.
Worked example, step by step
A parallel-plate capacitor has a uniform electric field between plates, with electric flux density changing in time at dE/dt = 2.0 × 10^6 V·m⁻¹·s⁻¹. Use the displacement-current term from Ampère-Maxwell law: ∇×B = μ₀ε₀(dE/dt) Substitute constants: μ₀ = 4π × 10⁻⁷ H·m⁻¹, ε₀ = 8.85 × 10⁻¹² F·m⁻¹ Compute μ₀ε₀ = (4π × 10⁻⁷)(8.85 × 10⁻¹²) ≈ 1.11 × 10⁻¹⁷ Then ∇×B ≈ (1.11 × 10⁻¹⁷)(2.0 × 10^6) = 2.22 × 10⁻¹¹ T·m⁻¹ Final answer: the changing electric field produces a magnetic-field curl of about 2.2 × 10⁻¹¹ T·m⁻¹.
Common mistakes to avoid
- Treating ∇·B = 0 as meaning the magnetic field is always zero, instead of meaning there are no magnetic monopole sources
- Forgetting the displacement-current term μ₀ε₀∂E/∂t in Ampère's law, especially in capacitors and wave problems
- Confusing field direction with field magnitude when interpreting divergence and curl
Did You Know?
When Maxwell computed the speed of his predicted waves and got the speed of light, he realised light itself is an electromagnetic wave.
Share this formula
More in Electricity & Magnetism
View allCoulomb's Law
IntermediateForce between two point charges.
Electric Field
BasicForce per unit charge at a point.
Field of a Point Charge
IntermediateElectric field around a point charge.
Electric Potential
IntermediatePotential due to a point charge.
Electric Potential Energy
IntermediateEnergy of two interacting charges.
Gauss's Law
AdvancedElectric flux through a closed surface.