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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

∇⋅E=ρε0∇×E=−∂B∂t∇⋅B=0∇×B=μ0J+μ0ε0∂E∂t\begin{aligned}\nabla\cdot\mathbf{E}&=\tfrac{\rho}{\varepsilon_0} & \nabla\times\mathbf{E}&=-\tfrac{\partial\mathbf{B}}{\partial t}\\ \nabla\cdot\mathbf{B}&=0 & \nabla\times\mathbf{B}&=\mu_0\mathbf{J}+\mu_0\varepsilon_0\tfrac{\partial\mathbf{E}}{\partial t}\end{aligned}

Variables

EElectric field
BMagnetic field
ρCharge density
JCurrent density
ε₀Permittivity of free space
μ₀Permeability of free space

Example

Together they predict electromagnetic waves travelling at c = 1/√(ε₀μ₀) ≈ 3×10⁸ m/s

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.

electromagnetismmaxwellfieldslightwaves

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