Skip to content

Maxwell's Equations

Maxwell's equations describe the local relationship between electric fields, magnetic fields, charge density, and current density.
MAXWELL'S EQUATIONS
In differential form, Maxwell's equations are
∇⋅E=ρε0,∇⋅B=0,∇×E=−∂B∂t,∇×B=μ0J+μ0ε0∂E∂t.\begin{align*}\nabla \cdot \mathbf{E} &= \frac{\rho}{\varepsilon_0},\\\nabla \cdot \mathbf{B} &= 0,\\\nabla \times \mathbf{E} &= -\frac{\partial \mathbf{B}}{\partial t},\\\nabla \times \mathbf{B} &= \mu_0\mathbf{J}+\mu_0\varepsilon_0\frac{\partial \mathbf{E}}{\partial t}.\end{align*}
Here E\mathbf{E} is the electric field, B\mathbf{B} is the magnetic field, ρ\rho is charge density, and J\mathbf{J} is current density. The constants ε0\varepsilon_0 and μ0\mu_0 are the electric permittivity and magnetic permeability of free space.
Electric field lines begin at positive charge and end at negative charge.
Gauss's law for electricity is
∇⋅E=ρε0.\begin{equation*}\nabla \cdot \mathbf{E} = \frac{\rho}{\varepsilon_0}.\end{equation*}
The divergence ∇⋅E\nabla \cdot \mathbf{E} measures how much the electric field flows outward from a point. Positive charge density ρ>0\rho>0 acts as a source of electric field lines, while negative charge density ρ<0\rho<0 acts as a sink. If a region contains no charge, then the electric field has zero net local divergence there.
Magnetic field lines form loops rather than starting or ending at isolated magnetic charge.
Gauss's law for magnetism is
∇⋅B=0.\begin{equation*}\nabla \cdot \mathbf{B} = 0.\end{equation*}
This says that the magnetic field has no local sources or sinks. Magnetic field lines do not begin or end at isolated magnetic charges; instead, they form closed loops or continue through the boundary of the region being studied.
Faraday's law is
∇×E=−∂B∂t.\begin{equation*}\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}.\end{equation*}
The curl ∇×E\nabla \times \mathbf{E} measures the tendency of the electric field to circulate around a point. Faraday's law says that a time-changing magnetic field creates a circulating electric field. The minus sign records Lenz's law: the induced electric circulation opposes the change in magnetic flux that produced it.
The Ampere-Maxwell law is
∇×B=μ0J+μ0ε0∂E∂t.\begin{equation*}\nabla \times \mathbf{B}=\mu_0\mathbf{J}+\mu_0\varepsilon_0\frac{\partial \mathbf{E}}{\partial t}.\end{equation*}
It says that magnetic circulation is produced by two effects: electric current J\mathbf{J} and a time-changing electric field. Maxwell's added term μ0ε0∂E∂t\mu_0\varepsilon_0\frac{\partial \mathbf{E}}{\partial t} is essential because it lets electromagnetic waves propagate even in empty space.

References