Definition
Maxwell's Equations
Maxwell’s equations are a set of four coupled partial differential equations that describe how electric charges and currents produce electric and magnetic fields, and how those fields evolve in time. Together with the Lorentz force law, they form the foundation of classical electromagnetism.
In differential form, in vacuum:
where
is the electric field, is the magnetic field, is the charge density, is the current density, is the vacuum permittivity, and is the vacuum permeability.
Physical Meaning
The four equations encode two kinds of physical content: two constrain the shape of the fields (divergence equations), and two govern their coupled time evolution (curl equations).
Gauss’s Law
Electric charges are the sources of the electric field. The flux of
Gauss’s Law for Magnetism
There are no magnetic monopoles. The flux of
Faraday’s Law
A changing magnetic field induces a circulating electric field. This is the principle behind electric generators and transformers. The minus sign encodes Lenz’s law: the induced field opposes the change that produced it.
Ampère–Maxwell Law
A steady electric current produces a circulating magnetic field (Ampère’s original law). Maxwell added the displacement current term