Conclusion
The law of conservation of charge applies locally at each point and time, so any variation of the total charge within a closed surface must be due to charges that flow across the surface of the region. This principle leads to the equation of continuity:
where ρ is the charge density and J is the current density at any given point and time. In magnetostatic situations, ∂ρ / ∂t = div J = 0.
Ampère's law, curl B = μ0J, is a law of magnetostatics. It applies when ∂ρ / ∂t = div J = 0. The appropriate generalisation, valid for time-dependent charge and current densities, is the Ampère–Maxwell law:
The extra term, ε0μ0∂E / ∂t, on the right-hand side is called the Maxwell term.
Maxwell's four equations
describe the dynamical behaviour of electromagnetic fields. They are the same in all inertial frames of reference and are unaffected by time-reversal. They are not valid in rotating frames of reference.
An electromagnetic wave is an oscillating disturbance of electric and magnetic fields that propagates in accordance with Maxwell's equations. We concentrate on linearly polarised monochromatic plane waves. In empty space, the electric and magnetic waves are in phase with one another, with B = E / c. They are mutually perpendicular and transverse to the direction of propagation. In empty space, electromagnetic waves travel at speed
Electromagnetic waves with frequencies in the visible range, 4 × 1014 Hz to 8 × 1014 Hz, all called light, but the known electromagnetic spectrum also embraces radio waves, microwaves, infrared, ultraviolet, X-rays and gamma rays. Electromagnetic waves transport energy. The amount of energy carried by the magnetic wave is the same as that carried by the electric wave. The energy flux is the total energy transported per unit area per unit time across a plane area perpendicular to the direction of propagation of the electromagnetic wave. Averaging over a complete cycle,
where E0 is the amplitude of the electric wave.