Maxwell's equations describe the behavior of electromagnetic waves and their relationship with electric and magnetic fields. These equations were formulated by James Clerk Maxwell in the 19th century and are a set of fundamental laws in classical electromagnetism. The equations mathematically link electric and magnetic fields, electric charges, and electric currents.
Maxwell's equations consist of four equations, known as:
Gauss's Law for Electric Fields: This equation relates the electric field to the distribution of electric charges. It states that the electric flux through a closed surface is proportional to the total electric charge enclosed by that surface divided by the permittivity of free space (ε₀). This equation helps us understand how electric charges create electric fields.
Gauss's Law for Magnetic Fields: This equation relates the magnetic field to the distribution of magnetic sources (magnetic charges or currents). It states that the magnetic flux through a closed surface is zero. This equation indicates that magnetic monopoles (isolated magnetic charges) do not exist and that magnetic field lines always form closed loops.
Faraday's Law of Electromagnetic Induction: This equation describes the electromagnetic induction phenomenon. It states that a changing magnetic field induces an electromotive force (emf) and thus an electric field. The induced emf is proportional to the rate of change of magnetic flux through a surface and is manifested as a circulating electric field.
Ampere's Law with Maxwell's Addition: This equation relates the magnetic field to electric currents and the rate of change of electric flux. Ampere's Law states that the circulation of the magnetic field around a closed loop is proportional to the sum of the electric current passing through the loop and the rate of change of electric flux through the loop. Maxwell's addition introduced the displacement current term to account for the changing electric field's contribution to magnetic fields.
These four equations together form Maxwell's equations and describe how electric and magnetic fields interact and propagate. When solving these equations, solutions emerge that exhibit wave-like behavior, resulting in electromagnetic waves. These waves consist of oscillating electric and magnetic fields that propagate through space at the speed of light. Thus, Maxwell's equations provide the theoretical foundation for understanding and predicting the behavior of electromagnetic waves, including their propagation, interference, and interaction with matter.