Maxwell's Equations
Maxwell's equations are a set of four fundamental equations that describe the behavior of electric and magnetic fields in classical electromagnetism. They were formulated by James Clerk Maxwell and are the basis for understanding various electromagnetic phenomena. Here are the key formulas associated with Maxwell's equations:
1. Gauss's Law for Electric Fields:
∇⋅E = ρ / ε₀
This equation relates the divergence of the electric field E to the electric charge density ρ and the electric permittivity of free space ε₀.
2. Gauss's Law for Magnetic Fields:
∇⋅B = 0
This equation states that the divergence of the magnetic field B is always zero, indicating that there are no magnetic monopoles.
3. Faraday's Law of Electromagnetic Induction:
∇×E = -∂B/∂t
This equation relates the curl of the electric field E to the time rate of change of the magnetic field B. It explains how a changing magnetic field induces an electric field.
4. Ampère's Circuital Law:
∇×B = μ₀J + μ₀ε₀∂E/∂t
This equation relates the curl of the magnetic field B to the electric current density J, the magnetic permeability of free space μ₀, and the time rate of change of the electric field E. It describes how a current or a changing electric field generates a magnetic field.
These four equations, together with appropriate constitutive relations, form Maxwell's equations. They provide a comprehensive description of the interplay between electric and magnetic fields, electric charges, and electric currents. Maxwell's equations have far-reaching implications, including the propagation of electromagnetic waves, the behavior of light, the operation of antennas, and the functioning of electronic devices.
