Understanding Maxwell's Equations

Maxwell’s Equations!
Understanding in depth the set of equations.

Maxwell's equations are a set of four partial differential equations that describe the behavior of electric and magnetic fields, and their interactions with charges and currents. 
They are named after the physicist James Clerk Maxwell, who first published them in the 1860s.

The four equations are:

1- Gauss's law for electric fields: This equation relates the electric field to the distribution of electric charges in space. It states that the total electric flux through any closed surface is proportional to the charge enclosed within that surface.

∇ ⋅ E = ρ / ε₀

where ∇ is the divergence operator, E is the electric field, ρ is the charge density, and ε₀ is the electric constant.

2- Gauss's law for magnetic fields: This equation relates the magnetic field to the distribution of magnetic charges (also known as magnetic monopoles) in space. It states that the total magnetic flux through any closed surface is zero.

∇ ⋅ B = 0

where B is the magnetic field.

3- Faraday's law of electromagnetic induction: This equation relates a changing magnetic field to the creation of an electric field. It states that the emf (electromotive force) induced around a closed loop is equal to the negative time rate of change of the magnetic flux through the loop.

∇ × E = - ∂B / ∂t

where ∂/∂t denotes the partial derivative with respect to time.

4- Ampere's law with Maxwell's correction: This equation relates a current-carrying wire to the magnetic field it produces. It states that the circulation of the magnetic field around a closed loop is proportional to the current passing through the loop, with an additional term that depends on the rate of change of the electric field.

∇ × B = μ₀ (J + ε₀ ∂E / ∂t)

where μ₀ is the magnetic constant, J is the current density, and ∂/∂t denotes the partial derivative with respect to time.
These equations describe the fundamental laws of electricity and magnetism, and they have many important implications for the behavior of electromagnetic waves and the properties of materials. For example, they imply that electromagnetic waves propagate through space at a constant speed (the speed of light), and that the speed of light is related to the properties of the electric and magnetic fields. They also imply that electromagnetic waves can be reflected, refracted, and diffracted, just like waves in other media. Finally, they imply that the behavior of materials can be understood in terms of their electromagnetic properties, such as their permittivity and permeability.