Explain Maxwells equations and its elegant equation

Question

Explain Maxwells equations and its four elegant equation?

Maxwell's equations (J.C. Maxwell; 1864):

The four elegant equations that explain classical electromagnetism in its entire splendor. They are:

Gauss law:
The electric flux via a closed surface is proportional to the arithmetical sum of electric charges encompassed within that closed surface; in its differential form,

div E = rho,

Here rho is the charge density.

Gauss law for magnetic fields:

The magnetic flux via a closed surface is zero (0); no magnetic charges exist. In the differential form,

div B = 0

Faraday's law:

The line integral of the electric field about a closed curve is proportional to the instant time rate of change of the magnetic flux via a surface bounded by that closed curve; in its differential form,

curl E = -dB/dt,

Here d/dt here symbolizes partial differentiation.

Ampere's law, modified form:

The line integral of the magnetic field about a closed curve is proportional to the addition of two terms: first, the arithmetical sum of electric currents flowing via that closed curve; and second, the instant time rate of change of the electric flux via a surface bounded by that closed curve; in its differential form,

curl H = J + dD/dt,

Here d/dt here symbolizes partial differentiation.

In addition to explaining electromagnetism, his equations too predict that waves can propagate via the electromagnetic field, and would for all time propagate at similar speed -- these are electromagnetic waves; the speed can be found by evaluating (epsilon₀ mu₀)^-1/2, that is c, the speed of light in vacuum.