Derivation of Maxwell's Equations in Vacuum

Gauss's Law for Electricity

The first of Maxwell's equations is Gauss's Law for Electricity, which states that the electric flux through any closed surface is equal to the enclosed charge divided by the permittivity of free space (ε₀).

Mathematically, it is expressed as:

∮_S E · dA = Q / ε₀

Gauss's Law for Magnetism

The second equation is Gauss's Law for Magnetism, which states that the magnetic flux through any closed surface is zero.

Mathematically, it is expressed as:

∮_S B · dA = 0

Faraday's Law of Induction

The third equation is Faraday's Law of Induction, which describes how a changing magnetic field induces an electromotive force (EMF) in a closed loop.

Mathematically, it is expressed as:

∮_C E · dl = -dΦ_B/dt

Ampère's Circuital Law with Maxwell's Addition

The fourth equation is Ampère's Circuital Law with Maxwell's addition, which states that the magnetic field circulation around a closed loop is proportional to the sum of the current passing through the loop and the rate of change of electric flux through the loop.

Mathematically, it is expressed as:

∮_C B · dl = μ₀(I + ε₀dΦ_E/dt)