Maxwell’s first equation or Gauss’s law in electrostatics

Statement. It states that the total electric flux φ_E passing through a closed hypothetical surface is equal to 1/ε₀ times the net charge enclosed by the surface:

Φ_E=∫E.dS=q/ε₀

∫D.dS=q

where D=ε₀E= Displacement vector

Let the charge be distributed over a volume V and p be the volume charge density .therefore                               q=∫ pdV

Therefore                                   ∫ D.dS=∫_vpdV                                   (1)

Equation (1) is the integral form of Maxwell’s first equation or Gauss’s law in electrostatics.

Differential form:

Apply Gauss’s Divergence theorem to change L.H.S. of equation(1) from surface integral to volume integral

That is                                           ∫ D.dS=∫( ∇.D)dV

Substituting this equation in equation (1), we get

∫ (∇.D)dV=∫_v pdV

As two volume integrals are equal only if their integrands are equal

Thus,                                                           ∇.D=p                                     (2)

Equation (2) is the Differential form of Maxwell’s first equation.

Maxwell’s second equation or Gauss’s law for Magnetism

Statement. It states that the total magnetic flux φ_m emerging through a closed surface is zero.

φ_m=∫B.dS=0                                                           (3)

The equation (3) is the Intergal form of Maxwell’s second equation.

This equation also proves that magnetic monopole does not exist.

Differential Form:

Apply Gauss’s Divergence theorem to equation (3)

That is                                   ∫_s B.dS=∫_v(∇.B)dV

As                                              ∫B.dS=0

Thus ,                        ∇ .B=0                                                                  (4)

The equation (4) is differential form of Maxwell’s second equation.

I hope you have understood the concept and how to derive Maxwell’s first and second equations.

Note: You can also read article on Maxwell third equation and its derivation.