# Maxwells first and second equations and their derivation

**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/ε_{0} times the net charge enclosed by the surface:

Φ_{E}=∫E.dS=q/ε_{0}

∫D.dS=q

where D=ε_{0}E= Displacement vector

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

Therefore ∫ D.dS=∫_{v}pdV (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.

i thik it is easy to learn

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