# Maxwell’s Equations and their derivations

Hello friends, today we will discuss the Maxwell’s fourth equation and its differential & integral form.

Let us first derive and discuss Maxwell fourth equation:

**1. Maxwell’s Fourth Equation or Modified Ampere’s Circuital Law**

Here the first question arises , **why there was need to modify Ampere’s circuital Law?**

To give answer to this question, let us first discuss Ampere’s law(without modification)

**Statement of Ampere’s circuital law (without modification).** It states that the line integral of the magnetic field H around any closed path or circuit is equal to the current enclosed by the path.

That is ∫H.dL=I

Let the current is distributed through the surface with a current density J

Then I=∫J.dS

This implies that ∫H.dL=∫J.dS (9)

Apply Stoke’s theorem to L.H.S. of equation (9) to change line integral to surface integral,

That is ∫H.dL=∫(∇ xH).dS

Substituting above equation in equation(9), we get

∫( ∇xH).dS=∫_{s}J.dS

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

Thus , ∇ x H=J (10)

This is the **differential form of Ampere’s circuital Law (without modification) for steady currents.**

Take divergence of equation (10)

∇.(∇xH)= ∇.J

As divergene of the curl of a vector is always zero ,therefore

∇ .( ∇xH)=0

It means ∇.J=0

Now ,**this is equation of continuity for steady current** but not for time varying fields,**as equation of continuity for time varying fields is**

∇ .J= – dp/ dt

**So, **there is inconsistency in Ampere’s circuital law. This is the reason, that led Maxwell to modify: Ampere’s circuital law.

**Modification of Ampere’s circuital law. **Maxwell modified Ampere’s law by giving the concept of displacement current D and so the concept of displacement current density J_{d} for time varying fields.

He concluded that equation (10) for time varying fields should be written as

∇ xH=J+j_{d} (11)

By taking divergence of equation(11) , we get

∇ .( ∇ xH)= ∇.J+ ∇.J_{d}

As divergence of the curl of a vector is always zero,therefore

∇ .( ∇ x H)=0

It means, ∇ .(J+ .J_{d)}=0

Or ∇. J= – ∇.J_{d}

But from equation of continuity for time varying fields,

∇.J= – dρ/ dt

By comparing above two equations of .j ,we get

∇ .j_{d} =d(∇ .D)/dt (12)

Because from maxwells first equation ∇ .D=ρ

As the divergence of two vectors is equal only if the vectors are equal.

Thus J_{d}= dD/dt

Substituting above equation in equation (11), we get

∇ xH=J+dD/dt (13)

Here ,dD/dt= J_{d}=Displacement current density

J=conduction current density

D= displacement current

The equation(13) is the **Differential form of Maxwell’s fourth equation** or Modified Ampere’s circuital law.

Intergal form

Taking surface integral of equation (13) on both sides, we get

∫( ∇xH).dS=∫(J+ dD/dt).dS

Apply stoke’s therorem to L.H.S. of above equation, we get

∫( ∇xH).dS=∫_{l} H.dL

Comparing the above two equations ,we get

∫H.dL=∫(J+dD/dt).dS

**Statement of modified Ampere’s circuital Law. **The line integral of the

Magnetic field H around any closed path or circuit is equal to the conductions current plus the time derivative of electric displacement through any surface bounded by the path.

Equation(14) is the **integral form of Maxwell’s fourth equation.**

This is all about the derivation of differential and integral form of Maxwell’s fourth equation that is modified form of Ampere’s circuital law.

**2. Maxwell first equation and second equation and Maxwell third equation are already derived and discussed.**