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
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
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)
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 Jd for time varying fields.
He concluded that equation (10) for time varying fields should be written as
∇ xH=J+jd (11)
By taking divergence of equation(11) , we get
∇ .( ∇ xH)= ∇.J+ ∇.Jd
As divergence of the curl of a vector is always zero,therefore
∇ .( ∇ x H)=0
It means, ∇ .(J+ .Jd)=0
Or ∇. J= – ∇.Jd
But from equation of continuity for time varying fields,
∇.J= – dρ/ dt
By comparing above two equations of .j ,we get
∇ .jd =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 Jd= dD/dt
Substituting above equation in equation (11), we get
∇ xH=J+dD/dt (13)
Here ,dD/dt= Jd=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.
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
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.