There are four fundamental equations of electromagnetism known as Maxwell’s equations
which may be written in differential form as
1. div D = p
2. div B = 0
3. curl E = – dB /dt
4. curlH =J + -dD / dt
In above equations the notations have the following meanings:
J =current density in Amp/met3
D =electric displacement vector in Coulomb/met2
p = free charge density in Coul / met3
B =magnetic induction in Weber/met2
E = electric field intensity in Volt/met or Newton/Coul
H =magnetic field intensity in Amp/met-tum.
7.16.1 Derivation of Maxwell’s Equations
1. Derivation of First Equation
div D = ∆.D = p
Proof: “The maxwell first equation .is nothing but the differential form of Gauss law of electrostatics.”
Let us consider a surface S bounding a volume V in a dielectric medium. In a dielectric medium total charge consists of free charge. If p is the charge density of free charge at a. point in a small volume element dV. Then Gauss’s law can be expresses as
“The total normal electrical induction over a closed surface is equal to – times of 1/ ε 0total charge enclosed.
where p = charge per unit volume
V = volume enclosed by charge.
By Gauss transformation formula
ʃV div E dv =1/ε0 ʃ v p dv [ʃ s A n ds = ʃ v div Adv]
div E =1/ε0 P
ε0 div E =P
div ε0 E =P
P=0 Then D =ε0 E [D =ε0 E+P]
2. Derivation of Second Equation
div B = ∆.B = 0
“It is nothing but the differential form of Gauss law of magnetostatics.”
Since isolated magnetic poles and magnetic currents due to them have no significance.Therefore magnetic lines of force in general are either closed curves or go off to infinity.Consequently the number of magnetic lines of force entering any arbitrary closed surface isexactly the same as leaving it. It means that the flux of magnetic induction B across anyclosed surface is always zero.
Gauss law of magnetostatics states that “Total normal magnetic induction over aclosed surface is equal to zero.”
i. e; ʃ S B n ds =0
Applying Gauss transformation formula we get
ʃ V div B dv =0
The integrand should vanish for the surface boundary as the volume is arbitrary.
div B =0
3. Derivation of Third Law
Curl E =- Db /dt
“It is nothing but the differential form of Faraday’s law of electromagnetic induction.”
According to Faraday’s law of electromagnetic induction, it is known that e.m.f.induced in a closed loop is defined as negative rate of change of magnetic flux i.e.,
where θ = magnetic flux
ϕ = ʃ S BndS
ϕ=B / A where S is any surface having loop as boundary.
ʃ l E .dl = – d ϕ/dt
Putting the value of ϕ in equation (1), we get
ʃl E .dl = – d /dt ʃS b .n ds
ʃl E .dl = ʃS-dB /dt .n ds
Applying Stoke’s transformation formula on L.H.S.
ʃs( curl EndS= ʃS =Db /DT n ds
or ʃ S (curl E +dB /dt ) n ds =0
Further validity of the equation
Curl E =Db /dt =0
Or curl E = 0 Db /dt
This is known as Maxwell’s third equation.
4. Derivation of Maxwell’s Fourth Equation:
“This is nothing but differential form of modified Ampere circuital law.”
Ampere’s modified circuital law
According to law the work done in carrying a unit magnetic pole once around closed arbitrary path linked with the current is expressed by
ʃl B dl =µ 0 x i
i = current enclosed by the path
ʃl B dl =µ 0ʃs J n ds
On applying Stoke’s transformation formula in L.H.S.
ʃ s curl B n ds = ʃs µ 0 J n ds
è ʃ s (curl B- µ 0 J) n ds =0
For the validity this equation
curl B- µ 0 J =0
curl B- µ 0 J
It is known as the fourth equation of Maxwell.
Taking divergence of both sides
Div .(curl B )= div (µ 0 J)
0= div(µ 0 J)
=µ 0 div J [div (cual A) =0]
Div J =0
Which means that the current is always closed and there are no source and sink. Thuswe arrive at contradiction equation (3) is also in conflict with the equation of discontinuity.
But the according to law of continuity
Div J = – d p / d t
So this equation fails and it need of little modification. So Maxwell assume that
curl B = µ 0 (div J ) +µ 0(div J d)
0= µ0 (div J ) +µ 0(div J d)
By putting div J d =dp/dt
Div J d =div dD/ dt
Jd =Dd /dt(By Maxwell first equation, div D = p in equation (4))
Putting in equation (4), we get
Curl B =µ0(J +Dd /dt)
B =µ0 =H
Curl (µ0 =H) =µ0( J +Dd /dt)
Curl H =j dD /dt