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Maxwell’s Equations

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.,

 

e=–dθ/dt

 

where θ = magnetic flux

or

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)

 

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