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MAXWELL’S EQUATIONS
INTRODUCTION
The electromagnetic theory was
developed on the basis of
electromagnetism with the help of four
vector differential equations. These
equations are known as Maxwell’s
equations.
Two of these relations are independent of
time and are called as steady state
equations. The other two relations depend
upon time and are called as time varying
equations.
Displacement current:
From Ampere circuital law it follows that
Curl H=J ……..(1)
J is current density and H is magnetic intensity
So , div(curl H)=divJ
But div(curl H)=0
So divJ=0……..(2)
From eqn. of continuity
divJ=-∂ρ/∂t…………..(3)
ρ is volume charge density
Using eqn.(2) in above eqn.
-∂ρ/∂t=0
This eqn. represents only a steady state condition in which ρ is
constant
Hence equation (1) represents only a steady state
condition.
For time dependent fields, it needs some modifications.
For this, Maxwell suggested that we must add some vector
J’ to R.H.S. of this equation to make it valid in general i.e.
Curl H=J+J’..........(4)
Where J’ is the displacement current density. the
corresponding current is called displacement current.
Hence div(curl H)=0 implies that
divJ+divJ’=0
Therefore divJ’=-divJ = -(-∂ρ/∂t) ………..(5)
But
D is electric field displacement vector
So eqn.(5) gives
divJ’=∂/∂t(div D)=div(∂D/∂t)
Therefore, J’= ∂D/∂t………(6)
Hence using eqn.(6) in eqn.(4),
curlH=J+ ∂D/∂t
obviously the displacement current density J’ arises from time
variation of electric displacement D.
Note: The conduction current is produced due to actual flow
of charged particles while the displacement current arises in the
region where electric displacement or electric field varies with
time.
Maxwell’s equations
Terms used in Maxwell’s
equations
1)D is electric displacement in Cm-2
2)ρ is free charge density in Cm-3
3)B is magnetic induction in Wbm-2 (or tesla)
4) E is electric intensity in Vm-1
5) H is magnetic intensity Am-1
6) J is current density in Am-2
Physical significance of Maxwell’s
equations:
1) Maxwell’s Ist equation i.e.
i.e.
divD=ρ
a) It is time independent equation.
b) Since divD is scalar, therefore charge density is a scalar quantity.
c) It relates space variation of div. of electric field with charge density.
d) It is statement of Gauss law of electrostatics.
2) Maxwell’s 2nd equation
a) It is time independent equation
b) According to this equation isolated magnetic poles do not exist
c) Since ∫B.dS=0 i.e.number of lines of magnetic force leaving and
entering a given volume are equal.
d) It is statement of Gauss law in magnetism.
3) Maxwell’s 3rd equation
a) It is time dependent equation.
b) It relates space variation of E with time variation of B.
c) It implies that time variation of magnetic field generates electric
field.
d) It is statement of Faraday’s law of e.m. induction and -ve sign
justifies Lenz’s law.
4) Maxwell’s 4th equation:
a) It is time dependent equation
b) It shows that magnetic field can be generated by current
density vector and time variation of D jointly or separately.
c) It relates magnetic field vector with electric displacement vector
and current density vector.
d) It is the statement of Ampere’s law.
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