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.
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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
From eqn. of continuity
ρ is volume charge density
Using eqn.(2) in above eqn.
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’=0Therefore divJ’=-divJ = -(-∂ρ/∂t) ………..(5)ButD 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/∂tobviously 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.
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