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MAXWELL’S EQUATIONS

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- 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.

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

Butdiv(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’=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.

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:
- Maxwell’s Ist equation i.e.
- i.e. divD=ρ
- a) It is time independent equation.
- Since divD is scalar, therefore charge density is a scalar quantity.
- It relates space variation of div. of electric field with charge density.
- It is statement of Gauss law of electrostatics.
- 2) Maxwell’s 2nd equation
- It is time independent equation
- According to this equation isolated magnetic poles do not exist
- Since ∫B.dS=0 i.e.number of lines of magnetic force leaving and entering a given volume are equal.
- It is statement of Gauss law in magnetism.

- 3) Maxwell’s 3rd equation
- It is time dependent equation.
- It relates space variation of E with time variation of B.
- It implies that time variation of magnetic field generates electric field.
- It is statement of Faraday’s law of e.m. induction and -ve sign justifies Lenz’s law.
- 4) Maxwell’s 4th equation:
- It is time dependent equation
- It shows that magnetic field can be generated by current density vector and time variation of D jointly or separately.
- It relates magnetic field vector with electric displacement vector and current density vector.
- It is the statement of Ampere’s law.