Maxwell’s Displacement current; Maxwell...

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Maxwell’s Displacement

current; Maxwell Equations

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Topics

� Laws of Electric & Magnetic Field

� Displacement Current and its Derivation

� Static equation and Faraday’s law

� Maxwell’s Equations with modification of Ampere’s law

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Displacement Current

In Electromagnetism, displacement current is a quantity appearing

in Maxwell's equations that is defined in terms of rate of change

of electric displacement field.

If the current carrying wire possess certain symmetry, the

magnetic field can be obtained by using Ampere's law

The equation states that line integral of magnetic field around the

arbitrary closed loop is equal to µ0Ienc

. Where Ienc

is the

conduction current passing through surface by closed path.

∫ ⋅=•enclosedo

IμdsB

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Derivation of Displacement Current

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Cont’d…

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Cont’d…

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Cont’d…

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Displacement Current

Maxwell realized that Ampere’s law is not valid when the current is

discontinuous as is true of the current through a parallel plate

capacitor: He concluded that when the charge within an enclosed

surface is changing it is necessary to add to Ampere’s law another

current called the displacement current

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Maxwell’s Equation



Differential form in VacuumDifferential form in Vacuum


Static Equation and Faraday’s Law

The two fundamental equations of electrostatics are shown below:The two fundamental equations of electrostatics are shown below:

∇⋅E = ρtotal

/ ε0

Coulomb's Law in Differential Form

Coulomb's law is the statement that electric charges create diverging electric

fields.

∇×E = 0 Irrotational Electric Fields when Static

This means that if everything is static, then the electric fields have no curl.

The two fundamental equations of The two fundamental equations of magnetostaticsmagnetostatics are shown below:are shown below:

∇⋅B = 0 No Magnetic Monopoles

Electric charges give rise to diverging electric fields, magnetic charges

would give rise to diverging magnetic fields. But there are no magnetic

charges (no magnetic monopoles). So there is no divergence to the

magnetic fields.


Cont’d…

∇×B = µ0J

totalAmpere's Law for Steady Currents

This equation states that steadily moving electric charges give rise to

curling magnetic fields.

These four equations completely specify all electromagnetic fields when

everything is static in time. But what happens if something changes in

time? Faraday was the first to show that these equations are not complete if

we want to include time-varying effects. He showed that changing

magnetic fields give rise to curling electric fields. The irrotational E field

equation of electrostatics became Faraday's law in electrodynamics. The

second equation now stood as :

∇×E= −∂ B/∂ t Faraday's Law of Induction


Cont’d

But these four equations are now logically and mathematically

inconsistent if we are no longer considering static situations,

To show this, take the divergence of Ampere's law:

∇⋅(∇×B) = µ0∇⋅J

total

Mathematically speaking, the divergence of the curl (shown on the left) is

always zero, leading to:

0 =∇⋅Jtotal

This equation was fine for static situations, but for non-static situations, the

continuity equation states:

-∂ ρtotal

/∂t = ∇⋅Jtotal



- It took the genius of Maxwell to realize this problem and figure out

how to fix it. For this accomplishment he is now honored with the

distinction of having the final four equations named after him.

- Maxwell realized that to remove the contradiction, he could add an

extra term to Ampere's law that would automatically make the

continuity equation hold true.

- Let us start with the continuity equation and work backwards to see

what the more complete form of Ampere's law should look like


How Maxwell fix Ampere's law

We can rewrite Ampere's law,

(We know B = µ0H)

∇×B = µ0

Jtotal

∇× µ0

H = µ0

Jtotal

∇×H = Jtotal

Taking Divergence on both sides,

∇⋅ Jtotal

= 0 (only valid for static (steady-state) problem)

But for non-static situations, the continuity equation states:

∇⋅ Jtotal

= -∂ ρtotal

/∂t

What Maxwell saw was that the continuity equation could be converted

into a vanishing divergence by using coulomb’s law.


Cont’d...

In term of partial field instead of total field, and in terms of free

current/charge instead of total current/charge,

∇⋅J = - ∂ ρ /∂t ∇⋅(J + ∂ D/∂t )= 0 (using coulomb’s law ∇⋅D = ρ )

Then Maxwell replaced J in Ampere’s law by its generalization

J→J + ∂ D/∂t

For time-dependent field. Thus Ampere’s law becomes,

∇×H = J + ∂ D/∂t

Still the same, experimentally verified, law for steady state phenomena, but

now mathematically consistent with the continuity equation for time

dependent field.


Cont’d…

We now have four equations which form the foundation of

electromagnetic phenomena:

∇⋅D = ρ ∇×H = J + ∂ D/∂t

∇⋅B = 0 ∇×E + ∂ B/∂ t =0

An important consequence of Maxwell’s equations, as we shall see, is

the prediction of the existence of electromagnetic waves that travel with

speed of light c2 =1/ μ0 ε

0. The reason is due to the fact that a changing

electric field produces a magnetic field and vice versa, and the coupling

between the two fields leads to the generation of electromagnetic waves.


Thanks!!Thanks!!For your patience……


Backup slides…

µ0 is the magnetic

constant

7 2

0 4 10 N/Aµ π−

= ×

ε0 is the electric

constant12 2 2

0 8.854 10 C /(N m )ε−

= × ⋅

7

0 0

12

1

2

2 2

27 2

4 10 [ ]

8.854 10 [ ]

1.1

N/A

13

C /(N m )

s10 [ ]/m

µ ε π−

−

−

= ×

× ×

= ×

⋅


……

8

0 0

12.998 10 m/s

µ ε= ×

From

7

0

2

0

211.113 1 s /m0 [ ]µ ε−

= ×

we can write

which is the speed of light in vacuum!

Maxwell’s Displacement current; Maxwell...

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