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Week 2

Electromagnetics 2 (EM-2)

Displacement Current

- - - - -

- -

- b

Relationship between J and E in a conductor with DC bias

V

L

DC voltage

- - - -

a

J +

-

E

σσσ σ ε

S

L E = V

E F

= − e Ea

E = v

= − µ

E

µ



 





sec

2

V m



 m V



 sec

m Drift Velocity:

Ja

J =

S J = I

I

ρρ

ρρ_e : volume electron density Review on EM-1

Ohm’s law

- - - - -

- -

- b

Relationship between J, E, and H in a conductor with DC bias

V

L

DC voltage

- - - -

a

J +

-

E

σσσ σ ε

S

I

X X

X X

X

X X X

X

X X X

J X

=

×

∇

= H

H

curl J

JS I

d

L

=

∫ H o L =

Review on EM-1

I X

- - - - -

- -

- b

H and Current Density in a conductor with AC bias

V

L

AC Voltage

- - - -

a J

+

σ

-

σσ σ ε

S

I+I

~

=

×

∇

= H

H

curl J

Still correct ?

J d

J

H = +

×

∇

= ? J d

J

Conduction current

Extreme case:

Current in insulator (ρ ρ ρ ρ

= 0) with AC bias

=

×

∇ H J d

= ? J d

b

V

L

AC Voltage

- - - -

a

J

+

-

σ = 0 σ = 0 σ = 0 σ = 0 ε

S

~

- - - -

I

J d

J

H = +

×

∇

No Conduction current

(No real flow of free carriers)

Electrons cannot flow thru insulator No real flow of free carriers (ρρρρ_v=0) No conduction current

Extreme case:

Current in insulator (ρ ρ ρ ρ

= 0) with AC bias

b

V

L

AC Voltage

- - - -

a

J

+

-

σ = 0 σ = 0 σ = 0 σ = 0 ε

S

~

- - - -

V

AC Voltage

I

~

C

dt C dV

I d = dt

dV S

J d = C

I

Displacement current

Relationship between

J

and E-field (E or D) in the insulator

U

d dt

dV S

C a

J =

s V m

unit = F 2

s V V

C m 2

= 1

s m

C 1

= 2

dt ??

d

d

J = D ∇ × H =

dt dD

] / [ D t

=

Mathematical Derivation of general form including the conduction current (J)

continuity equation

Divergence of curl = 0 (because it is

)

Derivation of Current Continuity Equation

) (

)

( I x x

t x Q

I ∆

∆

∆ + +

=

x S t Q x

S

x I x x

I

∆

∆

∆

∆

∆ ) ( ) 1

( + − = −

⇒

t x

x J x

x

J _v

∆ ρ

∆

∆

∆ − = −

⇒ ( + ) ( )

t x

x J x x

J _v

t

x ∆

ρ

∆

∆

∆

∆

∆

= −



 



 + −

⇒

→

→

lim

lim

) ( )

(

dt d dx

dJ ρ_v

−

⇒ =

dt dρ_v

−

=

•

⇒ ∇ J I(x)

t Q

∆

∆

Continuity Equation Or

Current Conservation Equation

Review on EM-1

Dielectric material

D vs. J

- - - - -

+ + + + +

D = ∆ Q / ∆ S

Q -Q

∫

∫ • = ∇ •

=

v S

dv d

Q D S D

Conductor -

- - - -

+ + + + +

J = ∆ I / ∆ S

Q -Q

∫

∫ • = ∇ •

=

v S

dv d

I J S J

Divergence theorem Divergence theorem

+ + + + +

ρ

=

•

∇ D

dt d ρ

−

=

•

∇ J

Electric Flux Density (Charge Effect Density) Charge Flow Density

Review on EM-1

Equations in two different cases

Conductor

(both of them exist)

Perfect insulator

(no-conduction current)

symmetry

Integral form

dt d

d

J = D

=

×

∇ H

dt J + dD

Stokes’ theorem

Example: Find I d (Midterm Exam ?)

I

C

insulator

•

•

•

•

•

•

•

•

•

•

•

•

•

B(t) V t

dt

emf d Φ ω

0

cos

=

−

=

∴

A t

B ( ) Φ =

given A t

t V

B ω L

ω sin

)

( = −

Circular area = A

Total flux thru ‘A’

- + emf [ V ]

t dt CV

C dV I

I =

= = − ω

sin ω

d C S

where =ε

d

t d V

S ω

ωε

sin

−

=

Example: Another solution

I

C

vacuum

t V

V =

cos ω

Circular area = A - + V

d t

d V d

E = V =

cos ω

d t E V

D = ε = ε

cos ω

t d V

S dt

S dD SJ

I

=

= = − ωε

sin ω Same result

as the capacitor concept

Ampere’s law modified for the insulator

I

C

Circular area = A - + V

L1

I d

L

∫ H o L =

1

Conduction current

L2

0

2

=

∫ d = I

L

L H o

No Conduction current

Contradiction in series connection

dt S I d

d

L

L D

H = =

∴ ∫ o

Ampere-Maxwell’s Law

=I