Displacement Current

By: Dr. Ahmed M. AttiyaBy: Dr. Ahmed M. Attiya

Recall Ampere’s Lawp

∫ encIsdB∫ =⋅ 0μrrenc∫ 0μ

Imagine a wire connected to a charging or discharging capacitor. The area in the Amperian

loop could be stretched into the open region of the capacitor. In this case there would be current

passing through the loop, but not through the area p g g p gbounded by the loop.

If Ampere’s Law still holds, there must be a magnetic field generated by the changing E-fieldmagnetic field generated by the changing E field

between the plates. This induced B-field makes it look like there is a current (call it the displacementlook like there is a current (call it the displacement

current) passing through the plates.

Ampere’s Law and the Continuity p yEquation

• The differential form of Ampere’s law in the static case is

Th i i i iJH =×∇

• The continuity equation is

0≠∂

∂−=⋅∇t

qJ ev

∂t

Ampere’s Law and the Continuity p yEquation (Cont’d)

• In the time-varying case, Ampere’s law in the above form is inconsistent with the continuity yequation

( ) !!!!!!!!!!!0=×∇⋅∇=⋅∇ HJ ( ) !!!!!!!!!!!0×∇∇∇ HJ

Ampere’s Law and the Continuity p yEquation (Cont’d)

• To resolve this inconsistency, Maxwell modified Ampere’s law to readp

D∂tDJH c ∂

∂+=×∇t∂

d i displacementconduction current density

displacement current density

Ampere’s Law and the Continuity p yEquation (Cont’d)

• The new form of Ampere’s law is consistent with the continuity equation as well as with y qthe differential form of Gauss’s law

( ) ( ) 0∇∇∇∂∇ HDJ ( ) ( ) 0=×∇⋅∇=⋅∇∂

+⋅∇ HDt

J c

qqev

Displacement Currentp

• Ampere’s law can be written as

dc JJH +=×∇ dc

where

)(A/mdensitycurrentntdisplaceme 2=∂= DJ )(A/mdensity current nt displaceme=∂

=t

J d

Displacement Current (Cont’d)p ( )

• Displacement current is the type of current that flows between the plates of a t at o s bet ee t e p ates o acapacitor.

• Displacement current is the mechanism• Displacement current is the mechanism which allows electromagnetic waves to propagate in a non conducting mediumpropagate in a non-conducting medium.

• Displacement current is a consequence of h h i l ill fthe three experimental pillars of

electromagnetics and wireless systems.

Displacement Current in a pCapacitor

id ll l l i i h l• Consider a parallel-plate capacitor with plates of area A separated by a dielectric of permittivity ε and thickness d and connected to an ac generator:

+iAz

tVtv ωcos)( 0=+

z = d εicA

id )( 0

-z = 0 d

Displacement Current in a Capacitor (Cont’d)

• The electric field and displacement flux• The electric field and displacement flux density in the capacitor is given by

tdVa

dtvaE zz ωcosˆ)(ˆ 0−=−= • assume

fringing is

Th di l t t d it i it

dVaED z ωεε cosˆ 0−==

fringing is negligible

• The displacement current density is given by

tdVa

tDJ zd ωωε sinˆ 0=

∂∂=

Displacement Current in a pCapacitor (Cont’d)

• The displacement current is given by

tVAAJsdJI −=−=⋅= ∫ ωωε sindS

dd

dv

tVd

AJsdJI −=−=⋅= ∫ ωsin0

cIdtdvCtCV ==−= ωω sin0 conduction

current incurrent in wire

Conduction to Displacement pCurrent Ratio

• Consider a conducting medium characterized by conductivity σ and permittivity ε.y y p y

• The conduction current density is given by

EJ• The displacement current density is given by

EJ c σ=p y g y

EJ d ∂∂= ε

td ∂

Conduction to Displacement pCurrent Ratio (Cont’d)

• Assume that the electric field is a sinusoidal function of time:

ThtEE ωcos0=

• Then, 0

tEJtEJ

d

c

ωωεωσsin

cos

0

0

−==

tEJd ωωε sin0

Conduction to Displacement pCurrent Ratio (Cont’d)

• We have

0max

EJ

EJc

ωε

σ=

• Therefore

0maxEJd ωε=

Therefore

σmaxcJωε

=max

max

d

c

J

Conduction to Displacement pCurrent Ratio (Cont’d)

• The value of the quantity σ/ωε at a specified frequency determines the properties of the q y p pmedium at that given frequency.

• In a metallic conductor the displacement• In a metallic conductor, the displacement current is negligible below optical frequencies.

• In free space (or other perfect dielectric), the conduction current is zero and onlyconduction current is zero and only displacement current can exist.

Conduction to Displacement

6Humid Soil (εr = 30, σ = 10-2 S/m)

pCurrent Ratio (Cont’d)

104

105

106

102

103

10

goodconductor

100

101

10

σ

10-2

10-1ωε

100

102

104

106

108

1010

10-4

10-3

good insulator10 10 10 10 10 10

Frequency (Hz)

Complex PermittivityComplex Permittivity

• In a good insulator, the conduction current (due to non-zero σ) is usually negligible.) y g g

• However, at high frequencies, the rapidly varying electric field has to do work against molecularelectric field has to do work against molecular forces in alternately polarizing the bound electronselectrons.

• The result is that P is not necessarily in phase with E and the electric susceptibility and hencewith E, and the electric susceptibility, and hence the dielectric constant, are complex.

Complex Permittivity (Cont’d)p y ( )

• The complex dielectric constant can be written as εεε ′′′ j

S b i i h l di l i

εεε −= jc

• Substituting the complex dielectric constant into the differential frequency-domain form of Ampere’s law, we have

+′+′′=×∇ EjEH εωεω

Complex Permittivity (Cont’d)p y ( )

• The term ωε′′ E2 is the basis for microwave heating of dielectric materials.g

I b l i h di l i i fσεω =′′

• In tabulating the dielectric properties of materials, it is customary to specify the real part of the dielectric constant (ε′ / ε0) and the loss tangent (tanδ) defined asloss tangent (tanδ) defined as

σεδ′′

tanεωε

δ′

=′

=tan

ExampleExample