Maxwell’s Equations in Differential Form

dcici

idi

JJJtDJJH

MMMtBE

++=∂∂

++=×∇

−−=−∂∂

−=×∇

mv

ev

BD

ρ

ρ

=⋅∇

=⋅∇

tBM d ∂∂

=

,

tDJ d ∂∂

=

≡E

Electric field intensity [V/m]

≡B

Magnetic flux density [Weber/m2 = V s/m2 = Tesla]

≡iM

Impressed (source) magnetic current density [V/m2]

≡dM

Magnetic displacement current density [V/m2]

≡H

Magnetic field intensity [A/m]

≡iJ

Impressed (source) electric current density [A/m2]

≡D

Electric flux density or electric displacement [C/m2]

≡cJ

Electric conduction current density [A/m2]

≡dJ

Electric displacement current density [A/m2]

≡evρ Electric charge volume density [C/m3]

≡mvρ Magnetic charge volume density [Weber/m3]

Remarks:

1. Impressed magnetic current density ( iM

) and magnetic charge density ( mvρ ) are unphysical quantities introduced through “generalized” current to balance Maxwell’s equations.

2. Although unphysical, iM

, mvρ similar to iJ

and evρ can be considered as energy sources that generate the fields.

3. Through “equivalent principle” iM

and mvρ can be used to simplify the solutions

to some boundary value problems.

4. tBM d ∂∂

=

(magnetic displacement current density [V/m2]) is introduced

analogous to tDJ d ∂∂

=

(electric displacement current density [A/m2])

iJ

Current source

cJ

dtDJ d

∂

=

iJ

generates the electric displacement current

Integral form of Maxwell’s Equations

Elementary vector calculus:

Stokes’ Theorem: ( ) ∫∫∫ ⋅=⋅×∇CS

ldAsdA

• It says that if you want to know what is happening in the interior of a surface bounded by a curve just go around the curve and add up the field contributions.

Divergence Theorem: ( ) ∫∫∫∫∫ ⋅=⋅∇SV

sdAdVA

• In simple words, divergence theorem states that if you want to know what is happening within a volume of V just go around the surface S (bounding volume V ) and add up the field contributions. Null Identities:

( )( ) φφ −∇=⇔=×∇⇔=∇×∇

×∇=⇔=⋅∇⇔=×∇⋅∇

EEABBA

atic)(electrost0000

• The Divergence and Stokes’ theorems can be used to obtain the integral forms of the Maxwell’s Equations from their differential form.

• ∫∫∫∫∫∫ ⋅−⋅∂∂

−=⋅×∇⇒−∂∂

−=×∇S

iSS

i sdMsdBt

sdEMtBE

volume V

surface S

S

C

∫∫∫∫∫ ⋅−⋅∂∂

−=⋅⇒S

iSC

sdMsdBt

ldE

• ∫∫∫∫∫∫∫∫ ⋅+⋅+⋅∂∂

=⋅×∇⇔++∂∂

=×∇S

iSSS

i sdJsdEsdtDsdHJE

tDH

σσ

∫∫∫∫∫∫∫ ⋅+⋅+⋅∂∂

=⋅⇒S

iS

cSC

sdJsdJsdDt

ldH

Where EJc

σ=

• eV

evSV

evV

ev QdvsdDdvdvDD ==⋅⇒=⋅∇⇔=⋅∇ ∫∫∫∫∫∫∫∫∫∫∫ ρρρ

• ∫∫∫∫∫∫∫∫∫∫∫ ==⋅⇒=⋅∇⇔=⋅∇

Vmvm

Vmv

Vmv dvQsdBdvdvBB ρρρ

Since 00 =⋅⇒= ∫∫S

m sdBQ

S

C

S

V

Helmholtz Theorem

• Traditionally, Newtonian mechanic is formulated in terms of force ( F

) and torque (τ ),

,dtPdF

=

dtLd

=τ where L

is the angular momentum.

However such an approach to classical electromagnetism will be unnecessarily cumbersome. Instead, the description of electromagnetics starts with Maxwell’s equations which are written in terms of curls and divergences. The question is then whether or not such a description (in terms of curls and divergences) is sufficient and unique? The answer to this question is provided by Helmholtz Theorem • A vector field is determined to within an additive constant if both its divergence and its curl are specified everywhere. • Equivalent statement: A vector field is uniquely specified by giving its divergence and its curl within a region and its normal component over the boundary, that is if: S and C

are known and given by SM =⋅∇

,

CM

=×∇ and nM

(the normal component of M

on the boundary) is also known; then M

is

uniquely defined. Remark: Helmholtz’s theorem allows us to appreciate the importance of the Maxwell’s equations in which E

and H

are defined by their divergence and curl.

Ex.: Bt

E

∂∂

−=×∇ and ερevE =⋅∇

Irrotational & Solenoidal Fields (Use of Helmholtz Theorem)

Definition: • A field is irrotational if its curl is zero

ii FF ≡=×∇ 0

is irrotational • A field is solenoidal (divergenceless) if its divergence is zero

ss FF

≡=⋅∇ 0 is solenoidal Theorem: • A vector field which its divergence and curl vanishes at infinity can be written as the sum of an irrotational & a solenoidal fields. • According to the theorem stated above, the vector field M

can be written as

si FFM

+= (1) • Since iF

is irrotational then VFF ii −∇=⇒=×∇

0 whereV is a scalar function.

• Since sF

is solenoidal then AFF ss

×∇=⇒=⋅∇ 0 then (1) AVM

×∇+−∇=⇒

Constitutive Relations

ED

ε=

0

0

µµµεεε

µ

r

r

HB

===

≡ε permittivity [F/m] ≡oε vacuum permittivity = 8.85 ×10-12

[F/m] ≡rε Relative permittivity or dielectric constant [#] ≡µ permeability [H/m] ≡0µ free space permeability = 4π × 10-7 [H/m] ≡rµ relative permeability [#]

• We also write

mr χµ += 1 (1)

er χε += 1 (2) Where mχ and eχ are the magnetic and electric susceptibility, respectively. mχ , eχ are dimensionless. • Index of refraction is defined as

rrn µε= (3) ≡n index of refraction or phase index [#]

• If we are mostly concerned with non-magnetic materials then

rr n εµµµ =⇒=⇒≈ 01

Polarization and Magnetization • Polarization vector P

and magnetization vector M

are related to D

and E

and B

and

H

according to:

( )MHMHBEPD

+=+=

+=

000

0

)5( (4)

µµµ

ε

• Assuming EP e

χε 0= , then:

( )ED

EEEEPED ree

ε

εεχεχεεε

=

⇒=+=+=+= 00000 1

• Assuming HM m

χ= then

( )HB

HHMHB rom

µ

µµχµµµ

=

⇒=+=+= 1000

• ε and µ describe the macroscopic response of the media.ε characterizes the electric response while µ describes the magnetic response. In the following we assume our medium is nonmagnetic.

Homogeneous vs. Inhomogeneous, Isotropic vs. Anisotropic, Linear vs. Non-Linear • Ifε depends on position, i.e. ( )rε , media is non-homogeneous. • Ifε depends on the direction of the applied field, i.e., D

and E

are not co-linear, then the

medium is said to be anisotropic. Examples of anisotropic materials are Calcite (uniaxial) or topaz (biaxial). • In the case of anisotropic mediumε is a tensor (for our purposes a matrix of rank 2). We then write:

ED

ε= (1) where

=

z

y

x

zzzyzx

yzyyyx

xzxyxx

z

y

x

EEE

DDD

εεεεεεεεε

ε0 (2)

• Ifε depends on the magnitude of the applied field, i.e. ( )E

ε , we say medium is nonlinear. Note that in this case even though permittivity is a function of the filed strength, it can still be a scalar function. • An example of non-linear medium is when

( )2/1

2222

0

11 )3(−

−+= EBc

bεε ,

Whereb is the maximum field strength.

• Interesting thing about (3) is the fact that it describes the response of the vacuum, (proposed by Born & Infeld) in order to address the problem of vacuum infinite self-energy.

Infinite Self Energy • A charge particle can be thought as the localization of the charge density. As a charge distribution localizes to a point charge, its electromagnetic energy grows more and more and becomes unbounded1

( )2/1

2222

0

11

−+= EBc

bεε

. To avoid this infinite self-energy we can think that some saturation of field strength takes place, i.e., field strength has an upper bound. This

classical non-linear effect is given by

• However, there are few problems with Born & Infeld classical non-linear vacuum response. (1) The theory suffers from arbitrariness in the manner in which the nonlinearities occur. (2) There are problems with transitions to the quantum domain. (3) So far, there has been no experimental evidence of the existence of this kind of classical nonlinearities. • As to the last point, we may note that in the orbits of electrons in atoms, field strengths of 1011-1017 V/m are present. For heavier atoms, these fields can be even as large as 1021 V/m at the edge of the nucleus; yet ordinary quantum theory with linear superposition is sufficient to describe the observed phenomena with a high degree of accuracy.

HW: Consider a hydrogen atom unexcited and in thermal equilibrium. Calculate the magnitude of the electric field due to its nucleus at the site of its electron.

Temporal dispersion • Ifε depends on frequency, i.e. ( )ωε , we say the medium is dispersive (frequency dispersion)

220

2

0

1 )1(ωωγω

ωεεε

−++==

jp

r

• Note that from (1) we can write εεε ′′−′= j )2(

• Remarks: Temporal dispersion means that the parameters describing the medium response (e.g. ε and µ ) are functions of time derivatives. Spatial dispersion means that the parameters describing the medium response (e.g. ε and µ ) are functions of space derivatives. • If a medium is linear, homogeneous, and isotropic, we say the medium is simple.

1 Recall that the potential energy (U) corresponding to two charges q1 and q2 separated by a distance r is given by: U = q1 q2 /4 π ε0 r

x

y

A r

ra

1q

Observation Point z

Electric Field • Electric field due to a point charge in origin

30

13

0

12

0

1

44ˆ

4 rrq

rrq

raqE r

πεπεπε=== where

rr

rrar

==ˆ and we use the shorthand notation

rr = . • Electric filed due to a point charge not at the origin

30

13

0

12

0

1

44ˆ

4 rrrrq

RRq

RaqE R

′−

′−===

πεπεπε

• Superposition principle

( ) ( ) ( )

+

′−

′−+

′−

′−+

′−

′−=

+++=

+++=

33

333

2

223

1

11

0

3

3

333

2

223

1

11

0

2

3

332

2

222

1

11

0

41

41

ˆˆˆ4

1

rrrrq

rrrrq

rrrrq

RRq

RRq

RRq

Raq

Raq

RaqE RRR

πε

πε

πε

∑= ′−

′−=

N

k k

kk

rrrr

qE1

304

1

πε

x

y

z A

r r ′ 1q

rrR ′−=

x

y

z

1r ′

2r ′ 3r ′

1q

2q

3q

A

1R

2R

3Rr

Observation Point

Observation Point

x

y

z

r′

r rrR ′−=

Differential surface charge density sρ′

A (Observation point)

Electric Field & Potential due to Continuous Charge Distribution • Volume charge density, ( ) ( )zyxr vv ′′′′=′′ ,,ρρ

vdrrrr

vdRavd

RREd

v

vR

v

′′′−

′−=

′′=′′=

ρπε

ρπε

ρπε

30

20

30

41

ˆ4

14

1

∫∫∫∫∫∫′′

′′′−

′−=′′=

vv

vv

R vdrrrrvd

R

aE ρπε

ρπε 3

02

0 41ˆ

41

∫∫∫ ∫∫∫∫∫∫′ ′′ ′−

′′=

′′=

′′=

v v

v

v

vv

rrvd

Rvd

RvdV

ρπε

ρπε

ρπε 000 4

14

14

1

• Surface charge density, ( ) ( )zyxr ss ′′′′=′′ ,,ρρ

sdrrrrsd

RRsd

RaE s

ss

sss

R ′′′−

′−=′′=′′= ∫∫∫∫∫∫

′′′

ρπε

ρπε

ρπε 3

03

02

0 41

41ˆ

41

∫∫∫∫∫∫′′′ ′−

′′=

′′=

′′=

s

s

s

s

s

s

rrsd

Rsd

RsdV

ρπε

ρπε

ρπε 000 4

14

14

1

• E

and V due to a line charge, ( ) ( )zyxr ll ′′′′=′′ ,,ρρ

ldrrrrld

RRld

RaE l

ll

lll

R ′′′−

′−=′′=′′= ∫∫∫

′′′

ρπε

ρπε

ρπε 3

03

02

0 41

41ˆ

41

∫∫ ∫′′ ′ ′−

′′=

′′=

′′=

l

l

l l

ll

rrld

Rld

RldV

ρπε

ρπε

ρπε 000 4

14

14

1 x

y

z

rrR ′−=

r

r ′ Differential line charge lρ ′

x

y

z A (Observation point)

Differential volume charge density vρ′

r ′

r rrR ′−=

Remark: If you have forgotten the differential length, surface, and volume elements for rectangular, cylindrical, or spherical, you may want to revisit these. See also the end of this note set.

Electric Field of a Dipole

drdrr

drr

rr

rrqE

>>+=

−=

−=

&2

2

4

2

1

32

23

1

1

0πε

Then

⋅−≈+

⋅+≈−

+

+−

−

−=

−−

−−

233

233

330

2312

2312

2

2

2

24

rdrrdr

rdrrdr

dr

dr

dr

drqE

πε

( ) ( )

⋅−+−

⋅+−≈ −

223

0 2312

2312

4 rdrdr

rdrdrrqE

πε

−

⋅=

−

drr

drrqE

20

3

34πε

Recall pdq = is the dipole moment, then

( )

−

⋅= pr

rpr

rE

230

34

1πε

• For our coordinate system zapp ˆ=

x

y

z

q+

q−

d

r

1r

2r

A

• r is the position vector in spherical coordinate, then let us express E

in the spherical coordinate

( ) ( ) ( )( ) ( ) ( )

θφθφθ

φφθφθφθθφθφθ

φθφθφθ

φ

θ

φφθθ

cossinsincossin

0cossinsinsincoscoscos

cossinsincossin

ˆ,,ˆ,,ˆ,,

ˆ,,ˆ,,ˆ,,

rzryrx

AAA

AAA

azyxAazyxAazyxAA

arAarAarAA

z

y

xr

zzyyxx

rr

===

−−=

++=

++=

with

( ) paaapppp

ppp

rzz

zr

θ

φ

θ θθθθ

ˆsinˆcosˆ0

sincos

−==⇒

−=

• Or finally from

( )

−

⋅= pr

rpr

rE

230

34

1πε

we get

( ) ( )

−−

−⋅= paaar

rpaaar

rE rr

rrθ

θ θθθθπε

ˆsinˆcosˆˆsinˆcosˆ314

123

0

⇒

[ ]θθθπε

aar

pE r ˆsinˆcos24 3

0

+=

, where rr =

HW: Show that potential at point A for an electric dipole is given by

20

20

4ˆ

4ˆ

rap

rapV rr

πεπε⋅

=⋅

=

r

q+

q−

A

d

E

Electric Polarization P

v

pP

vN

kk

v ′∆=

∑′∆

=

→′∆

1

0limt

p [C·m] Electric dipole moment P

[C/m2] Electric polarization vector N [#/m3] is the number of dipoles per unit volume • P

[C/m2] is the volume density of electric dipole moment p [C·m] Note P

and D

have the same units [C/m2]: PED

+= 0ε

• Polarization vector P

may come to exist due to (a) induced dipole moment, (b)

alignment of the permanent dipole moments, or (c) migration of ionic charges.

• In differential form: vdpdP′

=

Potential due to Bound (Polarized) Surface & Volume Charge Densities A dielectric of volume v′ is polarized. We want to calculate the potential V [Volt] set up by this polarized dielectric.

• Potential due to a single dipole

204ˆR

apV R

πε⋅

=

x

y

z

r ′

r

A

R

Differential volume element vd ′

v′

x

y

z

( )zyxSource ′′′ ,,

( )zyxAnObservatio ,,

R

r

r ′

• An elemental electric dipole, having a differential electric dipole moment of pd [C·m],

will set up a differential potential 204ˆRapddV R

πε⋅

=

But from our definition of polarization we ⇒′= vdPpd

vdR

aPRapddV RR ′⋅

=⋅

= 20

20 4

ˆ4

ˆπεπε

Total potential V is found by integrating the above:

vdR

aPVv

R ′⋅= ∫∫∫

′2

0

ˆ4

1

πε

Where ( ) ( ) ( )22222 zzyyxxRR ′−+′−+′−==

Note that 2

ˆ1Ra

RR=

∇′ (see Remarks below) then

∫∫∫′

′

∇′⋅=

v

vdR

PV 14

1

0

πε

and furthermore

( ) ⇒∇′⋅+⋅∇′=⋅∇′ fAAfAf ( ) AfAffA

⋅∇′−⋅∇′=∇′⋅

* Let PA

= and R

f 1= then

∫∫∫ ∫∫∫∫∫∫′ ′′

′

⋅∇′−′

⋅∇′=′

∇′⋅

v vv

vdPR

vdRPvd

RP

11

Use divergence theorem ⇒

vdPR

sdRPvd

RP

vv s

′

⋅∇′−′⋅=′

∇′⋅ ∫∫∫∫∫∫ ∫∫

′′ ′

11

The potential then can be written as

′⋅∇′

−′′⋅

= ∫∫ ∫∫∫′ ′S v

n vdR

PsdRaPV

ˆ

41

0πε,

Where na′ˆ is perpendicular to surface S ′ bounding volume v′ . Compare above to the previously obtained expressions for V due to surface and volume charge densities, i.e.:

∫∫′

′′=

s

s

RsdV ρ

πε041 and ∫∫∫

′

⇒′′

=v

v

RvdV ρ

πε041

v

sn

P

aP

ρ

ρ

′=⋅∇′−

′=′⋅

ˆ

Or in general, dropping the prim notation since we know that integration is carried with respect to the prim coordinate, we define

• Bound or polarized surface charge density: nsP aP ˆ⋅=

ρ [C/m2]

• Bound or polarized volume charge PvP

⋅−∇=ρ [C/m3]

• A polarized dielectric can be replaced by a bound (polarized) surface and volume charge densities ( sPρ & vPρ ). The potential setup by these bound charges then can be calculated. Remarks: Few useful identities

( ) ( )RRfaRf

RR

RRaR

Ra

R

Ra

R

R

R

R

R

∂∂

=∇

===∇

−=

∇

=

∇′

ˆ

ˆ

ˆ1

ˆ1

2

2

( )RR

32 41 πδ=∇− , or ( )rr

rr′−=

′−∇−

32 41 πδ

Generalized Gauss’ Law & Constitutive Relation ED

ε= • In free space

0ερvE =⋅∇

.

• When a medium is polarized we must take into account the effects of the bound charges, hence

( ) vv

vvpv

PEPE

PE

ρερε

εερ

ερ

ερ

=+⋅∇⇒=⋅∇+⋅∇

⇒⋅∇

−=+=⋅∇

00

0000

Let’s define PED

+= 0ε then

vD ρ=⋅∇

← Generalized Gauss’ Law • Also note that for PED

+= 0ε if EP e

χε0= then ( ) EED re

εεχε 00 1 =+=

Where er χε += 1 then

EED r

εεε == 0 where rεεε 0=

Magnetization & Permeability • Magnetic materials exhibit magnetic polarization ( M

, magnetization) when subjected

to an applied magnetic field • This magnetization is the result of alignment of the magnetic dipoles of material with the applied magnetic field. This is similar to electric polarization which is the result of alignment of electric dipoles of the material with the applied electric field.

Magnetic Dipole & Magnetic Dipole Moment • To accurately describe magnetic behavior of materials quantum theory of matter is needed. However, accurate qualitative and quantitative description can be found using simple atomic model (semi-classical) • The electron orbiting the nuclei can be thought of as a small current loop of area idswith current iI • As long as loop is small, its shape can be circular, square, or any other closed curve

• The magnetic dipole moment is given by

iiii dsInmd ˆ= [A·m2], where in is perpendicular to the loop surface. • The magnetic field of the current carrying loop at large distance is similar to the field of a linear bar magnet, i.e., a magnetic dipole. • For a material of volume v∆ which contains mN magnetic dipoles (orbiting electrons) per unit volume, the total magnetic moment is given by

∑∑∆

=

∆

=

==vN

iiii

vN

iit

mm

dsInmdm11

ˆ

+

-

- - −e

−e

−e

in

iI idsids

≡ ≡

in

• The magnetic polarization, i.e. magnetization ( M

) is given by

∆

=

∆

=

∆

= ∑∑∆

=→∆

∆

=→∆→∆

vN

iiii

v

vN

ii

vt

v

mm

dsInv

mdv

mv

M10100

ˆ1limt1limt1limt [A/m]

• Note that magnetization ( M

) is the volume density of the total magnetic dipole moment

( tm ), and also the fact that magnetization ( M

) has the same units as the magnetic field intensity, H

[A/m].

• In absence of an applied field ( 0=aB

) the magnetic dipoles point in random directions.

However, when 0≠aB

, the dipoles will experience a torque given by

iaiiaiaiiaiaiai BdsIBnBdsIBmdBmdBmd Ψ=∠=∠=×=∆ sin),ˆsin(),sin(τ

• Subjected to the above torque, the magnetic dipoles realign themselves such that their moment ( imd ) is collinear with aB

(see figure in the next page)

00 →∆⇒→Ψ τi Remark: Comparing the similarities between the torque & potential energy for electric & magnetic dipoles

aE

aB

EpdBmd

×=∆

×=∆

τ

τ

aE

aB

EpdUBmdU

⋅−=∆

⋅−=∆

• From next page figure we see that in absence of an applied magnetic field, we can write

aHB

0 )1( µ= . But, when a magnetic material is present, a magnetic polarization ( M

) is also present

and an additional term must be added to (1). In order to take into account the influence of the material, we write

( )MHMHB aa

+=+= 000 )2( µµµ

• However, M

is ultimately related to the applied field aH

. If we assume

am HM

χ= )3( , Where mχ is a scalar (or tensor) function then we have

[ ] aaram HHHB

µµµχµ ==+= 00 1 )4( , Where mr χµ += 1 is the relative permeability and µ is the permeability.

Bound Magnetization Current Density • Recall that for an electric field applied to a medium we had

nsP aP ˆ⋅=

ρ

PvP

⋅−∇=ρ , where P

is the electric polarization, vPρ and sPρ are the volume and

surface bound charges, and na is the normal to the surface. • Similarly, for magnetic field applied to a medium we have

nsm aMJ ˆ×=

MJvm

×∇=

• Here, smJ

is the bound magnetization surface current density [A/m], vmJ

is the bound magnetization volume current density [A/m2], and na is the normal to the surface. Remark: The origin of magnetization ( M

) can also be visualized by the following:

• When 0≠aB

, the magnetic moments line up with aB

to minimize the potential energy as shown in the figure. • Since the number of dipoles is very large and therefore they are closely packed, in the limit, the currents of the loops within the interior part of the medium will cancel each other and only a surface current ( smJ

) on the exterior of the slab remains.

• This bound magnetization surface current density ( smJ

) is responsible for producing the

magnetization ( M

). • So far we have only considered the magnetic moment of the orbiting electron; however, a magnetic moment can also be assigned to the spin of electron. • Only electrons in the atomic shell that are not completely filled will contribute to the spin magnetic moment. • In general the magnitude of the spin magnetic moment is 24109 −×±≈ [A·m2]. • There is also a magnetic moment associated with the nucleus .

DC Conductivity • Consider a small cylinder containing N electrons per unit volume, where electrons are moving with velocity v .

≡N Number of electrons per unit volume [1/m3] ≡v Velocity vector of electrons

≡e Electron charge ≡n Normal to the surface ≡∆V Volume of the cylinder

• The total chare ( Q∆ ) contained within the volume ( V∆ ) is given by

VeNQ ∆=∆ , where tvnSV ∆⋅∆=∆ ˆ hence

tvnSeNQ ∆⋅∆=∆ ˆ . This implies

• We define JveN

= , where J

is the current density vector [A/m2] and ItQ ∆=∆∆ ; then we have SnJI ∆⋅=∆ ˆ

Remark: • SnJI ∆⋅=∆ ˆ

can be written as dsnJdI ˆ⋅=

in differential form,

which implies

∫∫ ⋅=⇒ dsnJI ˆ

← This is our standard equation for calculating current from current density. • Let us assume a linear relationship between velocity ( v ) and electric filed ( E

), i.e.,

Ev µ−= , where µ is called mobility [m2/V·s] (note E

and v are anti-parallel)

• Then EeNveNJ

µ−== , for electron 1910602.1 −×−=−= qe [C]

ENqJ

µ=⇒ • Compare the above to ⇒= EJ s

σ µσ Nqs = . This says that static conductivity is the

product of electron charge, electron density, and electron mobility. • In our analysis so far we have only considered the electrons, however when positive charges (ions of holes) are present we must consider the contributions of both carriers to the conductivity. The static conductivity is then modified according to:

hhees NqNq µµσ += ≡eµ Electron mobility ≡hµ Hole mobility

eN and hN are electron and holes densities [1/m3]

tvn ∆⋅

S∆

v

n vnSeNItQ ⋅∆=∆=∆∆ ˆ

E

v -

Time Harmonic or Sinusoidal Steady State Electromagnetic Fields • Assuming time harmonic fields, the instantaneous field ( )tzyxE ,,,

and the complex

spatial field ( )zyxE ,,

are related by ( ) ( )[ ]( ) ( )[ ]

tj

tj

ezyxHtzyxHezyxEtzyxE

ω

ω

,,Re,,,,,Re,,,

==

Conductivity (DC & AC) • The Amper’s law given by dcisi JJJ

tDEJtzyxH

++=

∂∂

++=×∇ σ),,,( can be

written as ),,(),,(),,( zyxEjzyxEJJJJzyxH siDCi

ωεσ ++=++=×∇

• Whereε in general is complex: εεε ′′−′= j and sσ is the static (DC) conductivity (this is due to free carriers; i.e., electrons at 0=ω ). • The Ampere law then can be written as:

( ) ( ) EjEJEjEJEjjEJH eisisi

εωσεωεωσεεωσ ′++=′+′′++=′′−′++=×∇

where we have defined the followings: asse σσεωσσ +=′′+= ; where eσ is the equivalent (effective) conductivity [1/Ω·m]

≡′′= εωσ a Alternating (AC) conductivity [1/Ω·m]

+=

tors)semiconduc(for )conductors(for

qNqNqN

hhee

ees µµ

µσ = static (DC) conductivity [1/Ω·m]

• Note sσ is due to free charges at 0=ω (a signature of true conductors). • aσ is due to “resistance” of the dipoles as they attempt to align (rotate) themselves with the applied field. • The phenomenon of dipole rotation, which contributes to aσ is sometimes called dielectric hysteresis. • For good dielectrics such as glass or plastic 0≈sσ , but these materials when exposed to alternating fields ( 0≠aσ ) can dissipate large amount of energy. Example of large aσ and its application are: - microwave cooking - selective heating of human tissue - removing sulfur from mineral coal to produce clean coal (selective heating)

• Note deceiei JJJEjEJH

++=′++=×∇ εωσ ≡iJ

Impressed current density

( ) ( ) EEEJ sasece

εωσσσσ ′′+=+=≡ : Effective conduction current density

EjJde

εω ′≡ : Effective displacement current density

Loss Tangent • Note that Amper’s law given by EjEJH ei

εωσ ′++=×∇ can be rewritten as:

( ) EjjJEjjJH eie

i

δεω

εωσεω tan11 −′+=

′−′+=×∇ , where

≡eδtan Effective electric loss tangent

as

ssasee

δδεε

εωσ

εωεω

εωσ

εωσ

εωσ

εωσδ

tantan

tan

+=′′′

+′

=′′′

+′

=′

+′

=′

=

with

εωσ

δ′

= sstan : Static (DC) loss tangent

εεδ′′′

=atan : Alternating (AC) loss tangent

• Manufacturer usually provides loss tangent or the conductivity. • Note that in the above discussion we have expressed the conduction (DC) and dielectric losses (AC) in terms of effective conductivity ( eσ ) or effective loss tangent ( eδtan ). We could have also formulated the problem in terms of complex permittivity.

• To see this we write: EjJEjjJH cie

i

ωε

εωσεω +=

′−′+=×∇ 1 , where

)()(1 εωσε

ωσσε

εωσεε ′′+−′=

+−′=

′−′= sase

c jjj

• In the expression for cε the free carrier losses and dielectric losses are clearly evident. • Remark: The presence of static conductivity as a separate mechanism of loss in addition to the dielectric loss (ε ′′ ) can also be observed in the Kramers-Kronig relations which connects the real and imaginary parts of the dielectric constant. When a medium has static conductivity sσ then Kramers-Kronig relations are given by

222 ,, σµε

111 ,, σµε 1

2

n

y∆

x∆

0C

0Sz

x

y

( ) ( )

( ) ( ) ωωω

εωεωπ

ωεωε

ωωωεωε

πω

ωσωεωε

′−′

′′+==′

′−′

−′−==′′

∫

∫∞+

+∞

dP

dPs

022

0

022

0

}]{Im[21]Re[)(

1}]{Re[2]Im[)(,

where P stands for the principle value integral.

Boundary Conditions • Maxwell’s equations in differential forms are point equations; i.e. they are valid when fields are: single valued, bounded, continuous, and have continuous derivatives. • When boundaries are present, fields are discontinuous; hence to find the fields we must rely on their integral form. • Boundary conditions for tangential H

:

Assume finite conductivity ( )∞≠21 ,σσ and no sources on boundary ( )0,0 == ii JM

∫∫∫∫∫ ⋅∂∂

+⋅=⋅000 SSC

sdDt

sdEldH σ (1)

• Taking the limit of the both sides of Eq. (1), the Left hand side (LHS) can be written as: [ ]

( ) xxx

yC

y

axHHaxHaxH

ldHldHldH

ˆˆˆ

limlim

2121

2211000

∆⋅−=∆⋅−∆⋅=

⋅+⋅=⋅ ∫∫∫ →∆→∆

• The first term on the right hand side (RHS) of Eq. (1) can be written as:

[ ] 0ˆlim

ˆlimlim

0

0000

=⋅∆∆=

⋅=⋅

→∆

→∆→∆ ∫∫∫∫

zy

Szy

Sy

ayxE

adxdyEsdE

σ

σσ

• The second term on the RHS of Eq. (1) can be written as:

0)ˆ(limˆlimlim000

00

=∆∆⋅∂∂

=⋅∂∂

=⋅∂∂

→∆→∆→∆∫∫∫∫ zyS

zySy

ayxDt

adxdyDt

sdDt

• Putting it all together:

( ) ( ) 0ˆ0ˆ 1221 =−⋅⇒=∆−⋅ HHaxHHa xx

.

• Note that:

≡⋅ 2ˆ Hax Tangential component of 2H

WRT the interface,

≡⋅ 1ˆ Hax

Tangential component of 1H

WRT the interface.

• Also the fact that we can carry the same analysis in the y-z plane which results in

( ) 0ˆ 12 =−⋅ HHaz

, with 2ˆ Haz ⋅ and 1ˆ Haz ⋅ designating the tangential components of the

H

fields. The conclusion is then the following: tangential components of H

are continuous across the boundary between two dielectrics. This all can be summarized as

( ) 0ˆ 12 =−× HHn

Boundary Condition on Normal Components (Not corrected) Medium (1) and (2) are non conductors (dielectrics) ( ∞≠21 ,σσ ) and there are no sources at the boundary 0== mses ρρ

∫∫∫∫∫ =⋅v

vdvsdD ρ

LHS: [ ] [ ]

( )yyy

yyyyy

aADaAD

adxdzDadxdzDsdDsdDsdD

ˆˆlim

ˆˆlimlimlim

01020

1201200

−=

⋅−⋅=⋅+⋅=⋅

→∆

→∆→∆→∆ ∫∫∫∫∫∫ ∫∫∫∫

RHS: [ ] 0limlimlim 000000

==∆=∆=→∆→∆→∆ ∫∫∫ svyvy

vvy

AyAyAdv ρρρρ

Then ( ) ⇔=⋅− 0ˆ12 yaDD

( ) ( ) 0ˆ0ˆ 112212 =−⋅⇔=−⋅ EEnDDn

εε

222 ,, σµε

111 ,, σµε 1

2

n

z

x

y

0A

0A

222 ,, σµε

111 ,, σµε 1

2

n

z

x

y

Summary of boundary conditions General Case: • ( ) sMEEn

−=−× 12ˆ

sM : Fictitious magnetic current density [V/m] • ( ) sJHHn

=−× 12ˆ

sJ

: Electric surface current density [A/m] • ( ) esDDn ρ=−⋅ 12ˆ

esρ : Electric surface charge density [C/m2] • ( ) msBBn ρ=−⋅ 12ˆ

msρ : Fictitious magnetic surface charge density [Weber/m2] Boundary Conditions Between to Perfect Dielectrics:

( ) 0ˆ 12 =−× EEn

, ( ) 0ˆ 12 =−× HHn

,

( ) 0ˆ 12 =−⋅ DDn

, ( ) 0ˆ 12 =−⋅ BBn

Boundary Conditions for Two Media in which One Medium Is a Perfect Conductor ( ∞=1σ ), With no Sources Present ( 0=sM

, 0=msρ ):

• In medium-1, since perfect conductor 011 ==⇒ DE

then 111 0 Bt

Bt

E

∂∂

=⇒∂∂

−=×∇

. But this means that 1B

must be a constant function of time which contradicts the assumption of time varying electric and magnetic fields; i.e. the electrodynamics assumption. Therefore, 011 == HB

• ( ) 0ˆˆ 212 =×⇒−=−× EnMEEn s

Electric filed has no tangential component on the boundary between perfect conductor and dielectric. • ( ) ss JHnJHHn

=×⇒=−× 212 ˆˆ

Tangential component of H

is discontinuous by amount of surface current sJ

at the boundary between perfect conductor and dielectric. • sJ is the surface current due to the free charges on the metal (not the bound charges)

• ( ) eses DnDDn ρρ =⋅⇒=−⋅ 212 ˆˆ

Electric field has only normal component on the boundary between perfect conductor and dielectric. • ( ) 0ˆˆ 212 =⋅⇒=−⋅ BnBBn ms

ρ

Magnetic field has no normal component on the boundary between perfect conductor and dielectric. Boundary Conditions Between Two Medium one of which Is a Perfect Magnetic Material (the medium has infinite magnetic conductivity, i.e.

01 =tH

) and no sources are present (0=esρ , 0=sJ

)

• Here 00 11 =⇒= BH

, 011 == DE

• ( ) ss MEnMEEn

−=×⇒−=−× 212 ˆˆ

Electric filed is tangential to the boundary • ( ) 0ˆˆ 212 =×⇒=−× HnJHHn s

Magnetic filed has no tangential component on the boundary • ( ) 0ˆˆ 212 =⋅⇒=−⋅ DnDDn es

ρ

Electric filed has no normal component at the boundary • ( ) msms BnBBn ρρ =⋅⇒=−⋅ 212 ˆˆ

Magnetic field is normal to the boundary

Metal

E

E

Differential length elements Rectangular Coordinate System: Cylindrical Coordinate System: Spherical Coordinate System: