2-1

Chapter 2 Maxwell’s Equations

ECE 3317Dr. Stuart Long

https://learnodo-newtonic.com/maxwell-contributions

2-2Electromagnetic Fields

Four vector quantities

E electric field strength [Volt/meter]=[kg-m/sec3-Amp]

D electric flux density [Coul/meter2]=[Amp-sec/m2]

H magnetic field strength [Amp/meter]=[Amp/m]

B magnetic flux density [Weber/meter2] or [tesla] =[kg/Amp-sec2]

each are functions of space and timee.g. E(x,y,z,t)

J electric current density [Amp/meter2]

ρv electric charge density [Coul/meter3]=[Amp-sec/m3]

Sources generating electromagnetic fields

2-3

0

v

t

t

ρ

∂∇× = −

∂∂

∇× = +∂

∇⋅ =∇ ⋅ =

BE

DH J

BD

Maxwell’s Equations

(time dependant, differential form)

https://phys.libretexts.org

2-4MKSA units

length – meter [m]mass – kilogram [kg]time – second [sec]

current – Ampere [Amp]

Some common prefixes and the power of ten each represent are listed below

atto - a - 10-18

femto - f - 10-15

pico - p - 10-12

nano - n - 10-9

micro - μ - 10-6

milli - m - 10-3

mega - M - 106

giga - G - 109

tera - T - 1012

peta - P - 1015

exa - E - 10 18

centi - c - 10-2

deci - d - 10-1

deka - da - 101

hecto - h - 102

kilo - k - 103

André-Marie Ampère http://en.wikipedia.org/wiki/Andr%C3%A9-Marie_Amp%C3%A8re

2-6

Gradient ∇(Scalar) = Vector

“Del” Operator

ˆ ˆ ˆx y zφ φ φφ ∂ ∂ ∂

= + +∂ ∂ ∂

x y z∇

2 2 22

2 2 2x y z∂ ∂ ∂

= + +∂ ∂ ∂

∇Laplacian

ˆ ˆ ˆ x y z

∂ ∂ ∂⇔ + +

∂ ∂ ∂x y z∇

∇2 (Scalar) = Scalar

∇2 (Vector) = Vector

[2.10]

In symbolic form

2-7

Divergence ∇ (Vector) = Scalar

Curl

yx zAA A

x y z∂∂ ∂

⋅ = + +∂ ∂ ∂

A∇

ˆ ˆ ˆy yx xz zA AA AA A

y z z x x y∂ ∂ ∂ ∂∂ ∂ × = − + − + − ∂ ∂ ∂ ∂ ∂ ∂

A x y z∇

In Cartesian coordinates with ˆ ˆ ˆx y zA A A= + +A x y z

ˆ ˆ ˆ x y z

∂ ∂ ∂⇔ + +

∂ ∂ ∂x y z∇

( )is not really a vector does not existNote: All of above only true for Cartesian coordin .

ates

⋅A∇ ∇

“Del” Operator

∇ x (Vector) = Vector

2-8

( )

( ) ( ) ( )

( ) ( ) 2

0

0φ

⋅ × ≡

× ≡

⋅ × ≡ ⋅ × − ⋅ ×

× × = ⋅ −

A

A B B A A B

A A A

∇ ∇

∇ ∇

∇ ∇ ∇

∇ ∇ ∇ ∇ ∇

Vector Identities

2-9Curl Operator

z

y∆y

∆z

Curl measures the tendency of a vector field to rotate (or circulate) about an axis.

𝛁𝛁 × 𝑨𝑨 ⋅ �𝒙𝒙 =�𝑨𝑨 ⋅ 𝐝𝐝𝒍𝒍𝚫𝚫𝒚𝒚𝚫𝚫𝒛𝒛

2-10

xˆzA z

ˆyA yy

z

ˆxA x

Divergence Operator

𝛻𝛻 ⋅ 𝐴𝐴 =∮𝐴𝐴 ⋅ �𝑛𝑛 𝑑𝑑𝑆𝑆Δ𝑥𝑥Δ𝑥𝑥Δ𝑥𝑥

The Divergence measures the net outward flux density per unit volume.

2-11Gradient Operator

The Gradient of a function (at any point) is a vector that points in the direction of greatest increase of the function at that point.

mathinsight.org

ˆ ˆ ˆx y zφ φ φφ ∂ ∂ ∂

= + +∂ ∂ ∂

x y z∇

2-12

( )

( )

0

t

t

t

∂× = +

∂∂ ⋅ × = ⋅ + ⋅ ∂

∂= ⋅ + ⋅

∂

DH J

DH J

J D

∇

∇ ∇ ∇ ∇

∇ ∇

Law of Conservation of Electric Charge (Continuity Equation)

v

tρ∂

⋅ = −∂

J∇

2 3

1 1 1sec sec

coul coulm m m

⋅ ⋅ = ⋅

Flow of Electric Current out of volume

(per unit volume)

Rate of decrease of Electric Charge (per unit

volume)

[2.20]

2-13Maxwell’s Equations

Deco

Time Dependant

Time Independ

0

0

ent (Static)

0

v

v

t tρ

ρ

∂ ∂× = − × = + ⋅ = ⋅ =

∂ ∂

× = × = ⋅ = ⋅ =

B DE H J B D

E H J B D

∇ ∇ ∇ ∇

∇ ∇ ∇ ∇

uples and is a function of ; is a function of vρ⇒E H E H J

daviddarling.info

2-14

0

0

Source-free

Region

0

t t

=

∂ ∂× = − × = ⋅ = ⋅ =

∂ ∂

J

B DE H B D∇ ∇ ∇ ∇

Note: [ lim 0 (DC case) gives static equations]

0

0

Time Harmonic Ca

se v

vj ρ

j j ρ

ω ω

ρ

ω ω

→ ⇒ ∇ ⋅ = −

=

× = − × = + ⋅ = ⋅ =

J

E B H D J B D

v

∇ ∇ ∇ ∇

Maxwell’s Equations

rampantscotland.com

2-15

Negative charge

Positive charge

max0

vρ = +⋅ =J∇

0max out

vρ =⋅ =J∇

0)0

(t

tω ==

)2

T4

(

t

t πω =

=

v

tρ∂

⋅ = −∂

J∇ Continuity equation

2-16

0max in

vρ =⋅ =J∇

3(

3T4

)2

t

t πω =

=

max0

vρ = −⋅ =J∇

( )

T2

t

t

ω π=

=

Continuity equation

Negative charge

Positive chargev

tρ∂

⋅ = −∂

J∇

animation

2-17

[ ]

( )

0 ( )

0 ( )

0

t

t

tρt t

ρtρ

∂× = +

∂∂

⋅ × = ⋅ + ⋅∂

∂= ⋅ + ⋅

∂∂ ∂

= − + ⋅∂ ∂

∂= ⋅ −

∂⋅ =

v

v

v

DH J

DH J

J D

D

D

D

∇

∇ ∇ ∇ ∇

∇ ∇

∇

∇

∇

Usual EM fields problem:Sources and are known and sa

tisfy the

+ = 0

.

v

v

ρ

ρt

∂⋅

∂

continuity equatiJ

J

on

∇

Lets find out:

Are Maxwell’s 4 equations

independent?

2-18

Similarly from:

[ ]

( )

0

0

t

t

t

∂× = −

∂∂ ⋅ × = ⋅ − ∂

∂= ⋅

∂

= ⋅

BE

BE

B

B

∇

∇ ∇ ∇

∇

∇

So divergence equations are NOT independent

2-19

} 3 equations (x, y, z components)t∂

× = −∂BE∇

} 3 equations (x, y, z components)6 equationst

∂× = +

∂DH J∇

} 3 unknowns ( Ex , Ey , Ez )} 3 unknowns ( Dx , Dy , Dz )} 3 unknowns ( Hx , Hy , Hz )} 3 unknowns ( Bx , By , Bz )

12 unknowns

EDHB

Need 6 more equations or 2 more vector equations…

2-20Constitutive Relations

Characteristics of media relate D to E and B to H.

0

0

( = permittivity )

( = permea

=

bili t= y)

r

rµ µ

ε ε ε

µ µ

εD E =

B H =

-12 -90

-70

[F/m1 = = 8.85 x10 1036

= 4 x

]

1 [ / ]0 H m µ µ

ε επ

π

≈

=

[2.24]

[2.25]

[p. 35]For free space, vacuum, or air

2-21

μ or ε Independent of Dependent on

space homogenous inhomogeneous

frequency non-dispersive dispersive

time stationary non-stationary

field strength linear non-linear

direction of isotropic anisotropicE or H

Terminology

2-22Isotropic Materials

ε (or μ) independent of direction

It is a scalar quantity which means E||D (and H||B )

x xD = Eε

y yD = Eε

yE

xE

x

y x xB = Hµ

y yB = Hµ

yH

xH

x

y

2-23

results in E and DNOT parallel

x x

y

x x

y y

zz z z

y

D 0 0 ED 0 0 ED 0 0 E

D ED ED E

x x

zz

yy

εεε

εεε

=

= =

=

Anisotropic Materials

dependent on directionε (or μ) is a tensor (can be

written as a matrix)

x x xD = Eε

y y yD = Eε

yE

xE

x

y x x xB = Hµ

y y yB = Hµ

yH

xH

x

y

2-24Poynting's Theorem

( )

(

( ) ( ) ( )

0)

t t

t t

∂ ∂ ⋅ − − ⋅ + ∂ ∂

⋅ × = ⋅ × − ⋅ ×

∂ ∂+ ⋅ ⋅

∂

=

+ ⋅ =

⋅

×∂

×

⋅

E H

E

B DH E

B DH + EH

E H H E E

J

E J

H

∇

∇

∇

∇ ∇

Outward Flux of Rate of increase of Power lost tostored electromagnetic ohmic heating.

energy density= Poynting vector

= ×S E H

Vector Identity

Each term has units of power density 3 3

Joules Wattssec m m

= ⋅

2-25

[ ]3

3

2

2

1 Joules

sec m

1 Joules

sec m

12 2

2

t t t t

t t

εε ε

µ

∂ ⋅ ∂ ∂ ∂⋅ = ⋅ = = ∂ ∂ ∂ ∂

∂ ∂⋅ = ∂ ∂

E ED E EE E

B HHMagnetic Energy

Density

Electric Energy Density

> 0

< 0

⋅

⋅

E J

E J

Power flow

) 0(t t

∂ ∂+ ⋅ ⋅

∂ ∂+ ⋅⋅ × =

B DH + E EH JE∇

Poynting's Theorem

(=σE2) Ohmic losses in material

Source gives power to volume

http://www.microwaves101.com/encyclopedia/halloffame1.cfm

John Henry Poynting

S = (E x H) = Poynting Vector Wattsm3

2-26

( ){ }

( ){ }( ){ }

( , , , ) ( , , , ) ( , , , )

( , , , )

( ) Re cos sin

( ) Re cos si

For and

Re

n

j t

j t

j t

e

x y z t x y z t x y z t

x y z t

j j

t j e t t

t j e t t

ω

ω

ω

ω ω

ω ω

×

= ×

= + = +

= + = −

= + = −

≠ E H

E H

R I R

R I R I

R I R I

I

S E H

S

E E H H

E E E E E

H H H H H

( ) ( )2 2( ) cos sin

( ) sin cos

t t t

t t

ω ω

ω ω

= × + ×

− × + ×

R R I I

R I I R

S E H E H

E H E H

Instantaneous Poynting Vector

2-27

{ }

2

0

1( , , , ) ( )

2

1( )

21

Re *2

x y z t d tπ

ωπ

=

= × + ×

= ×

∫

E H

R R I I

S S

S E H E H

S

Time Average Poynting Vector

{ } ( ) ( ) ( ) ( ){ }

{ } ( ) ( )

2 2cos sin 0.5 siSin n coce an s 0

Re * Re

Re

*

d

j j

= = =

× = × + × − × + ×

× = × + ×

∫ ∫ ∫

E H

E H

R R I R R I I I

R R I I

E H E H E H E H

E H E H

Where is the complex Poynting Vector* ×E H

[2.38]