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Prof. Ji Chen

Adapted from notes by Prof. Stuart A. Long

Notes 4 Maxwell’s Equations

ECE 3317

Spring 2014

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Electromagnetic Fields Four vector quantities

E electric field strength [Volt/meter]

D electric flux density [Coulomb/meter2]

H magnetic field strength [Amp/meter]

B magnetic flux density [Weber/meter2] or [Tesla]

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]

Sources generating electromagnetic fields

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MKS units

length – meter [m]

mass – kilogram [kg]

time – second [sec]

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

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

centi - c - 10-2

deci - d - 10-1

deka - da - 101

hecto - h - 102

kilo - k - 103

4

0

v

BE

tD

H Jt

B

D ρ

Maxwell’s Equations

(Time-varying, differential form)

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Maxwell

James Clerk Maxwell (1831–1879)

James Clerk Maxwell was a Scottish mathematician and theoretical physicist. His most significant achievement was the development of the classical electromagnetic theory, synthesizing all previous unrelated observations, experiments and equations of electricity, magnetism and even optics into a consistent theory. His set of equations—Maxwell's equations—demonstrated that electricity, magnetism and even light are all manifestations of the same phenomenon: the electromagnetic field. From that moment on, all other classical laws or equations of these disciplines became simplified cases of Maxwell's equations. Maxwell's work in electromagnetism has been called the "second great unification in physics", after the first one carried out by Isaac Newton.

Maxwell demonstrated that electric and magnetic fields travel through space in the form of waves, and at the constant speed of light. Finally, in 1864 Maxwell wrote A Dynamical Theory of the Electromagnetic Field where he first proposed that light was in fact undulations in the same medium that is the cause of electric and magnetic phenomena. His work in producing a unified model of electromagnetism is considered to be one of the greatest advances in physics.

(Wikipedia)

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Maxwell’s Equations (cont.)

0

v

BE

tD

H Jt

B

D ρ

Faraday’s law

Ampere’s law

Magnetic Gauss law

Electric Gauss law

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0

D

H Jt

DH J

t

J Dt

Law of Conservation of Electric Charge (Continuity Equation)

vJt

Flow of electric

current out of volume (per unit volume)

Rate of decrease of electric charge (per unit volume)

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Continuity Equation (cont.)

vJt

Apply the divergence theorem:

Integrate both sides over an arbitrary volume V:

v

V V

J dVt

ˆ v

S V

J n dVt

V

S

n̂

ˆV S

J J n

Hence:

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Continuity Equation (cont.)

Physical interpretation:

V

S

n̂

ˆ v

S V

J n dVt

vout v

V V

i dV dVt t

enclout

Qi

t

(This assumes that the surface is stationary.)

enclin

Qi

t

or

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Decouples

0

0 0

v

v

E

B DE H J B D

t t

E D H J B

Time -Dependent

Time -Independent (Static s)

and is a function of and is a function of vH E H J

Maxwell’s Equations (cont.)

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E B

H J D

B 0

D v

j

j

Time-harmonic (phasor) domain jt

Maxwell’s Equations (cont.)

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Constitutive Relations

The characteristics of the media relate D to E and H to B

00

0 0

( = permittivity )

(

= permeability)

D E

B µ H µ

-120

-70

[F/m] 8.8541878 10

= 4 10 H/m] ( ) [µ

exact

0 0

1c

c = 2.99792458 108 [m/s] (exact value that is defined)

Free Space

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72 2 10 N/m when 1 mxF d

Definition of I =1 Amp:

I

I

d

Fx2 x

# 1

# 2

Two infinite wires carrying DC currents

Definition of the Amp:

20

2 2x

IF

d

70 4 10 [A/m]

From ECE 2317(see next slide):

Constitutive Relations (cont.)

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Force calculation:

Constitutive Relations (cont.)

2 2

2 2 1ext

C C

F I d l B I d l B (Lorentz force law for force on a wire)

I

I

d

Fx2

x

# 1

# 2

Two infinite wires carrying DC currents

y

2

02

2 0 0ˆ ˆˆ2 2C L

I IF I y dy z x dy

d d

The contour C runs in the reference direction of the current on the wire.

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2 2x

IF

d

Hence we have

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Constitutive Relations (cont.)

Free space, in the phasor domain:

00

0 0

( = permittivity )

(

D

=

= E

B pe= rmeability ) H µµ

This follows from the fact that

VaV t a

(where a is a real number)

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Constitutive Relations (cont.)

In a material medium:

( = permittivity )

( = permeability

D = E

B = ) H µµ

0

0

= r

rµ µ

r = relative permittivity

r = relative permittivity

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Variation Independent of Dependent on

space homogenous inhomogeneous

frequency non-dispersive dispersive

time stationary non-stationary

field strength linear non-linear

direction of isotropic anisotropic E or H

Terminology

Properties of or

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Isotropic Materials

ε (and μ) are scalar quantities,

which means that E || D (and H || B )

x xB = H

y yB = H

yH

xH

x

yx xD = E

y yD = E

yE

xE

x

y

D E

B µH

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ε (or μ) is a tensor (can be written as a matrix)

This results in E and D being NOT proportional to each other.

0 0

0 0

0 0

x x x

y y y

x x x

z z z

y y y

z z z

D E

D E

D E

D E

D E D E

Anisotropic Materials

Example:

D E or

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0 0

0 0

0 0

x h x

y h y

z v z

D E

D E

D E

Anisotropic Materials (cont.)

Practical example: uniaxial substrate material

Teflon substrate

Fibers

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Anisotropic Materials (cont.)

The column indicates that v is being measured.

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