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Electromagnetic Fields

Four vector quantities

E

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

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]

v

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

Sources generating

electromagnetic fields


3/18

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 -

-

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


4/18

0

v

BE

t

DH J

t

B

D

=

= +

=

=

Maxwells Equations

(time-varying, differential form)


5/18

Maxwells Equations

James Clerk Maxwell (18311879)

James Clerk Maxwell was a Scottish mathematician andtheoretical physicist. His most significant achievement was thedevelopment of the classical electromagnetic theory, synthesizingall previous unrelated observations, experiments and equations

of electricity, magnetism and even optics into a consistent theory.His set of equationsMaxwell's equationsdemonstrated thatelectricity, magnetism and even light are all manifestations of thesame phenomenon: the electromagnetic field. From that momenton, all other classical laws or equations of these disciplines

became simplified cases of Maxwell's equations. Maxwell's workin electromagnetism has been called the "second greatunification in physics", after the first one carried out by IsaacNewton.

Maxwell demonstrated that electric and magnetic fields travel

through space in the form of waves, and at the constant speed oflight. Finally, in 1864 Maxwell wroteA Dynamical Theory of theElectromagnetic Field

where he first proposed that light was infact undulations in the same medium that is the cause of electricand magnetic phenomena. His work in producing a unified modelof electromagnetism is considered to be one of the greatest

advances in physics.

(Wikipedia)


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Maxwells Equations (cont.)

0

C S

C S S

S

v

S V

dE dr B ndS

dt

ddr J ndS D ndS

dt

B ndS

D ndS dV

=

= +

=

=

Faradays law

Amperes law

Magnetic Gauss law

Electric Gauss law

(Time-varying, integral form)

The above are the most fundamental form of Maxwells equations sinceall differential forms, all boundary forms, and all frequency domain forms

derive from them!


7/18

Maxwells Equations (cont.)

0

v

BE

tD

H Jt

B

D

=

= +

=

=

Faradays law

Amperes law

Magnetic Gauss law

Electric Gauss law

(Time-varying, differential form)


8/18

( )

( ) 0

D

H Jt

D

H J t

J Dt

= +

= +

= +

Law of Conservation of Electric

Charge (Continuity Equation)

vJ

t

=

Flow of electriccurrent out of volume

(per unit volume)

Rate of decrease of electriccharge (per unit volume)

[2.20]


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Continuity Equation (cont.)

vJt

=

Apply the divergence theorem:

Integrate both sides over an arbitrary volume V:

v

V V

d V d V t

=

v

S VJ n d S d Vt

= V

S

n


10/18

Continuity Equation (cont.)

Physical interpretation:

V

S

n

v

S V

n d S d V t

=

v

o u t vV V

i d V d V t t

= =

e n c l

o u t

Q

i t

=

(This assumes that the

surface is stationary.)

e n c l

in

Q

i t

= or


11/18

Maxwells Equations

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 ofv

H E H J


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E BH J D

B 0D v

jj

=

= +

=

=

Maxwells Equations

Time-harmonic (phasor) domain jt


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Constitutive Relations

Characteristics of media relate D

to E

and H

to B

00

0 0

( = permittivity )

( = permeability)

D E

B H

=

=

-12

0

-7

0

[F/m]8.8541878 10

= 4 10 H/m] ( )[

exact

[2.24]

[2.25]

[p. 35]

0 0

1c = c = 2.99792458

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

Free Space


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Constitutive Relations (cont.)

Free space, in the phasor domain:

00

0 0

( = permittivity )

(

D

=

= E

B pe= rmeability)H

This follows from the fact that

( ) VaV t a

(where a

is a real number)


15/18

Constitutive Relations (cont.)

In a material medium:

( = permittivity )

( = permeability

D = E

B = )H

0

0

= r

r

=

r

= relative permittivity

r

= relative permittivity


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or

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

I i M i l


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Isotropic Materials

(or ) is a scalar quantity,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

A i t i M t i l


18/18

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

This results in E

and D

being NOT parallel to each other; they are

in different directions.

0 0

0 0

0 0

x x x

y y y

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

DE

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