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]

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

Sources generating electromagnetic fields

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

0

v

BE

tD

H Jt

B

D ρ

Maxwell’s Equations

(time-varying, differential form)

Maxwell’s Equations

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)

The Four Laws of Electromagnetism

Law Mathematical Statement

Physical Meaning

Gauss for E 0

qd

E A How q produces E;

E lines begin & end on q’s.

Gauss for B

Faraday

Ampere(Steady I

only)

4 Laws of EM (incomplete)

0d B A

Bdd

d t

E r

0d IB r

No magnetic monopole;B lines form loops.

Changing B gives emf.

Moving charges give B.

Note E-B asymmetry between the Faraday & Ampere laws.

29.3. Maxwell’s EquationsLaw Mathematical

StatementPhysical Meaning

Gauss for E 0

qd

E A How q produces E;

E lines begin & end on q’s.

Gauss for B

Faraday

Ampere-Maxwell

0d B A

Bdd

d t

E r

No magnetic monopole;B lines form loops.

Changing B gives emf.

Moving charges & changing E give B.

0 0 0Ed

d Id t

B r

Maxwell’s Eqs (1864).Classical electromagnetism.

Maxwell’s Equations

Gauss's law electric

0 Gauss's law in magnetism

Faraday's law

Ampere-Maxwell lawI

oS

S

B

Eo o o

qd

ε

d

dd

dtd

d μ ε μdt

E A

B A

E s

B s

•The two Gauss’s laws are symmetrical, apart from the absence of the term for magnetic monopoles in Gauss’s law for magnetism•Faraday’s law and the Ampere-Maxwell law are symmetrical in that the line integrals of E and B around a closed path are related to the rate of change of the respective fluxes

Electromagnetic Fields

• Oersted and Ampere showed how an electric current could create a magnetic field, causing ‘action at a distance’.

• Faraday showed how a magnetic field could create a current, but only if it was varying in time.

• Maxwell generalised Faraday’s and Ampere’s Laws, combined them, and discovered an equation for travelling electromagnetic waves.

“Maxwell’s Equations”

Ddiv

0div B

t

B

Ecurl

JD

H

t

curl

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

(Time-varying, differential form)

Electromagnetic Waves

Electromagnetic (EM) waves

Faraday’s law:

Ampere-Maxwell’s law:

changing B gives E.

changing E gives B.

First law

• Gauss’s law (electrical):• The total electric flux through any closed

surface equals the net charge inside that surface divided by o

• This relates an electric field to the charge distribution that creates it

Second law

• Gauss’s law (magnetism): • The total magnetic flux through any closed

surface is zero• This says the number of field lines that

enter a closed volume must equal the number that leave that volume

• This implies the magnetic field lines cannot begin or end at any point

• Isolated magnetic monopoles have not been observed in nature

THIRD LAW

• Faraday’s law of Induction:• This describes the creation of an electric

field by a changing magnetic flux• The law states that the emf, which is the

line integral of the electric field around any closed path, equals the rate of change of the magnetic flux through any surface bounded by that path

• One consequence is the current induced in a conducting loop placed in a time-varying B

FOURTH LAW

• The Ampere-Maxwell law is a generalization of Ampere’s law

• It describes the creation of a magnetic field by an electric field and electric currents

• The line integral of the magnetic field around any closed path is the given sum

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)

[2.20]

Continuity Equation (cont.)

vJt

Apply the divergence theorem:

Integrate both sides over an arbitrary volume V:

v

V V

J dV dVt

ˆ v

S V

J n dS dVt

V

S

n̂

Continuity Equation (cont.)

Physical interpretation:

V

S

n̂

ˆ v

S V

J n dS dVt

vout v

V V

i dV dVt t

enclout

Qi

t

(This assumes that the surface is stationary.)

enclin

Qi

t

or

Maxwell’s 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 of vH E H J

E B

H J D

B 0

D v

j

j

Maxwell’s Equations

Time-harmonic (phasor) domain jt

Constitutive Relations

Characteristics of 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

[2.24]

[2.25]

[p. 35]

0 0

1c

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

Free Space

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)

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

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 anisotropic E or H

Terminology