om maxwell equations for students
DESCRIPTION
MAXWELL EQUATIONSfour equations of maxwell equations that can describes the electromagnetic theories of emft which help the sudents to know about the variousTRANSCRIPT
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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]
ρv electric charge density [Coul/meter3] = [Amp-sec/m3]
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
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0
v
BE
tD
H Jt
B
D ρ
Maxwell’s Equations
(time-varying, differential form)
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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)
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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.
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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.
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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
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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.
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“Maxwell’s Equations”
Ddiv
0div B
t
B
Ecurl
JD
H
t
curl
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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
(Time-varying, differential form)
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Electromagnetic Waves
Electromagnetic (EM) waves
Faraday’s law:
Ampere-Maxwell’s law:
changing B gives E.
changing E gives B.
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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
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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
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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
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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
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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)
[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
J dV dVt
ˆ v
S V
J n dS dVt
V
S
n̂
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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
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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
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E B
H J D
B 0
D v
j
j
Maxwell’s 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 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
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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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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