maxwell’s equations dr. alexandre kolomenski. maxwell (13 june 1831 – 5 november 1879) was a...
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![Page 1: Maxwell’s equations Dr. Alexandre Kolomenski. Maxwell (13 June 1831 – 5 November 1879) was a Scottish physicist. Famous equations published in 1861](https://reader035.vdocuments.us/reader035/viewer/2022062313/56649cb75503460f9497c776/html5/thumbnails/1.jpg)
Maxwell’s equations
Dr. Alexandre Kolomenski
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Maxwell(13 June 1831 – 5 November 1879) was a Scottish physicist.
Famous equations published in 1861
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Maxwell’s equations: integral form
Gauss's law
Gauss's law for magnetism:no magnetic monopole!
Ampère's law(with Maxwell's addition)
Faraday's law of induction (Maxwell–Faraday equation)
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Relation of the speed of light and electric and magnetic vacuum constants
ε0
permittivity of free space, also called the electric constant
As/Vm or F/m (farad per meter)
μ0permeability of free space, also called the magnetic constant
Vs/Am or H/m (henry per meter)
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Differential operators
t
the divergence operatordiv
the curl operatorcurl, rot
the partial derivative with respect to time
yx zAA AA
x y z
ˆ ˆ ˆ
x y z
x y z
Ax y z
A A A
x x
Other notation used
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Transition from integral to differential form
Gauss’ theorem for a vector field F(r)Volume V, surrounded by surface S
Surface , surrounded by contour
Stokes' theorem for a vector field F(r)
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Maxwell’s equations: integral form
Gauss's law
Gauss's law for magnetism:no magnetic monopoles!
Ampère's law(with Maxwell's addition)
Maxwell–Faraday equation(Faraday's law of induction)
![Page 9: Maxwell’s equations Dr. Alexandre Kolomenski. Maxwell (13 June 1831 – 5 November 1879) was a Scottish physicist. Famous equations published in 1861](https://reader035.vdocuments.us/reader035/viewer/2022062313/56649cb75503460f9497c776/html5/thumbnails/9.jpg)
Maxwell’s equations (SI units)differential form
density of charges
j density of current
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Electric and magnetic fields and units
E electric field, volt per meter, V/m
Bthe magnetic field or magnetic induction
tesla, T
Delectric displacement
fieldcoulombs per square meter,C/m^2
H magnetic field ampere per meter, A/m
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Constitutive relationsThese equations specify the response of bound charge and current to the applied fields and are called constitutive relations.
P is the polarization field,
M is the magnetization field, then
where ε is the permittivity and μ the permeability of the material.
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Wave equation
2 2( ) ( )B B B B
2
2 2 21 1
( ) ( ) ( )B
E E Bt t t tc c t
22
2 21
0B
Bc t
22
2 21
0E
Ec t
0
Double vector product rule is used a x b x c = (ac) b - (ab) c
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2 more differential operators
2 Laplace operator or Laplacian
22 22
2 2 2yx zAA A
Ax y z
),
=
d'Alembert operator or d'Alembertian
or
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Plane waves
0 [ ( )]B B Exp i k r t
)]([0 trkiExpEE
B i k B
E i k E
is parallel to
is parallel to B
E
Thus, we seek the solutions of the form:
From Maxwell’s equations one can see that k
B
E
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Energy transfer and Pointing vector
Integral form of Pointing theorem
Differential form of Pointing theorem
|| S is directed along the propagation direction
u is the density of electromagnetic energy of the field
kH
E
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Energy quantities continued
max [ ( )]B B exp i k r t
max [ ( )]E E exp i k r t
Observable are real values:
max cos[ ( )]B B i k r t
max cos[ ( )]E E i k r t
2 2max 0 max 0/ 2 / 2av avI S E c cB cu
2 2 2max 0 max 0/ (4 ) / 4avu E c B
max max /B E c