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Physique math´ematique : r´esum´e
1
Magnetostatics
Biot and Savart law
~
~ = k. I dl × ~x
dB
|x|3
(1)
~ = I dl
~ ×B
~
dF
(2)
Z
~ =µ
~ dl
B.
∂S
Z
~
~ dS
J.
(3)
S
~ =∇
~ × A(~
~ x)
B
(4)
~ = −µ0 J~
∆A
(5)
Coulomb gauge
Magnetic moment
m
~ =
1
2
Z
~ x0 ))
d3 x0 (~x0 × J(~
(6)
Density of magnetic moment
1
~
~
M(x)
= (~x × J(x))
2
(7)
µ0 3~x(m~
~ x) − m.~
~ x2
~
B(x)
=
4π
x5
(8)
Torque in classical mechanics
T~ =
X
~ri × F~i
(9)
i
Torque in magnetostatics
~
T~ ' m
~ ×B
(10)
Faraday induction law
E =−
with φ =
R
S
~ and E =
~ dS
B.
R
∂S
~
~ dl
E.
1
dφ
dt
(11)
Energy of magnetic field
δW = I.δφ
Z
1
d3 x.B 2
2µ0
Z
1
~A
~
W =
d3 x.J.
2
W =
2
(12)
(13)
(14)
Maxwell Equations
~ E
~ = ρ
∇.
0
~
~ ×B
~ − µ0 0 ∂ E = µ0 J~
∇
∂t
~
~ ×E
~ + ∂B = 0
∇
∂t
~
~
∇.B = 0
Conservation law
(I)
(II)
(III)
(IV)
∂ρ ~ ~
+ ∇.J = 0
∂t
(15)
~ =∇
~ ×A
~
B
(16)
~
~ = −∇ϕ
~ − ∂A
E
∂t
(17)
Potentials
Relativistic form
1 ν
j
c 0
3
∂µ F µν =
with F µν
0
E1
=
E2
E3
−E 1
0
cB 3
−cB 2
−E 2
−cB 3
0
cB 1
−E
cB 2
~
and j µ = (cρ, J)
−cB 1
0
2
(18)
3
Electric and magnetic fields in media
Polarization
Ptot
=σ≡P
V
(19)
~ P~ (x)
ρP (x) = −∇.
(20)
σP (x) = ~n.P~ (x)
(21)
~ D
~
ρext = ∇.
(22)
~
P~ ' 0 χe .E
(23)
~ ' E
~
D
(24)
~ E
~ 'ρ
∇.
(25)
~2 − E
~ 1) = 0
~n21 × (E
(26)
~2 −D
~ 1 ) = σext
~n21 .(D
(27)
~ = 0 E
~ + P~
with D
with = 0 (1 + χe )
Continuity
Energy
1
2
Z
~ D)
~
d3 x(E.
(28)
~ ×M
~ (x)
J~M (x) = ∇
(29)
~ ×H
~ = J~ext
∇
(30)
~ = µH
~
B
(31)
W =
Magnetostatics in media
~ =
with H
1 ~
µ0 B
~
−M
Relation between B and H
for isotropic diamagnetic and paramagnetic substances
~ = F (H)
~
B
for ferromagnetic substances
3
(32)
4
Charged particle in electromagnetic field
Equation of motion
e
dˆ
pµ
uν
= 2 Fˆ µν .ˆ
ds
c
(33)
– Space component :
d~
p
~ + ~v × B]
~
= e[E
dt
with p~ =
(34)
v
q m~
2
1− vc2
– Time component :
dE
~
= e~v .E
dt
p
E = m2 c4 + p2
–
–
p~ =
~v E
c2
(35)
(36)
(37)
Gyration frequency
ω
~B =
~
~
eB
ec2 B
=
mγ
E
(38)
Drift in non-uniform magnetic field
~vdrif t =
a2
~
.~
ω0 × ∇B
2B0
(39)
Drift in a constant electric and magnetic field
~ ×B
~
E
2
B
(40)
~2 − E
~2
I 1 = c2 B
(41)
~ B
~
I2 = E.
(42)
~vdrif t =
Field invariants
5
Electromagnetic waves
D’Alembert equations
[
1 ∂2
− ∂i2 ]Ei = 0
c2 ∂t2
(43)
[
1 ∂2
− ∂i2 ]Bi = 0
c2 ∂t2
(44)
4
Monochromatic wave
with ~k =
~ = Re[E~0 eiωt−i~k.~x ]
E
(45)
~ = Re[B
~0 eiωt−i~k.~x ]
B
(46)
~= 1E
~ ×B
~ = ~n.cW
S
µ0
(47)
1 1 ~2 1 ~2
B + 0 E
2 µ0
2
(48)
~
nω
c
Poynting vector
Energy
W =
−J.E =
Snell law
with n =
∂w ~ ~
+ ∇.S
∂t
(49)
n
sinβ
=
0
n
sinα
(50)
c
v
Brewster angle
cos2 α =
6
n02
n2
1
+1
(51)
Emission of electromagnetic waves
1 ∂2
ρ
− ∇2 ]ϕ =
2
2
c ∂t
0
(52)
1 ∂2
~ = µ0 J~
− ∇2 ]A
c2 ∂t2
(53)
[
[
Potentiels retard´
es
ϕ(~x, t) =
1
4π 0
~ x, t) = µ0
A(~
4π
Larmor formule
I=
~0
Z
dV
Z
dV
ρ(t − |x c−~x| , ~x)
|x~0 − ~x|
~ − |x~0 −~x| , ~x)
J(t
c
|x~0 − ~x|
µ0 e2 ~a2 − ( ~vc × ~a)2
6πc (1 − vc32 )3
5
(54)
(55)
(56)





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