Physique math´ematique : r´esum´e
1 Magnetostatics
Biot and Savart law
dB~ =k.I ~dl×~x
|x|3 (1)
dF~ =I ~dl×B~ (2)
Z
∂S
B. ~~ dl=µ Z
S
J . ~~dS (3)
B~ =∇ ×~ A(~~ x) (4)
Coulomb gauge
∆A~ =−µ0J~ (5)
Magnetic moment
~ m=1
2 Z
d3x0(~x0×J(~~ x0)) (6)
Density of magnetic moment M(x) =~ 1
2(~x×J~(x)) (7)
B~(x) = µ0
4π
3~x(~m~x)−m.~~ x2
x5 (8)
Torque in classical mechanics T~ =X
i
~ri×F~i (9)
Torque in magnetostatics
T~ 'm~ ×B~ (10)
Faraday induction law
E=−dφ
dt (11)
withφ=R
SB. ~~ dS andE=R
∂SE. ~~ dl
Energy of magnetic field
δW =I.δφ (12)
W = 1 2µ0
Z
d3x.B2 (13)
W = 1 2 Z
d3x. ~J . ~A (14)
2 Maxwell Equations
∇. ~~ E= ρ 0
(I)
∇ ×~ B~ −µ00∂ ~E
∂t =µ0J~ (II)
∇ ×~ E~ +∂ ~B
∂t = 0 (III)
∇. ~~ B= 0 (IV)
Conservation law
∂ρ
∂t +∇. ~~ J = 0 (15)
Potentials
B~ =∇ ×~ A~ (16)
E~ =−∇ϕ~ −∂ ~A
∂t (17)
Relativistic form
∂µFµν = 1 c0
jν (18)
withFµν =
0 −E1 −E2 −E3 E1 0 −cB3 cB2 E2 cB3 0 −cB1 E3 −cB2 cB1 0
andjµ= (cρ, ~J)
3 Electric and magnetic fields in media
Polarization
Ptot
V =σ≡P (19)
ρP(x) =−∇. ~~ P(x) (20)
σP(x) =~n. ~P(x) (21)
ρext=∇. ~~ D (22)
withD~ =0E~ +P~
P~ '0χe. ~E (23)
D~ ' ~E (24)
with=0(1 +χe)
∇. ~~ E' ρ
(25)
Continuity
~
n21×(E~2−E~1) = 0 (26)
~
n21.(D~2−D~1) =σext (27) Energy
W = 1 2
Z
d3x(E. ~~ D) (28)
Magnetostatics in media
J~M(x) =∇ ×~ M~(x) (29)
∇ ×~ H~ =J~ext (30) withH~ = µ1
0
B~ −M~
Relation between B and H
B~ =µ ~H (31)
for isotropic diamagnetic and paramagnetic substances
B~ =F(H~) (32)
for ferromagnetic substances
4 Charged particle in electromagnetic field
Equation of motion
dˆpµ ds = e
c2
Fˆµν.ˆuν (33)
– Space component :
d~p
dt =e[E~ +~v×B]~ (34) with~p=qm~v
1−v2
c2
– Time component :
dE
dt =e~v. ~E (35)
–
E =p
m2c4+p2 (36)
–
~ p=~vE
c2 (37)
Gyration frequency
~
ωB = e ~B
mγ = ec2B~
E (38)
Drift in non-uniform magnetic field
~vdrif t= a2 2B0
.~ω0×∇B~ (39)
Drift in a constant electric and magnetic field
~vdrif t= E~ ×B~
B2 (40)
Field invariants
I1=c2B~2−E~2 (41)
I2=E. ~~ B (42)
5 Electromagnetic waves
D’Alembert equations [1
c2
∂2
∂t2 −∂i2]Ei = 0 (43) [1
c2
∂2
∂t2 −∂2i]Bi= 0 (44)
Monochromatic wave
E~ =Re[E~0eiωt−i~k.~x] (45) B~ =Re[B~0eiωt−i~k.~x] (46) with~k=~nωc
Poynting vector
S~ = 1 µ0
E~ ×B~ =~n.cW (47)
Energy
W =1 2
1 µ0
B~2+1
20E~2 (48)
−J.E= ∂w
∂t +∇. ~~ S (49)
Snell law
n
n0 = sinβ
sinα (50)
withn=vc Brewster angle
cos2α= 1
n02
n2 + 1 (51)
6 Emission of electromagnetic waves
[1 c2
∂2
∂t2 − ∇2]ϕ= ρ 0
(52)
[1 c2
∂2
∂t2− ∇2]A~ =µ0J~ (53) Potentiels retard´es
ϕ(~x, t) = 1 4π0
Z
dV ρ(t−|x~0−~cx|, ~x)
|x~0−~x| (54)
A(~~ x, t) = µ0
4π Z
dV
J~(t−|x~0−~cx|, ~x)
|x~0−~x| (55) Larmor formule
I=µ0e2 6πc
~a2−(~vc ×~a)2
(1−vc32)3 (56)
