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
1
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)
Relativistic form
∂µFµν = 1
c0jν (16)
3 Electric and magnetic fields in media
Polarization
Ptot
V =σ≡P (17)
ρP(x) =−∇. ~~ P(x) (18)
σP(x) =~n. ~P(x) (19)
ρ=∇. ~~ D (20)
withD~ =0E~ +P~
P~ '0χe. ~E (21)
D~ ' ~E (22)
2
with=0(1 +χe)
∇. ~~ E' ρ
(23)
Continuity
~
n21×(E~2−E~1) = 0 (24)
~
n21.(D~2−D~1) =σext (25) Energy
W = 1 2
Z
d3x(E. ~~ D) (26)
Magnetostatics in media
J~M(x) =∇ ×~ M~(x) (27)
∇ ×~ H~ =J~ext (28) withH~ = µ1
0
B~ −M~
Relation between B and H
B~ =µ ~H (29)
for isotropic diamagnetic and paramagnetic substances
B~ =F(H~) (30)
for ferromagnetic substances
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