Phys math

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

(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)

1 ν
j
c0

(16)

Relativistic form
∂µ F µν =

3

(12)

Electric and magnetic fields in media

Polarization

Ptot
=σ≡P
V

(17)

~ P~ (x)
ρP (x) = −∇.

(18)

σP (x) = ~n.P~ (x)

(19)

~ D
~
ρ = ∇.

(20)

~
P~ ' 0 χe .E

(21)

~ ' E
~
D

(22)

~ = 0 E
~ + P~
with D

2

with  = 0 (1 + χe )
~ E
~ 'ρ
∇.


(23)

~2 − E
~ 1) = 0
~n21 × (E

(24)

~2 −D
~ 1 ) = σext
~n21 .(D

(25)

Continuity

Energy
1
W =
2

Z

~ D)
~
d3 x(E.

(26)

~ ×M
~ (x)
J~M (x) = ∇

(27)

~ ×H
~ = J~ext
∇

(28)

~ = µH
~
B

(29)

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

(30)