Phys math help

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
c0

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
4π0

~ 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)