Phys math help.pdf

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