Formulas in Electrodynamics

Formulas in Electrodynamics
Based on course by Yuri Lyubarsky and Edited By Eitan Rothstein
Department of Physics, Ben-Gurion University, Beer-Sheva 84105, Israel
R 0 n
P∞
1
1
0
0 3 0
Electric multipole expansion: V (r) = 4π
n=0 r (n+1) (r ) Pn (cos θ )ρ(r )d r .
0
R 0 0 3 0 P
1 p·r̂
Electric dipole moment potential Vdip (r) = 4π
r ρ(r )d r = i qi ri .
2 , with the electric dipole moment p =
0 r
1 1
The electric field of an electric dipole is Edip (r) = 4π
[3(p
·
r̂)
·
r̂
−
p].
3
0 r
µ0 1
The magnetic field of a magnetic dipole
is
B
(r)
=
dip
4π r 3 [3(m · r̂) · r̂ − m],
R
with the magnetic moment: m = 21 r × JdV
Boundary conditions:
1 E1⊥ = 2 E2⊥ ,
k
k
E1 = E2 ,
B1⊥ = B2⊥ ,
1 k
1 k
B =
B .
µ1 1
µ2 2
Poynting vector: S = µ1 E × B.
1
Energy density: u = 2 E 2 + 2µ
B2.
Radiation angular distribution of a dipole: dI =
Liénard-Wiechert potentials:
V (r, t) =
1
qc
,
4π0 r̃c − r̃ · v
A(r, t) =
µ0
16π 2 c
2
(p̈ × n̂) dΩ.
v
V (r, t),
c2
where r̃ is the vector from the retarded position to the field point r and v is the velocity of the charge at the retarded
time.
Lorentz transformation
x̄ = γ(x − vt),
v
t̄ = γ(t − 2 x),
c
1
γ = p
.
1 − v 2 /c2
eB
.
Relativistic cyclotron frequency: ωB = mcγ
The field tensor:


0 −Ex /c −Ey /c −Ez /c
∂Aν
∂Aµ
0
Bz
−By 
 E /c
Fµν =
−
= x
µ
ν
E
/c
−B
0
Bx 
∂x
∂x
y
z
Ez /c By
−Bx
0
Transformation of fields
Ēk = Ek ,
B̄k = Bk ,
Ē⊥ = γ(E⊥ + v × B⊥ ),
1
B̄⊥ = γ(B⊥ − 2 v × E⊥ ).
c
Energy momentum four vector: P = mcγ(1, v/c).