(A · B) = A × (∇ × B) + B

Four-current (p. 96)
Z
Z
1
i 4
i
Ai j d x
e
u Ai ds =
c
Z
X
j
i
j
i 4
j
j (x ) =
ceA
uA δ (x − xA (τ )) ds
Vector identities (Griffiths)
A × (B × C) = B(A · C) − C(A · B)
∇(f g) = f ∇g + g∇f
∇(A · B) = A × (∇ × B) + B × (∇ × A)
+ (A · ∇)B + (B · ∇)A
∇ · (f A) = f (∇ · A) + A · ∇f
∇ · (A × B) = B · (∇ × A) − A · (∇ × B)
∇ × (f A) = f (∇ × A) − A × (∇f )
∇ × (A × B) = (B · ∇)A − (A · ∇)B
+ A(∇ · B) − B(∇ · A)
Four-current for EM field (p. 101): j = (cρ, j)
Action for scalar field (5) and EM field (6)
Z
Z
i
S = −mc
ds +
φ(x ) ds
(5)
Z
Z
Z
e
1
i
ij
4
S = −mc
ds −
u Ai ds −
F Fij d x (6)
c
16πc
Vector identities (Jackson)
(a × b) · (c × d) = (a · c)(b · d) − (a · d)(b · c)
Z
Z
3
∇ · Ad x =
A · da
VZ
ZS
3
∇ψ d x =
ψ da
V
Z
ZS
3
∇ × Ad x =
da × A
V
S Z
Z
2
3
(φ∇ ψ + ∇φ · ∇ψ) d x =
(φ∇ψ) · da
V
S
Z
Z
2
2
3
(φ∇ ψ − ψ∇ φ) d x =
(φ∇ψ − ψ∇φ) · da
V
S Z
Z
(∇ × A) · da =
A · dl
SZ
Z C
da × ∇ψ =
ψ dl
(1)
2F ij δ(∂i Aj − ∂j Ai )
(3)
(4)
identity;
ˆ dφ
dl = ˆ
r dr + r θˆ dθ + r sin θ φ
ˆ
ˆ = sin θ cos φ ˆ
x
r + cos θ cos φ θˆ − sin φ φ
ˆ
ˆ = sin θ sin φ ˆ
y
r + cos θ sin φ θˆ + cos φ φ
∇·v =
1
r2
1
∂θ t θˆ +
1
2
∂r (r vr ) +
1
r sin θ
1
=
=
=
∂B
Implications: ∇ · B = 0 and ∇ × E = − 1
c ∂t
Equation of motion for EM field (p. 107): ∂i F ij = 4π
jj
c
∂E
j+ 1
Implications: ∇ · E = 4πρ and ∇ × B = 4π
c
c ∂t
∂χ
Gauge invariance: φ0 ← φ + 1
; A0 ← A − ∇χ i.e.
c ∂t
Ai0 ← Ai + ∂ i χ (χ is a Lorentz scalar field)
Properties of Levi–Civita symbol (HW3)
p=
F
αβ
ij
αβγ
2
γ
Fkl ∝ B − E
0123
2
F
ij
F
kl
ijkl ∝ E · B (p. 85)
F ij F kl ijkl is a boundary term (4-divergence)
Approximate fields of localized charge configuration (x0 r, x0 λ) (p. 158)
ˆ
r
r
q
1 ˆ
˙ 0)
+
· d(t0 ) +
· d(t
r
r2
c r
1 1
˙
A(r, t) ≈
d(t0 )
c r
where t0 = t − r/c, and d is the dipole moment.
E and B fields for pure dipole (p. 163), Poynting vector
(p. 164), and total power radiated (p. 165)
2 2
(7)
E
Ev
hPtot i = e a3 ω 4
3c
Electromagnetic stress-energy tensor
(8)
c2
Motion in uniform E-field: Integrate equations of motion
to obtain energy and momentum, then use (7) to get velocity.
Motion in uniform B-field: Note that energy is constant.
Use (8) to write momentum in terms of velocity, then integrate equations of motion to get velocity.
r sin θ
r sin θ
1
ˆ
r[∂θ (vφ sin θ) − ∂φ vθ ]
2
r sin
θ
Coulomb gauge: φ = 0; ∇ · A = 0; −∇2 A + 12 ∂ A
=0
1
1
1
c
∂t2
ˆ r (rvθ ) − ∂θ vr ]
+ θˆ
∂φ vr − ∂r (rvφ ) + φ[∂
i
i
Lorentz gauge: ∂i A = 0; ∂i ∂ Aj = Aj = 0
r
sin θ
r
Fourier series
1
1
1
2
2
∂r (r ∂r t) +
∇ t=
∂θ (sin θ ∂θ t) +
∂φφ t
∞
X
1
r2
r 2 sin θ
r 2 sin2 θ
ik x
f (x) =
f˜(kn )e n
L n=−∞
Cylindrical coordinates (Griffiths)
Z L/2
ˆ dφ + z
ˆ dz
dl = ˆ
s ds + sφ
−ikn x
f˜(kn ) =
f (x)e
dx
ˆ
ˆ
ˆ = cos φ ˆ
ˆ = sin φ ˆ
x
s − sin φ φ
y
s + cos φ φ
−L/2
2πn
ˆ = − sin φ x
ˆ
ˆ + sin φ y
ˆ
ˆ + cos φ y
ˆ
s = cos φ x
φ
kn =
L
1
ˆ + ∂z t z
ˆ
∇t = ∂s t ˆ
s + ∂φ t φ
Fourier transform
s
Z ∞
1
1
1
ikx
∇ · v = ∂s (svs ) + ∂φ vφ + ∂z vz
f (x) =
f˜(k)e
dk
s
2π
−∞
s
Z
∞
1
−ikx
ˆ z vs − ∂s vz ]
∇×v =ˆ
s
∂φ vz − ∂z vφ + φ[∂
f˜(k) =
f (x)e
dx
s
−∞
Z ∞
1
0 )x
i(k−k
0
ˆ[∂s (svφ ) − ∂φ vs ]
+ z
e
dx = 2πδ(k − k )
s
−∞
1
1
2
∇ t = ∂s (s∂s t) +
∂φφ t + ∂zz t
Need f˜(k) = f˜∗ (−k) in order for f (x) to be real.
s
s2
Solution of wave equation for EM field in Coulomb gauge
Useful variation
(p. 121)
(dxi /dθ)(dδxi /dθ)
ZZZ
3
δ(ds/dθ) =
(54.2)
i(k·x−ωt) d k
ds/dθ
~ ∗ (−k)ei(k·x+ωt) + β(k)e
~
A(x, t) =
β
Primed frame moves with velocity vˆ
x relative to unprimed
(2π)3
frame.



 
Monochromatic plane wave
ct0
γ
−βγ
0
0
ct
~
~ 3 (k − p)(2π)3 which yields
β(k)
= βδ
 x0 
−βγ
x
γ
0
0
 0 = 
 
~ cos(p · x − ωt) (p. 122)
A(x, t) = 2β
y 
 0
0
1
0  y 
k · (E × B) > 0 (p. 123)
0
0
0
1
z
z0
ω = ckkk (p. 124)
Transformation of the field (HW3, problem 2.1)
Poynting vector and EM energy density (p. 126)
0
0
0
Ex = Ex Ey = γ(Ey − βBz ) Ez = γ(Ez + βBy )
c
S=
E×B
0
0
0
Bx = Bx By = γ(By + βEz ) Bz = γ(Bz − βEy )
4π
1
2
2
Basic objects of relativistic electrodynamics
Eem =
(E + B )
8π Ai = (φ, A) Ai = (φ, −A) Fij = ∂i Aj − ∂j Ai
∂
1
2
2
E − ∇φ − ∂A/∂t B = ∇ × A
(E + B ) = −j · E − ∇ · S


∂t 8π
0
Ex
Ey
Ez
Z
Z
Z
2
2
d
E +B
−Ex
3
3
0
−Bz
By 
d x=−
j · Ed x −
S · da

Fij = 
−Ey

dt V
8π
Bz
0
−Bx
V
S
−Ez
−By
Bx
0
Laplace Green function (GF1)
1


G(x) = 4πkxk
; ∇2 G(x) = −δ 3 (x)
0
−Ex
−Ey
−Ez
d’Alembert
Green
function (GF5)


Ex
0
−Bz
By 
ij
F
=
δ(ct−r)
Ey
Bz
0
−Bx 
G(xi ) = − 4πr ; G(xi ) = −δ 4 (xi )
Ez
−By
Bx
0
ˆ em
Energy of plane wave (p. 128): S = kcE
~ em = S
EM momentum density (p. 129): P
F
= −
B (p. 80)
0123 = +1 = −1
∇×v =
2
¨ c (tr ))]
[(c − vr )u + rr × (u × x
1
¨ 0) × ˆ
(d(t
r) × ˆ
r
c2 r
B=ˆ
r×E
2
¨
2 kd(t0 )k
S=ˆ
r sin θ
4πc3 r 2
2
¨ 0 )k2
Ptot =
kd(t
3c2
Power of oscillating dipole, d = qa cos ωt (p. 165)
αβγ
pc2
2
(rr · u)3
B=ˆ
rr × E
E=
αβγ µνλ = δµνλ
αβγ µνγ = δαµ δβν − δαν δβµ
v=
u = cˆ
rr − vr
rr
E=e
0
Useful relations for relativistic mechanics
∂φ vφ
˙ c (tr ); and tr is the retarded
where rr = x − xc (tr ); vr = x
time and satisfies c(t − tr ) = kx − xc (tr )k.
E and B fields of point charge (p. 148)
A (r, t) ≈
Bianchi identity (p. 88): ijkl ∂j Fkl = 0
αβγ µβγ = 2δαµ
ˆ
∂φ t φ
∂θ (vθ sin θ) +
2F ij δFij
2F ij ∂i δAj − 2F ij ∂j δAi
αβγ µνλ Aαµ Aβν Aγλ = 6 det A
ˆ = cos θ ˆ
z
r − sin θ θˆ
ˆ
ˆ + sin θ sin φ y
ˆ + cos θ z
ˆ
r = sin θ cos φ x
ˆ − sin θ z
ˆ
θˆ = cos θ cos φˆ
x + cos θ sin φ y
ˆ = − sin φ x
ˆ + cos φ y
ˆ
φ
r
=
4F ij ∂i δAj = 4∂i (F ij δAj ) − 4(δAj )∂i F ij
(2)
Li´
enard–Wiechert potentials (p. 146)
ec
0
A (x, t) =
crr − vr · rr
evr
A(x, t) =
crr − vr · rr
du
F u
Equation of motion for charged particle: mc dsi = e
c ij j
j = 0 gives: dE
=
eE
·
v
dt
j = 1, 2, 3 gives: dp
=e E+ v
×B
dt
c
Useful variation
δ(F ij Fij ) = F ij δFij + Fij δF ij
C
∇t = ∂r t ˆ
r+
3
˙ c (t)δ (x − xc (t))
j(x, t) = ex
i
2
(1) divergence theorem; (2) Green’s first
(3) Green’s theorem; (4) Stokes’s theorem
Spherical coordinates (Griffiths)
3
ρ(x, t) = eδ (x − xc (t))
A
∇ × (∇ × A) = ∇(∇ · A) − ∇ A
S
Four-current for point particle (p. 142)
T
η km ij
1
kj m
F F j +
F Fij
4π
16π
km
∂k T
= 0 (172.1)
=−
T
km
mk
=T
(p. 172)
1
2
2
(E + B ) = Eem
T
=
8π
S
α0
0α
T
=T
=
c
1
αβ
T
=−
Eα Eβ + Bα Bβ
4π
1
2
2
− δαβ (E + B )
(p. 176)
2
αβ
σαβ = −T
00
σ is Maxwell stress tensor. T α0 is c times momentum density. T 0α is 1/c times energy flux density. T αβ is momentum flux.
Total electromagnetic force on volume (p. 178)
Z
β
α αβ
FV =
n T
da
S
Z
1 d
β
β
3
Sd x
FV = fV +
c2 dt V
β
n is inward unit normal. fV is total force on charges and
currents,
Z
j
β
3
ρE +
fV =
× Bd x
c
V
R
3
1
2 V S d x is total electromagnetic momentum.
c
Classical electron radius (p. 187): re =
Radiation reaction (p. 198): ma = F +
Electromagnetic duality:
e
me c2
2e2 a
˙
3c3
≈ 10−13 cm
E→B
B → −E
µ
~
d→
c
µ
~ → −cd
d is electric dipole moment, µ
~ is magnetic dipole moment
Potential of uniformly moving charge (x0 = (vt, 0, 0),
HW2)
γe
φ= p
γ 2 (x − vt)2 + y 2 + z 2
Ax = βφ Ay = 0 Az = 0
Field of uniformly moving charge (HW3)
Ex =
Ey =
Ez =
c2
Inhomogeneous wave equation in Lorenz gauge (p. 133):
Ai = 4π
ji
c
Retarded four-potentials in Lorenz gauge (142.1)
Z i 0
1
j (x , t − kx − x0 k/c) 3 0
i
A (x, t) =
d x
c
kx − x0 k
km
γe(x − vt)
Bx = 0
(γ 2 (x − vt)2 + y 2 + z 2 )3/2
γey
By = −βEz
(γ 2 (x − vt)2 + y 2 + z 2 )3/2
γez
Bz = βEy
(γ 2 (x − vt)2 + y 2 + z 2 )3/2
~×E
i.e., B = β
Boundary conditions for E and B fields
+
−
+
E⊥ − E⊥ = 4πσ
−
B⊥ − B⊥ = 0
4π
−
ˆ
−
=0
− Bk =
~
κ×n
c
σ is surface charge density, ~
κ is surface current density
+
Ek
−
Ek
+
Bk

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