8.07 – Fórmulas básicas ϵijkϵpqk = δipδjq − δiqδjp , a · b = aibi , (a

8.07 – Fórmulas básicas
a · b = ai bi ,
ijk pqk = δip δjq − δiq δjp ,
(a × b)i = ijk aj bk
det A = i1 i2 ···in A1,i1 A2,i2 · · · An,in
F · da ,
F · dl
∇ · F =
∇ × F =
V
S
Γ
S
=0
∇ × (∇φ) = 0,
∇ · (∇ × A)
δ(x − xi )
, g(xi ) = 0
δ(g(x)) =
(x )|
|g
i
i
∇r = er
∇2
1
= −4πδ(x)
r
1 ∂Φ
1 ∂Φ
∂Φ
er +
eθ +
eφ
∂r
r ∂θ
r sin θ ∂φ
∂2Φ
1
∂ ∂Φ 1
1 ∂ 2 ∂Φ 2
r
+ 2
sin θ
+ 2 2
∇ Φ= 2
r ∂r
∂r
r sin θ ∂θ
∂θ
r sin θ ∂φ2
∇Φ =
|E(r)|
=
1 q
4π0 r2
V (r) =
1 q
4π0 r
ρ
· da = Qencl ,
= −∇V
E
E
∇2 V = −
0
0
S
x)ρ(x )
1 qi qj
1
0 2
3
3 ρ(
W =
→
→ w = |E|
d xd x
8π0
|xi − xj |
8π0
|xi − xj |
2
= ρ,
∇·E
0
i=j
V (x) =
1
4π0
Q = CV → Qi = Cij Vj
1
∂V
∂G(x, x ) 3 G(x, x ) − V (x )
da
d x G(x, x )ρ(x ) +
4π S
∂n
∂n
∇2x G(x, x ) = −4πδ(x − x )
G(x, x ) = “potencial” en x debido a una unidad de carga en x , cuando ...
a
Imagen fijada para esfera: (q , y), (q , y ) con y y = a2 , q = − q
y
cos αx
cos βy
cosh γz
una solución cartesiana de Laplace
γ 2 = α2 + β 2
sin αx
sin βy
sinh γz
{rl , r−(l+1) } Pl (cos θ)
soluciones acimutales de Laplace
∞
∞
rl
1
<
P (cos γ) ,
=
l+1 l
|x − x |
r>
1
√
=
hl Pl (x)
1 − 2hx + h2
l=0
l=0
1
1
l
Pl (1) = 1
Pl (−x) = (−1) Pl (x)
Si1 ···in = (det R) Ri1 j1 · · · Rin jn Sji ···jn
−1
dx Pl (x)Pl (x) =
2
δll
2l + 1
(pseudo) tensor bajo rotación
p · x 1 xi xj
1 Q
+
Q
+
·
·
·
+
ij
4π0 |x|
|x|3
2 |x|5
3
pi = d x ρ(x) xi Qij = d3 x ρ(x)(3xi xj − δij |x|2 ) ,
V (x) =
Q=
dip =
E
d3 x ρ(x) ,
1 1
(3(
p · r̂)r̂ − p )
4π0 r3
,
da ⇔ I dl
J d3 x ⇔ K
→
dip =
E
J = ρv ,
1 p
(2 cos θ r̂ + sin θ θ̂) ,
4π0 r3
p = pẑ
d
p
= q v × B
dt
∇ · J = 0,
× (x − x )
µ0 I dl
= µ0 I ∇Ω
,
B
3
4π |x − x |
4π
x )
µ
0
3 J(
= 0,
B = ∇ × A,
A(x) =
d x
, ∇·A
4π
|x − x |
· dl = µ0 Ienlazado ,
B
∇·B =0
∇ × B = µ0 J ,
x) =
dB(
Γ
× x
= µ0 m
,
A
4π |x|3
· x = − µ0 ∇ m
B
,
4π
|x|3
1
m
=
2
x) ,
d3 x x × J(
dip = µ0 1 (3(m
dip = µ0 m (2 cos θ r̂ + sin θ θ̂) ,
B
· r̂)r̂ − m
) → B
3
4π r
4π r3
d
· da
E = (E + v × B) · dl = −
B
dt
Γ
S
1
· J d3 x
·B
d3 x = 1 A
B
W =
2µ0
2
µ0
Mij =
4π
Ci
Cj
dEmec
dt
i · dl
j
dl
1
,
ij (x) · da
=
B
|xi − xj |
Ij Si
Mij = Mji
= µ0 J + µ0 0 ∂ E
∇×B
∂t
×B
,
= 1E
d3 x J · E
S
=
µ
0
V
2
m
= IA
m
= mẑ
dEmec
dt
d
+
dt
1 · B)
=−
d x (E
·D+H
2
+ J × B)
d3 x (ρE
=
· da
S
3
V
dPmec
dt
S
V
1 2 Tαβ = 0 Eα Eβ + c Bα Bβ − (E · E + c B · B)δαβ
2
dP
d
mec
3
d x 0 (E × B) = + Tij nj da
+
dt
dt V
i
S
2
=∇×A
= −∇Φ − ∂A , B
E
∂t
|
x−
x | x , t − |x−x |
ρ
x
,
t
−
J
µ
1
0
c
c
x, t) =
,
A(
d3 x
d3 x
Φ(x, t) =
4π0
|x − x |
4π
|x − x |
x, t) = Ee
i(k·x−ωt)
E(
x, t) = Be
i(k·x−ωt)
B(
= √µ n × E
× n ,
B
E = Z H
= 0,
n · E = n · B
Z=
µ/
∗ )
◦ B
= 1 Re(E ◦ B
E
2
x)
x) = 1 ∇ × A(
H(
µ0
Radiación : e−iωt
dP
c2 Z0 4
k |(n × p) × n|2 ,
=
dΩ
32π 2
ikr
= − iµ0 ω p e ,
A
4π
r
radiación dipolar magnética
dP
dP
=
(
p → m/c)
,
dΩ
dΩ
= 0 E
+ P ,
D
=ρ
∇·D
ρpol = −∇ · P ,
,
= B −M
H
µ0
= J ,
∇×H
,
Jb = ∇ × M
= −∇ΦM
H
→
x) = iZ0 ∇ × H(
x)
E(
k
P =
c2 Z0 4 2
p|
k |
12π
P = P (
p → m/c)
= E
material lineal : D
σpol = n · P
= µH
material lineal : B
b = M
× n
K
∇2 ΦM = −ρM ,
3
,
ρM = −∇ · M
· n
σM = M