Induction électromagnétique et relativité

t,⃗
i,⃗ j,⃗ k
tt
R(Oo, τ,⃗ i,⃗ j,⃗ k)t
o+tτ τ ⃗
i
Ett
DτE(Ot)t
t
−→
EτBs(
⃗
i,⃗ j,⃗ k)RτE
Rτ⊕−→ Eτ
B(τ, Bs
B B′
B′B Bs
τ
Ro, τ,⃗ i,⃗ j,⃗ kR′0, τ′,⃗ i′,⃗ j,⃗ k
τ′=λτ +α
⃗
i , ⃗ i′=µτ +β⃗ i
t=λt′+µx′, x =αt′+βx′, y =y′, z =z′
λ=β , α =c2µ λβ −µα = 1
λ α λ2−α2/c2= 1 R R′
−→
E,−→ B−→ E′,−→ B′BsBs
E′
x′= Ex,E′ y′=λEy−αBz,E′ z′=λEz+αBy
B′
x′= Bx,B′ y′=λBy+α
c2Ez,B′
z′=λBz−α
c2Ey
c2t−x2=c2t′2−x′2
Q(t, x, y, z)

R R′0, τ′,⃗ i′,⃗ j′,⃗ k′
τ′
τ
Q
E
Q L
B(τ,⃗ i,⃗ j,⃗ k) Φ Q
Q(τ) = c2,Q(
⃗
i) = 0,Φ(τ,⃗ i) = 0,Φ(
⃗
i,⃗ j) = 0 ⃗ j,⃗ k
Q E
B Q
τΦτ−Q ττt
τ τ′Q
τ[τ, τ′]
⃗
i⃗ i=a(τ′−γτ ) (γ2−1)a2c2= 1
τ′=γτ +α
⃗
i α = 1/a γ2(1 −α2/c2) = 1 |α|< c
τ∼
=τ′Φ(τ, τ′)>0
τ∼
=τ′τ′∼ =τ′′ τ′
τ=γτ ′+α
⃗
i1τ′′ =γ′τ′+α′
⃗
i2Φ(τ, τ′′) = c2γγ′−αα′
⃗
ii·⃗ i2
|αα′|< c2Φ(τ, τ′′)>0
E
τ τ′
Φ
τ τ′
(⃗ j, ⃗ k)⃗ i−→ i′τ τ′
τ′=γτ +α
⃗
i, τ =γτ ′+α′−→ i′Φ(τ, τ′) = γc2, γ2(1 −(α/c)2=γ21−(α′/c)2= 1
|α|=|α′|| ̸= 0 τ′α′−→ i′=−α(γ⃗ i+α/c2τ=
τ τ′(
⃗
i,⃗ j,⃗ k) (−→ i′,⃗ j,⃗ k)
−→
i′α′=−ατ τ′
L
L+L
(aµ
λ)a1
1
Φ
′
D,D′τ, τ′o

′−→
AA′
Φτ τ′
τ τ′
⃗
j,⃗ k
τ′=γτ +α
⃗
i , ⃗ i′=α
c2τ+γ⃗
i τ =γτ ′−α
⃗
i′,⃗ i=−α
c2τ+γ⃗
i′
′
′t t′
−−−→
OtO′
t′=t′τ′−tτ = (γt′−t)τ+αt′
⃗
iou (t′−γt)τ′+αt
⃗
i′
•O′
t′tγt′=t−−−→
OtO′
t′=αt′
⃗
i−−−→ OtO′
t′= (α/γ)t
⃗
i
v=α/γ −−−→ OtO′
t′=vt
⃗
i v⃗ i′
•′−−−→ O′
t′Otγt =t′
−−−→
O′
t′Ot=−vt′
⃗
i′−v⃗ i i ′
⃗
i̸=⃗ i′
τ
νRτ⊕Eτν=
ω(τ+c⃗u)⃗u
⃗u ω
-
*
6
 τ
τ′
⃗
i
−v⃗ i′
⃗
i′
s
s
s-
′
t′
γ t′
t′
γ
o
v⃗ i
s
s s
s
s
o
t
′
t′
1
′
t/γ
′
t′
2
6
τ
τ′
−−−→
OtO′
t′
c2(t−γt′)2−v2t′2= 0
t′t t′
1=ct/(γc +v), t′ 2=ct/(γc −v)
v
′
t′
1t′
t′
2
∆′=t′
2−t′ 2′∆′=2vct
γ2c2−v2=2vt c
′
t/γ t
−−−−→
OtO′
t/γ =vt
⃗
i d t
′
t′/γ d=c∆′/2′
′
t′
1
′
t′
2′
′d=c∆/2 ∆

−→
E−→ Bτ−→ E′,−→ B′
′τ′=γ(τ+v⃗
i)
E′
x′= Ex,E′ y′=γEy−γvBz,E′ z′=γEz+γvBy
B′
x′= Bx,B′ y′=γBy+γv
c2Ez,B′
z′=γBz−γv
c2Ey
τ⃗v ∧−→
B⃗v ∧−→ E
τ′
−→
E′= Ex
⃗
i′+γEy⃗ j+ Ez⃗ k+⃗v ∧−→ B,−→ B′= Bx
⃗
i′+γBy⃗ j+ Bz⃗ k−⃗v
c2∧−→
E
−→
E = −→ E∥+−→ E⊥,−→ B = −→ E′
∥+−→
E′
⊥
−→
E∥= Ex
⃗
i , −→ E′
∥= Ex
⃗
i′
⃗
i′⃗ i
−→
B′
∦=−→
B∥
′
Rτ⊕Eτ⃗ i′=γ⃗ i+v
c2τ
−→
E′=γ⃗v ·−→
E
c2τ+γ−→
E + γ ⃗v ∧−→ B−→ B′=γ⃗v ·−→
B
c2τ+γ−→
B
−→
E′=γ⃗v ·−→
E
c2τ+γ−→
E + γ ⃗v ∧−→ B,−→ B′=γ⃗v ·−→
B
c2τ+γ−→
B−γ⃗v ∧−→
E
c2
−→
E˙
−→
B−→ E2−c2−→ B2
−→
E′−→ E′=γ ⃗v ∧−→ B
dΦ
e=−dΦ/dt
dφ −→ AB
-
6
−→
H
′
′de =−dφ
dt
dφ :
′l
′o′
τ τ′v⃗ i−v⃗ i′ ′
′⃗
i⃗ i′[τ, τ′]
τ τ′
−→
AB = l⃗ j⃗ jEτ∩Eτ′⃗ k=⃗ i∧⃗ j
⃗
i,⃗ j⃗ i′,⃗ jτ τ′

-
6
3
+
o
o
o
r
r
r
r
r
r
r
t′
t′
t′
t′
′
t′
6
τ
τ′
−→
B
⃗
jvt
⃗
i
−→
E′
(t′=t/γ)
(C0)
(Ct)
+
−→
B
′−→
E′−→ E′=γ⃗v ∧−→ B
−→
E′⃗ i∧−→ B
l· ||−→ E′|| t′t′
−→
Bt l·||vt
⃗
i∧−→ B||
−→
B′ ′
t′t′−→
B′=
γBx
⃗
i′+γBy⃗ j+ Bz⃗ kl· ||vt′
⃗
i′∧−→ B|| =l· ||vt′
⃗
i∧γ−→ B|| t=γt′
t′l· ||v⃗ i∧γ−→ B|| =l· ||−→ E′||
ΓC2
sΓdM
ds
d2M
ds2
τ
QdM
ds =c2Φ
d
dsQdM ds = 2ΦdM ds ,d2M ds2= 0
R
Ms= Ot+⃗x(t)t=tτ
dM
ds =γ(τ+˙
⃗x)γ=dt
ds
⃗v =˙ ⃗x
τ
ΦdM
ds , τ=γc2>0γ=dt ds >0
c2=γ2(c2−v2)dt
ds2
(v=||⃗v||
ΦdM
ds , τ=γ > 0
dt
ds =γ=1
1−v2/c2
Γ