Magnétostatique : Notes de cours (Physique)

dq S dt I
A
I=dq
dt=ZZM∈S
j(M)·−−→ d2S
ji
qivini
j=X
i
niqivi
j j
A·m−2
M j(M)/R1R1
j(M)/R2R2
j(M)/R1= j(M)/R2+ρ(M).v(M)R2/R1
v(M)R2/R1M R2R1
I C u
I=ZM∈C jS(M)·u d
jSA·m−1
•ρ(x, y, z) = ρ(x, y) j(x, y, z) =  j(x, y)
•ρ(r, θ, z) = ρ(r, z) j(r, θ, z) =  j(r, z)
•ρ(r, θ, z) = ρ(r)
◦

•ρ(r, θ, ϕ) = ρ(r) j(r, θ, ϕ) =  j(r)
(xOy)⇔
jx(x, y, −z)=+jx(x, y, z)
jy(x, y, −z)=+jy(x, y, z)
jz(x, y, −z) = −jz(x, y, z)
(xOy)⇔
jx(x, y, −z) = −jx(x, y, z)
jy(x, y, −z) = −jy(x, y, z)
jz(x, y, −z)=+jz(x, y, z)
(Oz)R j//uz
(Oz)
j(r) (ur, uz) (ur, uθ)
div  B= 0
◦

φ1 B=ZZM∈S1 B(M). d2S1=φ2 B=ZZM∈S2 B(M). d2S2
S1S2 B
Σ
ZZ
B.−−→ d2ΣZZ B.−−→ d2Σ = ZZZ div  B.d3τ= 0
ZZ
B.−−→ d2Σ = −φ1 B+φ2 B
⇒
−→
rot  B=µ0 j+µ0ε0
∂ E
∂t
µ0= 4π×10−7H·m−1
−→
rot  B=µ0. j+µ0ε0
∂ E
∂t =µ0.(
j+ jD) = µ0. j+1
c2
∂ E
∂t
jD=ε0.∂~
E
∂t µ0.ε0=1 c2
IM∈C
B(M). d =ZZM∈S
−→
rot  B.−−→ d2S=µ0.ZZM∈S j(M).−−→ d2S+ZZM∈S jD(M).−−→ d2S
⇒
C S C C
IM∈C
B(M)·−→ d=µ0Iint
Iint =sM∈S j(M)·−−→ d2S
◦

B
BΠ
Π∗
(Oz)R j//uz
B
j
j(r) (ur, uz) (ur, uθ) B(r) = Bθ(r)uθ
◦

(Oz)R
I
j=jz(r).urRr=R
r=0 Rθ=2.π θ=0 
j.  d2S=I
j(r > R) =  0
j(r≤R) = I
π.R2uz
j(r)⇒ B(r) j(ur, uz) j(ur, uθ)
B=Bθ(r).uθ(Oz)r
dl =r.dθ.uθ d2S=dr0.r0.dθ.uz
I
B. dl =Bθ(r).2.π.r =µ0.Iint =µ0.Zr0=r
r0=0 Zθ=2.π θ=0
jz(r0).dr0.r0.dθ
r < R ⇒Iint =I.r2
R2⇒Bθ(r) = µ0.I.r 2.π.R2
r≥R⇒Iint =I⇒Bθ(r) = µ0.I
2.π.r
◦