Feuille d`exercices : Électrostatique, Équations de Maxwell

−−→
gradf=∂f
∂r ~ur+1 r
∂f
∂θ ~uθ+∂f ∂z ~uz
div ~ A=1
r
∂
∂r (rAr) + 1 r
∂Aθ
∂θ +∂Az ∂z
−→
rot( ~ A) = 1
r
∂Az
∂θ −∂Aθ ∂z ~ur+∂Ar ∂z −∂Az ∂r ~uθ+1 r∂ ∂r (rAθ)−∂Ar ∂θ ~uz
∆f=1
r
∂
∂r r∂f ∂r +1 r2
∂2f
∂θ2+∂2f ∂z2
−−→
gradf=∂f
∂r ~ur+1 r
∂f
∂θ ~uθ+1 rsin θ
∂f
∂ϕ ~uϕ
div ~ A=1
r2
∂
∂r (r2Ar) + 1 rsin θ
∂
∂θ (sin θAθ) + 1 rsin θ
∂Aϕ
∂ϕ
−→
rot( ~ A) = 1
rsin θ∂ ∂θ (sin θAϕ)−∂Aθ ∂ϕ ~ur+1 rsin θ
∂Ar
∂ϕ −1 r
∂
∂r (rAϕ)~uθ+1 r∂ ∂r (rAθ)−∂Ar ∂θ ~uϕ
∆f=1
r2
∂
∂r r2∂f ∂r +1 r2sin θ
∂
∂θ sin θ∂f ∂θ +1
r2sin2θ
∂2f
∂ϕ2

q
xA=−a xB=a
m q0
R= 50 x y
Q= 1,0.10−9
λ=Q
2πR
q
z x y z 6= 0
λ
q
x−q
a
v0
v0,min v0

+q
b v0
V0= 0
V1
me= 9,1.10−31
V1
V1
|V1|= 10
a
He2+ q(r, t)
r > a
r r + dr r > a
ρ(r, t) = 1
4πr2
∂q(r, t)
∂r
r > a
∂q(r,t)
∂t
~
j(, t) = −1
4πr2
∂q(r, t)
∂t ~er

r > a
~
E(, t) = q(r, t)
4πε0r2~er
a
t= 0
α
~v =v0~erv0
ρ(r, t)
r r + dr
~
j(, t)
~
j(, t) = ~ 0r > v0t~ j(, t) = −αe
4πr2~er0< r ≤v0t
~
E
V(r, t)V(r, t)
ε0∂~
E
∂t
ρ= 0 ~ j=~ 0
~
E=f(z) exp −t
τ~ex~
B=g(z) exp −t
τ~ey
~
E=−
−−→
gradV

f g
f~ E(x=0,y=0,z=0,t=0) =E0~ex
z a
q ~v =v0~ez
I0
I0a v0
z~ E=E(r)~er
~
B=B(r)~eθ
~
E, ~ B
~
B=ε0µ0~v ∧~ E
ε0µ0
ε0∼1
36π.109−1µ0= 4π.10−7−1
1
√ε0µ0
~v
~
B0
q