Feuille d`exercices : Champ électrique en régime stationnaire

−−→
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

a Q
δQ
δQ
δW
Q
a Q
Ep(Q) = 3
20π
Q2
ε0a
a
∆Uea
∆Ue∼2,79 11
5
11
6a=
10−15
ρ( ) = 
0 0 ≤r < a (a > 0)
ρ0a≤r < a +ε(ε > 0)
0a+ε≤r
ε ρ0ε
σ0a σ0
ρ0ε

lim
1→
2→~
E(2)−~ E(1)=σ( )
ε0
~n1→2
R
ρ
~go
R0< R
h>R0ρ0< ρ
δgo
go
λ0
~
E( )
V

~
E
z
2a−λ0
(−a, 0,0) x y
+λ0(+a, 0,0)
V(r, θ)x y
r θ r a
x y x
x0V=V0
x y
Er(r, θ)Eθ(r, θ)
r
sin θ=
y
V(M) = q
4πε0rexp −r ar
~
E(M)
Φ(r)~ E(M)
r
r→0~ E(M) Φ(r)
r r + dr
q0

m q
d t = 0