DS N 4 : Électrostatique

◦4
−−→
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

+e−e
r
ρe=αexp [−β r]
~
E( ) V( )
~
E( )
~
E( ) V( )
~
E+( ) V+( )
α β
q
r r +r α β r r
λ(r)
q=λ(r)r
r=r0r0

e α β
ρ(r)e r0
λ(r) = −e
r02r r02
exp −2r
r0
λ:r→λ(r)
r= 0 r=r0r→ ∞ ~
E−( )
rr0rr0
Q(r)
r
Q(r=r0) = (−e) [1 −5 exp(−2)]
~
E−(r=r0)
e= 1,6.10−19 r0= 0,53.10−10 1
ε0= 36π.109
k~ E−(r=r0)k
~
E( ) V( )
r→0r→ ∞
σ0
r0
σ0
~
E( ) V( ) r > r0
~
E( ) 0 < r < r0e r ε0
r=r0
∆V=−ρ
ε0
ρ0< r < r0
0< r < r0a b
R+∞
0rnexp [−β r]r=n!
βn+1

r→r0
V( ) 0 < r < r0e r r0ε0
r V
k~ Ekr= 0 r=r0
r→ ∞
d
Uc= 0
VA>0
ρ V
v I
x
V(x)ρ(x)

q V
v(x)
V(x)
I(x)S
x < d I
ρ(x)v(x)
S
I x
V
a=I
Sε0rme 2e
V(x) = 3
2√a x4
3
dV
dx
x= 0
I VA
VA
I
I=f(VA)VA
d= 3,00 S= 3,00 2
VA= 10,0
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