Corrections TD Électricité 3 SMP4

−→
p
−→
E0=E0−→ ex
(r, θ)
V0
−→
E0
V0
V
−→
E0
V= 0
−→
p−→ exx
Q
x
−→
E
−→
F
W
−→
E′
W′
W′W

−→
p1−→ p2
r
W2−→ p2
−→
p1
θ1θ2
−→
p2
εr−→ E0
−→
E
−→
E0εr−→ P
−→
P−→
Ep
−→
Ep
−→
Ep

−→
p
−→
E0=E0−→ ex
(r, θ)
V0=−∫−→ E0.d−−→ OM =−∫E0dx =−E0x+
V0
V0=−E0x=−E0rcos θ
V
−→
E0−→ p
V(M) = pcos θ
4πε0r2−E0rcos θ= cos θ(p 4πε0r2−E0r)
V= 0 ⇒θ=±π/2r3=p
4πε0E0
V= 0 θ=±π/2
r= ( p
4πε0E0
)1/3
−→
E(M) = −−−→ gradV =−∂V
∂r −→
er−1
r
∂V
∂θ −→
eθ
= cos θ(p
2πε0r3+E0)−→
er+ sin θ(p
4πε0r3−E0)−→
eθ

[r= ( p
4πε0E0
)1/3, θ =π/2] [r= ( p
4πε0E0
)1/3, θ =−π/2]
−→
p−→ exx
Q
x
−→
E=p
2πε0x3−→
ex
−→
F=Q−→ E=pQ
2πε0x3−→
ex
x
W
−→
F=−−−→ gradW =−∂W
∂x −→
ex
W=−pQ
2πε0∫dx x3=pQ 4πε0x2+
W→0x→ ∞ ⇒ = 0
W=pQ
4πε0x2
−→
E′=−Q
4πε0x2−→
ex
W′
W′=−−→ p .−→ E′=pQ
4πε0x2