a b> a
Q
−Q
ℓ
C = Q
Q ∆V
b
∆V = V b
− V a =
− a
d E r
·
r E
ℓ ℓ
≫a
ℓ
≫b
ℓ
r
rˆ a
φ=
ndS E n ˆ dS =
·
S
q int int
ndS E n ˆ dS = E 2πrℓ
·
manto
Q
E 2πrℓ = = E
Q rˆ 2πǫ0 rℓ
r
b d E r = E ˆ E rˆ d r = Edr =
·
·
Q dr 2πǫ0 rℓ
r=a b
V b
− V a =
− a
V b
Q dr = 2πǫ0 rℓ
−
r=b Q 2πǫ 0 ℓ
− V a = − 2πǫQ0ℓ ln( ab )
∆V
C =
Q = ∆V
Q Q ln( ab ) 2πǫ 0 ℓ
=
b
2πℓǫ 0 ln(b/a ln(b/a))
a
1 dr r
Q ǫ0
φ=
ndS E n ˆ dS =
·
S
q int int
ndS E n ˆ dS = E 2πrℓ
·
manto
Q
E 2πrℓ = = E
Q rˆ 2πǫ0 rℓ
r
b d E r = E ˆ E rˆ d r = Edr =
·
·
Q dr 2πǫ0 rℓ
r=a b
V b
− V a =
− a
V b
Q dr = 2πǫ0 rℓ
−
r=b Q 2πǫ 0 ℓ
− V a = − 2πǫQ0ℓ ln( ab )
∆V
C =
Q = ∆V
Q Q ln( ab ) 2πǫ 0 ℓ
=
b
2πℓǫ 0 ln(b/a ln(b/a))
a
1 dr r
Q ǫ0
R
2d
d >> R
V 0
Q
d >> R
(r ) = E ρˆ
0
· σ ǫ0
R r
r
≤
x<0 V 0 =
△V
− · − · − − − − · − − − − − − − − − − · (r) d E r
=
Γ
d+R
=
ˆ σRˆ σRx ǫ0 (d + x)
d R
σR ǫ0
=
2σR ln ǫ0
=
dxˆ dxx ˆ
1 1 + dx d+x d x
d R
σR ln ǫ0
=
σ=
d+R
ˆ x x)
σR ǫ0 (d
d+R
x+d d x 2d
d R
R
R
Q (2πR (2πR))L
Q V = ln ǫ0 πL
△
·
2d
−R
R
⇒ |△ |
Q V = ln ǫ0 πL
=
·
C =
⇒ C = ln ǫ20dπ−R =∼ lnǫ0π2d
=
R
R
2d
−R
R Q
|△V |L
−Q
d >> R
A d Q V 0
C (x) = ǫ0
A x
U =
= F
C (x) = ǫ0
U =
−∇
−
1 Q2 = 2 C
2
Q ⇒ U (x) = 12 Aǫ x 0
dU x ˆ= dx
−
1 Q2 xˆ 2 Aǫ0
1 C (x)(V 0 )2 = 2
⇒ U (x) = 12 Ax ǫ0(V 0)2
A x
= + U = + dU x F ˆ= dx
∇
−
Aǫ0 (V 0 )2 1 x ˆ 2 x2
x=d = F
−
Aǫ0 (V 0 )2 1 x ˆ= 2 d2
−
V 0 CV 0 xˆ = 2d
−
1 Q2 x ˆ 2 Aǫ0
F /A =
−
σ2 n ˆ 2ǫ0
a
b
a
rˆ
d +Q
−Q
(r) = E (r)ˆ E r a
a
△V 1 =
Q rˆ ⇒ E (r) = 4πǫ 2 0r
·
−
(r) d E r=
·
b
−
Q 4πǫ0
C i =
a
b
dr Q 1 = r2 4πǫ0 r
Q = V 1
△
a b
Q = 4πǫ0
− 1 a
1 b
4πǫ 0 1 a
1 b
− −Q
Q + Qc (r) n E ˆ dS = 0 = = ǫ0 Γ
⇒ Qc = −Q
·
Q d
a r r>b
≤
a
(r) = E
0 Q rˆ 4πǫ 0 r2
b < r, a > r c < r < b, a < r < c
△V 2
= a
=
− b
c
=
c
= = =
Q 4πǫ 0 Q 4πǫ 0
b
(r) d E r+
c
(r) d E r
·
d
b
(r) d E r+
a
(r) d E r
a
d
(r) d E r+
a
b
· · · · · · −− (r) d E r=
(r) d E r
d
c
b dr + 2 a r d 1 1 1 + a d c
C f =
Q = V 2
dr r2 1 b
4πǫ0
△
C i < C f
1 a
+
1 d
− 1c − 1b
4πǫ 0 0 ⇒ d1 − 1c < 0 =⇒ a1 − 1b + d1 − 1c < a1 − 1b =⇒ 14πǫ 1 = C i < C f = 1 1 1 1 a − b a − b + d − c
c
C 2
C 1 C 2 =
1 1 1 1 = + = C f C 1 C 2 4πǫ0
1 a
4πǫ0 1 d
−
−
1 b
, C 1 =
1 1 + b d
−
1 c
4πǫ0 1 a
− 1c
⇒ =
C f =
1 a
−
4πǫ0 + d1
1 b
− 1c
a
θ
θ C =
ǫ0 a2 (1 d
− aθ ) 2d
dx
C =
ǫ0 A H
H A = adx
H
h(x) a sin(θ) = x a
x H = H (x)
a
h(x) = x sin(θ) h(x) = xθ
θ
sin(θ) = θ H (x) = d + h(x)
h(x) = d + θx
ǫ0 adx d + θx
dC = x=0
x=a a
C =
0
ǫ0 adx ǫ0 a d + θa = ln( ) d + θx θ d d+θa d
θ ln( d+θa d )
ln(x) = (x x=
≈1
ln(x)
− 1) − 12 (x − 1)2
d+θa d
d + θa aθ ln( )= d d
−
1 (aθ)2 2 d2 C
C =
ǫ0 a aθ ( θ d
2
− 12 (aθ) d2
)=
ǫ0 a2 C = (1 d
ǫ0 a aθ ( (1 θ d
− aθ ) 2d
− aθ )) 2d
a
V 0 C
a
V =
kQ a
Q
V 0 V 0 =
kQ0 a
(1)
Q0 Qf V f =
kQf a
Qc C = V f =
(2) Qc V
V f
Qc C
(3)
Q0 = Qf + Qc Qc = Q0 = Qf +
kCQ f a kCQ f a
aQ ⇒ Qf = a+kC
=
0
V C
V f = Qc
kQ 0 a+kC
kQ0 a + kC Q0 QC = kC a + kC
V C = V f =
(4)
1 U = ǫ0 2
E 2 dV
V
Q0 E =
kQ0 r2
dV = 4πr 2 dr 1 1 kQ0 2πk 2 Q0 2 2 2 dU i = ǫ0 E dV = ǫ0 2 4πr dr = ǫ0 dr 2 2 r r2
U i =
∞ a
U C =
ǫ0
2πk 2 Q0 2 2πk 2 Q0 2 ǫ0 kQ0 2 dr = = r2 a 2a kQf 2 2a
Q2C 2C
kQf 2 QC 2 U f = + 2a 2C Q0 kQ0 2 U f = 2(a + kC )
∆U = U f
−
kQ0 2 1 U i = ( 2 a + kC
− a1 )
r>a
C 1 S 1
C 2
S 2
S 2 S 1 S 1
S 2
S 1 C 1
∆V Q1 = C 1 ∆V C 2
Q2 = 0 1 E i = C 1 ∆V 2 2
Q1 E i = S 2
1 Q21 2 C 1 S 1
∆V Q′1 = ∆V C 1 Q′2 = ∆V C 2 C 1 E f S 1 C 2
− E i = 12 C 2∆V 2
S 2
∆V Q′
1
C 1
C 2 E f = 12 C 1 ∆V 2 + 12 C 2 ∆V 2
Q′
2
C 2
V 1 = V 2
Q1
Q′1 Q′ = 2 C 1 C 2 C 1
Q1 Q′1 + Q′2 = Q1
E f
E f
− E i
Q′1 =
C 1 Q1 C 1 + C 2
Q′2 =
C 2 Q1 C 1 + C 2
1 Q′12 1 Q′22 = + 2 C 1 2 C 2 2 2 1 C 1 1 C 2 2 Q1 2 Q1 = ( ) + ( ) 2 C 1 + C 2 C 1 2 C 1 + C 2 C 2 1 C 1 1 C 2 = Q21 ( + ) 2 (C 1 + C 2 )2 2 (C 1 + C 2 )2
= =
Q21 C 1 C 2 1 Q21 ( + ) 2 (C 1 + C 2 )2 (C 1 + C 2 )2 2 C 1 2 Q1 C 1 C 2 1 ( + ) 2 2 2 (C 1 + C 2 ) (C 1 + C 2 ) C 1
−
−
C 1 = C 0 C 2 = 2C 0 V 0
C 3 = 4C 0 S S S
C 1
S
C 3
V 0 Q3 = V 0 C 3 C 3 = 4C 0 Q3 = 4C 0 V 0 S
C 3 Q3 = 4C 0 V 0 Q1
Q2
V 0
C 1
C 2
V 0 V 1 + V 2 = V 0 V 1 =
Q1 C 1
V 2 =
Q2 C 2
Q1 Q2 + = V 0 C 1 C 2 C 1
C 2
−Q1 Q2
− Q1 = 0
Q1 = Q2 = C 1 = C 0
C 1 C 2 V 0 C 1 + C 2
C 2 = 2C 0 2 Q1 = Q2 = C 0 V 0 3
Q′1 + Q′2 = Q1 + Q2
(1)
− Q1 = Q′3 − Q′1
(2)
Q3
V 1′ + V 2′ + V 3′ = 0 Q′1 Q′3 Q′2 + + =0 C 1 C 3 C 2
(3)
Q2
Q′2 = Q1 + Q2 Q′3 = Q3
Q′1 Q3 + C 1 Q′1 (
− Q′1
− Q1 + Q′1
− Q1 + Q′1 − Q1 + Q2 − Q′1 = 0 C 3
C 2
1 1 1 Q3 Q1 + + )+ C 2 C 2 C 3 C 3
−
− Q1 C +2 Q2 = 0
2 Q1 = Q2 = C 0 V 0 3 Q3 = C 3 V 0 = 4C 0 V 0
Q′1
7 4C 0
− 56 V 0 − 23 V 0 = 0
Q′1 =
2 C 0 V 0 21
10 C 0 V 0 7 24 Q′3 = C 0 V 0 7 Q′2 =
Q
A Q
−
d ǫ1 = 2ǫ0
ǫ2 = 4ǫ0
C 1 =
A/2 3A ǫ1 = ǫ0 2/3d 2d
C 2 =
A/2 6A ǫ2 = ǫ0 1/3d 2d
C 3 =
A/2 A ǫ0 = ǫ0 d 2d
C 1
C 2
1 1 1 = + = ′ C eq C 1 C 1
′ = C eq
′ C eq
1 + 3A ǫ 0 2d
1 5 d = 6A 6 Aǫ0 d ǫ0
6A 5 d ǫ0 C 3
6A A 17 A ǫ0 + ǫ0 = ǫ0 5d 2d 10 d Q Q q 3 C 1 C 2
′ + C 3 = C eq = C eq
−
C 3
q
q 1 + q 2 = Q
∆V 3 q 3 C 1 q 3 = Q
− ∆V 1 − ∆V 2 = 0 − C q 2 − C q 3 = 0
(1)
(2)
− q 1 (Q C 3
− q ) − q ( C 11 + C 12 ) = 0
−q ( C 13 + C 11 + C 12 ) + C Q3 = 0 12 Q 17 5 q 3 = Q 17 q =
a
b
r) = k rˆ P ( r k
r
(r) = E (r)ˆ E r r
Ω
r ) n D( ˆ dS = 4πr 2 D(r) = Qlibre = 0 =
r ) = ⇒ D( 0
·
(r) = E
r ) − ǫ10 P (
r ) = k rˆ P ( r (r) = E
− ǫk0r rˆ
Qencerrada (r) n E ˆ dS = ǫ0 Ω
ρ p
·
σ p
Ω
r ) n σ p = P ( ˆ
a
·
ρ p =
r) −∇ · P (
a) n σ p (a) = P ( ˆ k = rˆ rˆ a k = a
· ·−
−
σ p (b) = P ( b) n ˆ k = rˆ rˆ b k = b
·
·
ρ p (r) = =
r) −∇ · P ( 2 ∂ (sen(φ)P φ ) 1 1 ∂ (P θ ) − r12 ∂ (r∂rP r ) − rsen(φ) − ∂φ rsen(φ) ∂θ 2
= = =
− r12 ∂ (r∂rP r ) − r12 ∂rk ∂r k − r2
2
Qencerrada = σ p (a)4πa + = = = =
−4πka + 4π
·
ρ p (r)d3 x
Ω r
a
ρ p r2 dr
r
−4πka − 4πk dr a −4πka − 4πk(r − a) −4πkr
Qencerrada (r) n E ˆ dS = 4πr 2 E (r) = = ǫ0 Ω
−4πkr =⇒ E (r) = − k ǫ0
ǫ0 r
rˆ
r>b
r) = P ( 0 (r) = E
r ) = − ǫ10 P ( 0
Qencerrada (r) n E ˆ dS = ǫ0 Ω
·
2
2
Qencerrada = σ p (b)4πb + σ p (a)4πa + b
= 4πkb
− 4πka + 4π
= 4πk(b =
ρ p (r)d3 x
Ω
ρ p r2 dr
a b
− a) − 4πk dr a 4πk(b − a) − 4πk(b − a)
= 0
Ω
(r) n E ˆ dS = 4πr 2 E (r) = 0 =
⇒ E (r) = 0
·
(r) = E
−
0 k ˆ ǫ0 r r
0
rb
A d
≫L
d
≫ W κ1 = κ2 = κ C =
κǫ0 A d
d
d z)
d ǫ(z) = ǫ0 (1 +
a
b
L >> b,a
Q ǫ
m L
L >> a,b (r) = E (r)ˆ E r rˆ
(r) = E
△
Q V = 2πǫ 0 L
b
a
0
rb
Q rˆ 2πǫ 0 L r
0
dr Q = ln(b/a) = r 2πǫ0 L
2πǫ0 L ⇒ C = ln(a/b)
x x ˆ x C 1 =
C (x) = C 1 + C 2 =
2πǫ 0 x ln(b/a)
C 2 =
2πǫ(L x) ln(b/a)
2πǫ 0 x 2πǫ(L x) 2π + = (ǫL ln(b/a) ln(b/a) ln(b/a)
−
0
−
−∇ U = F
− dU x ˆ + mgˆ x = 0 dx
− x(ǫ − ǫ0))
mg = = =
⇒ (ǫL − x(ǫ − ǫ0))2
=
=
dU dx 1 Q 2 dC 2 C dx Q2 ln(b/a)(ǫ ǫ0 ) 4π (ǫL x(ǫ ǫ0 ))2 Q2 ln(b/a)(ǫ ǫ0 ) 4πmg
−
− − − −
−
Q
ln(b/a)(ǫ ǫ0 ) 4πmg
|ǫL − x(ǫ − ǫ0)|
= Q
(ǫL
=
±Q
=
ǫL (ǫ ǫ0 )
− x(ǫ − ǫ0)) =
⇒x
ln(b/a)(ǫ ǫ0 ) 4πmg
−
∓ (ǫ − ǫ0)
−
ln(b/a)(ǫ ǫ0 ) 4πmg
−
0
−
−
Q
=
⇒ (ǫ − ǫ0)
ln(b/a)(ǫ ǫ0 ) 4πmg
−
ln(b/a)(ǫ ǫ0 ) 4πmg
−
Q (ǫ ǫ0 )
−
x0 =
x0 = =
⇒L
>
ǫL (ǫ ǫ0 )
−
Q ǫ
−
< L 1
ln(b/a)(ǫ ǫ0 ) 4πmg
−
− (ǫ −Qǫ0)
ǫL (ǫ ǫ0 )
−
< L
− ǫ0 )
=
ln(b/a)(ǫ ǫ0 ) 4πmg
−
ln(b/a)(ǫ ǫ0 ) 4πmg
−
(ǫ
>0
− (ǫ −Qǫ0)
ǫ
ln(b/a)(ǫ ǫ0 ) >0 4πmg
−
0 − (ǫ Lǫ − ǫ0 ) < 0
x0 = =
⇒L
Q ǫ
<
ǫL (ǫ ǫ0 )
− − − Q ln(b/a)(ǫ − ǫ0 )
ǫ0
−
⇒
<
ǫL (ǫ ǫ0 )
−
ln(b/a)(ǫ ǫ0 )
−
4πmg
ln(b/a)(ǫ ǫ0 ) 4πmg Q = ǫ
x0 =
Q (ǫ ǫ0 )
− (ǫ −Qǫ0)
Q ǫ0 Q ǫ0
ln(b/a)(ǫ ǫ0 ) 4πmg
−
ln(b/a)(ǫ ǫ0 ) 4πmg
−
a 2a
q ǫ(r) = (1 + ar )ǫ0
2, 5a
•r
si r < a
= ǫ0 E = 0, D
r
• a < r < 2a q enc n E ˆ dS = ǫ0 S
= E
·
q
q rˆ, si a < r < 2a 4πǫ0 r2
= ǫ0 E = D
q rˆ, si a < r < 2a 4πr 2
• 2a < r < 2, 5a
S
n D ˆ dS = q enc
·
q rˆ 4πr 2
= D = ǫE D
ǫ
r = E
q
rˆ,si 4πǫ0 r2 (1 + ar )
2a < r < 2, 5a
• r > 2, 5a q rˆ 4πǫ0 r 2
= E
q rˆ 4πr 2
= D P D = ǫ0 E +
D = ǫE
= (ǫ P
− ǫ0)E
ǫ = ǫ0 (1 + ar ) = (ǫ0 (1 + r ) P a
= P
− ǫ0) 4πǫ0(1q + r )r2 rˆ a
q rˆ 4πar(1 + ar )
σ p (ˆ σ p = P n)
·
r = 2a σ p int = P
q · −r = 24πa 2
r = 2, 5a r = σ p ext = P
·
q 35πa2
ρ p ρ p =
−∇ · P
ρ p =
−∇ · P = − r12 ∂r∂ (r2P )
ρ p =
− 4πr 2(aqa+ r)2
d ǫ ǫ(x) ǫ(x) = ǫ0 (
1 1
)
2
− 3dy
2
C =
Q V
σL =
Q S
S
q enc = σL S
S
dS = D 1 S 1 + D 2 S 2 = q enc = σL S D
·
·
·
1 S 1 + D 2 S 2 = D
·
· −D1s + D2S = σLS −D1 + D2 = σL D1 = 0
D2 = ǫE 2 = σL E 2 = σǫL σL ǫ E 2 =
E 2 =
Q Sǫ 0 (
1 1
2
− yd
)
3 2
Q (3d2 2 3Sǫ 0 d
− y2 )
d
V =
0
d
E s (y)dy =
V =
C =
0
Q (3d2 2 3Sǫ 0 d
8 Qd 9 Sǫ 0
Q 9 Sǫ 0 = V 8 d
− y2 )
A
d
y
V 0
y ǫ = ǫ0 (1 + ) d
ρL = 0
∇ · D = ∂y∂ D = ρP = 0 V 0
V 0 = = E V 0 = D ǫ
D ǫ
d E l
·
ǫ = ǫ(y)
D d l=D ǫ
·
V 0 =
1 dy ǫ
Dd ln(2) ǫ0
D D=
E =
V 0 ǫ0 d ln(2)
D V 0 = ǫ d ln(2)(1 + yd )
dy = dl
= D P
y 0 ǫ0 − ǫ0E = dV ln(2) y+d
Q V 0
C = Q
V 0
S
dS = D 1 S 1 + D 2 S 2 = q enc = σL S D
·
·
·
1 S 1 + D 2 S 2 = D
·
· −D1s + D2S = σLS −D1 + D2 = σL D1 = 0
D2 = σL
V 0 ǫ0 Q = Sσ L = SD 2 = S d ln(2)
C =
Q ǫ0 = S V 0 d ln(2)
a b
c
κ b
a
c
q 1 c
q 2
S κ = 1
d X Q X κ = κ0 (1 + xd )
C 0
r1 r2
ℓ
+q
ℓ
−q κ = κ(r)
r1
ℓ
r2 κ(r)
r = r1 r = r2
κ(r)
a)
∇ · A =
1 ∂ r ∂r (rA r )
g(r) =
a
g0 b r
b
J = J (r)ˆr J
J I =
S
dS = J J
·
S
dS = J 4πr 2
S I rˆ 4πr 2
= J g(r) =
g0 b r
= g(r)E J J = E
b
V 0 =
a
R=
I rˆ 4πbg 0 r
I d E r= 4πbg0
·
b
a
1 I b dr = ln( ) r 4πbg0 a
V 0 I
R=
ln( ab ) V 0 = I 4πbg 0
I
Q0
a
d
a >> d ǫ
µ
RC =
ǫ µ
σ = σ x E ˆ 2ǫ0 x ˆ a >> d ǫ Q = σ x E ˆ= x ˆ 2ǫ 2ǫa2 = µE J = µQ x J ˆ = J ˆ x 2ǫa2
Q0
i =
n J ˆ dS
Ω
·
= Ja2 µQ = 2ǫ dQ = dt
−
a
− µQ 2ǫ ssi
=
0 =
ssi
0 =
ssi
0 =
=
⇒ Q(t)
=
dQ dt dQ µQ + dt 2ǫ dQ µt µQ µt exp + exp dt 2ǫ 2ǫ 2ǫ d µt Q exp dt 2ǫ µ Q0 exp t 2ǫ
· · −
Q(0) = Q0
i(t) =
−
dQ µQ0 = exp dt 2ǫ
·
µ t 2ǫ
−
= Q(t) x E ˆ = E (t)ˆ x 2ǫa2
µ
V = i R
·
V (t) i(t)
R
d
V (t) =
E (t)ˆ x x ˆdx = E (t)d =
·
0
i(t) =
V (t) =
A
Q(t)d 2ǫa2
µ Q(t) = 2ǫ
⇒ Q(t) = 2ǫµ i(t)
Q(t)d 2ǫd 1 d = i(t) = i(t) = R i(t) = 2ǫa2 2µǫa2 µ a2
·
d
ǫ
a2 C = ǫ d
RC =
1 d a2 ǫ ǫ = 2 µa d µ
·
+
U =
∞
i(t)V (t)dt
0
= = = =
⇒ R = µ1 ad2
+∞ µd Q2 (t)dt 4a2 ǫ2 0 µQ0 2 d +∞ µ exp t dt 4a2 ǫ2 0 ǫ Q0 2 d µ 0 exp t 4a2 ǫ ǫ +∞ Q0 2 d 4a2 ǫ
·
− −
a < b V 0 µ RC =
ǫ µ
(r) = E (r)ˆ E r J = µE V 0 = iR
i=
Ω
n J ˆ dS = J (r)
·
r J = J (r)ˆ r µ i
R
= i rˆ ⇒ J (r) 4π r 2
dS = J (r)4πr 2 =
Ω
(r) = 1 J (r) = i rˆ E µ 4πµ r2
V 0 =
− · − − dˆ E r
Γ
b i dr 4πµ a r2 i 1 a = 4πµ r b i 1 1 = 4πµ a b = iR
=
=
⇒R
=
i=
(r) = E
1 a
1 b
4πµ
4πµV 0 1 a
1 b
−
i rˆ = 4πµ r2
V 0 1 a
1 b
−
rˆ r2
a
b
a
ǫ
C =
RC =
4πǫ 1 a
1 b
− − · − 1 a
1 b
4πµ
4πǫ
1 a
1 b
=
ǫ µ
a
b
µ
V 0 = E (r, θ)θˆ E
θˆ
r θ E = E (r) ∂J z θ ∇ · J = 1r ∂r∂ (rJ r ) + 1r ∂J + =0 ∂θ ∂z
= µE = µE (r, θ)θˆ = J θˆ = J
⇒ J θ = J
∇ · J = 1r ∂J = 0 =⇒ J = f (r) + C = J (r) ∂θ = J (r)θˆ J
= E (r)θˆ E
i =
·
n J ˆ dS
Ω a
=
0
b+a
J (r)drdz
b b+a
= aµ
E (r)dr
b
V 0
V 0 =
dˆ E r
Γ π
=
·
E (r)θˆ rdθ θˆ
0
=
⇒
= E (r)rπ V 0 E (r) = πr
·
dˆ r = rdθ θˆ
b+a
i = aµ
E (r)dr
b
= = =
⇒R
V 0 = iR
=
aµV 0 b+a dr π r b aµV 0 a ln 1 + π b π aµln 1 + ab
A µ1 µ2
d
ǫ1 ǫ2
√ A >> d
d
V 0
= E (z )ˆz E
= J (z )ˆz J
∂J y ∂J z ∂J z x ∇ · J = ∂J + + = = 0 =⇒ J = J z = C ∂x ∂y ∂z ∂z = J ˆ J J zˆ = µE J
1 = J zˆ E µ1
2 = J zˆ E µ2
√ A >>
V 0 =
−
drˆ E
·
Γ
d2
=
d2 +d1
E 2 dz +
0
E 1 dz
d2
J d2 J d1 + µ2 µ1 d2 d1 = J + µ2 µ1 V 0 J = d d µ + µ =
=
⇒
i=
Ω
2
1
2
1
ndS J n ˆ dS = J A =
·
1 R= A
V 0 A
d1 µ1
+
d1 d2 + µ1 µ2
d2 µ2
=
V 0 R
zˆ
S D = ǫE
Ω
= D(z )ˆ D z
Ω
(D1
ndS D n ˆ dS = Qlibre = σS
n ˆ = zˆ
·
− D2)S =⇒ σ
= σS = D1 = = =
σ =
− D2 ǫ1 E 1 − ǫ2 E 2 ǫ1 ǫ2 − J µ1 µ2
ǫ1 µ1
− µǫ22
ǫ1 µ2 ǫ2 µ1 d1 µ2 + d2 µ1
−
V 0 d2 µ2
+
V 0
d1 µ1
2L
i
r
z
r = rrˆ r ′ = z zˆ
= idl di (r) = dB =
⇒ B (r)
=
= = = = = = =
(r r ′ ) µ0 idl 4π r r ′ 3 µ0 idz zˆ (r rˆ zzˆ) 4π (r2 + z 2 )3/2 L µ0 rdz θˆ i 4π −L (r2 + z 2 )3/2
× − | − | × −
· · − · ·
L
µ0 irθˆ 4π µ0 irθˆ 2π µ0 iθˆ 2π
r iθˆ
(r2
dz + z 2 )3/2
L arctan( arctan(L/r) L/r )
0 arctan( arctan(L/r) L/r)
z = rtan( rtan(α)
rsec2 (α)dα r 3 sec3 (α)
cos( cos(α)dα
0
µ0 arctan( arctan(L/r) L/r) sen( sen(α) 0 2π r µ0 iθˆ L/r 2π r 1 + (L/r ( L/r))2
· ·
µ0 iθˆ 2π r
·
|
√ L r 2 + L2
L
→∞
r) = µ0 i θˆ B( 2π r
·
d
i1 i2
λ d
v
zˆ x ˆ
yˆ
2
1
i r ) = µ0 i θˆ B( 2π r
·
1 2
2 dF 21 = i2 dl = =
⇒ ddzF 221
=
i=
=
× B 1 µ 0 i1 · xˆ i2 dz2 zˆ × 2π a µ0 · i1i2yˆdz2 2πa µ0 i1 i2 yˆ 2πa
·
dq λdz dz = =λ = λv dt dt dt
i = i1 = i2 = λv 1 dF m µ0 µ0 = i2 yˆ = (λv)2 yˆ dz 2πa 2πa
·
·
v
λ (r) = E
λ rˆ 2πǫ0 r
e = dq E (r) dF (r) = λdz E
⇒ ddzF e
=
=
−
=
−
λ2 yˆdz 2πǫ0 a λ2 yˆ 2πǫ0 a
· ·
e dF m dF + = 0 dz dz
λ2 µ0 = (λv)2 = 2πǫ 0 a 2πa
·
c
⇒ v = √ ǫ10µ0 = c
i
Γ r ) = µ0 B( 4π
idl
Γ
× (r − r ′) |r − r ′|3
= dsθˆ = Rdθθˆ r = z zˆ r ′ = Rˆ ρ dl
r ) = B( = = = = =
µ0 4π µ0 4π
idl
(r r ) r r 3 iRdθθˆ (z zˆ
× − ′ | − ′| × Γ
2π
− Rˆρ)
(R2 + z 2 )3/2 2π µ0 dθ(z ρˆ + Rˆ z) iR 4π (R2 + z 2 )3/2 0 2π µ0 dθ(z ρˆ) µ0 iR + iR 2 2 3/2 4π 4π (R + z ) 0 µ0 2 2πˆ z iR 4π (R2 + z 2 )3/2 µ0 iR2 zˆ 2 (R2 + z 2 )3/2 0
2π
0
ρˆ = cos(θ)ˆ x+sen(θ)ˆ y
dθ(Rˆ z) 2 (R + z 2 )3/2
σ w
σ
λ = σdr
i=
dq λds rdθ dθ = =λ = λr = λrw dt dt dt dt
i
r
r) = dB(
µ0 ir2 zˆ µ0 λwr 3 zˆ 1 r3 drˆ z = = µ σw 0 2 (r2 + z 2 )3/2 2 (r2 + z 2 )3/2 2 (r 2 + z 2 )3/2 w
r ) = B( = = = = =
1 µ0 σw zˆ 2 1 µ0 σw zˆ 2 1 µ0 σw zˆ 2 1 µ0 σw zˆ 2 1 µ0 σw zˆ 2 1 µ0 σw zˆ 2
R
r3 dr (r 2 + z 2 )3/2
√ | √ | | | √ 0
R2 +z 2
(u2
− z2)udu u3
z
R2 +z 2
z
u2 = r2 + z 2
2
1
− uz 2
R2 + z 2
− |z| +
R2 + z2 +
2z 2 + R2 R2 + z 2
z2
z2
u
√ | | − || R2 +z 2
z
√ R2 + z2
− || 2z
2z
→ udu = rdr
= J 0 yˆ J
r ′ = xˆ x + y yˆ
r = zˆ z
B(x,y,z) = B(z)
= J 0 yˆ J yˆdS = J 0 dS di = J
yˆ
·
r) = dB(
µ0 diˆ y (zˆ z xˆ x yyˆ) µ0 (xˆ z + zx ˆ) = J dS 0 4π (x2 + y 2 + z 2 )3/2 4π (x2 + y2 + z 2 )3/2
×
− −
dS = dxdy
∞ ∞ (xˆz + z xˆ)dxdy µ0 B(r) = J 0 4π −∞ −∞ (x2 + y 2 + z 2 )3/2 ∞ ∞ µ0 xˆ zdxdy µ0 = J 0 + J 0 2 2 2 3/2 4π 4π −∞ −∞ (x + y + z ) ∞ ∞ µ0 dxdy = J 0 z x ˆ 2 4π −∞ −∞ (x + y 2 + z 2 )3/2 = = =
∞ ∞
∞ ∞
−∞ −∞
(x2
zxˆdxdy + y 2 + z 2 )3/2
2π µ0 rdrdθ J 0 z x ˆ 4π (r 2 + z 2 )3/2 0 0 µ0 rdr J 0 z x ˆ 2 (r 2 + z 2 )3/2 0 1 z µ0 J 0 xˆ 2 z
||
0 (r) = E
σ σ z 2ǫ0 z
| | zˆ
x ˆ yˆ ˆ x
a g1
g2 J
ǫ1
ǫ2
V 0 I
r=b V 0
I =
n J ˆ dS
·
S
(r) = J (r)ˆ J r I = J 2πrL
= J
I rˆ 2πrL
1 = E
I 2πrLg1
2 = E
I 2πrLg2
r=a
r=c c
∆V =
a
d E r
·
c
b
∆V =
a
r=
∆V I
c
I b I c ln( ) + ln( ) 2πLg1 a 2πLg2 b
·
R=
1 b 1 c ln( ) + ln( ) 2πLg1 a 2πLg2 b
1 d E r+
2 d E r=
·
b
δV = V 0 I = RV 0 I =
V 0 2πL ln( ab ) g1
+
ln( cb ) g2
n D ˆ dS = q enc
·
S
2 (Aˆ 1 ( Aˆ D n) + D n) = σA
·
·−
2 (D
− D 1) · nˆ = σ
1 = ǫ1 E 1 D 2 = ǫ2 E 2 D
1 = D
ǫ1 I rˆ 2πrLg1
2 = D
ǫ2 I rˆ 2πrLg2 rˆ = n ˆ
σ=
I ǫ2 ( 2πbL g2
V 0
σ= b(
ln( ab ) g1
− gǫ11 )
ǫ2 ln( c ) + g b ) g2 (
2
− gǫ11 )
ρ A R=ρ
R= dx
H (x)
L w(y2
− y1 )
B ln(
y2 ) y1
ρℓ A
A(x) = wH (x) dx
ρ
H (x)
y2
− y1 = h(x) L
x (y2 y1 )x h(x) = L
−
H (x) = h(x)+y1 =
dR =
(y2 y1 )x +y1 L
ρdx ρdx ρL = = (y y )x − A(x) w (y2 w( L + y1 ) 2
x=0
− y1) ρL − y1) w
L
0
(y2
−
1
−
2
−y L
1
)x
+y1 )
dx y1 )x + y1 L
x=L
ρL R= w
(y2 R= (y2
A(x) = w( (y
−
L
0
(y2
−
dx y1 )x + y1 L
dx ρL = y1 )x + y1 L w(y2 y1 )
−
R=ρ
L w(y2
− y1 )
L
0
ln(
(y2 (y2
y2 ) y1
− y1)dx
− y1)x + y1L
=
ρL w(y2
− y1 )
ln(
(y2 y1 )L + y1 L ) (y2 y1 ) 0 + y1 L
− − ·
S
