m
q
R R
N = N cos(π/3)ˆi + N sin(π/3) jˆ N e F kq 2 F e = 2 R
2
F X = N cos(π/3)
− kq R2
=0
(1)
F Y = N sin(π/3)
− mg = 0
(2)
N =
mg sin(π/3)
mg cos(π/3) sin(π/3)
−
kq 2 =0 R2
q q =
±R
mg cot(π/3) k
mg = k
q 2 x0 2
x0 =
⇒ x0 = q ·
k mg
(1) x0
d2 x F = m 2 = dt
kq 2 mg + (x0 + x)2
−
|x| << |x0| =⇒ | xx | << 1 0
F = m
d2 x = dt2
2
2
2
kq x kq x −mg + (x0kq = −mg + 2 · (1 + )−2 ∼ = −mg + 2 · (1 − 2 ) 2 + x) x0 x0 x0 x0 mg =
kq 2 x0 2
d2 x m 2 = dt
∼ −2kq 32x
=
⇒
2
x0
2
2kq x ⇒ ddtx2 + mx · =0 2 x 0 0
= mx0 2 =
kq 2 g
=
⇒
d2 x 2g + x = 0 dt2 x0
·
w =
2g x0
x ˆ yˆ
2 1 = k q (cos( π )ˆ F x L2 3
·
−
π q 2 x ˆ sen( )ˆ y ) = k 2 ( 3 L 2
· −
√ 3 2
yˆ)
2
2 = k q ˆ F x L2 3 q 2 F = F 1 + F 2 = k 2 ˆ x 2 L
−
√ 3
q 2 k ˆ y = 2 L2
√
√
q 2 3 3 k 2 ( x ˆ L 2
·
·
−
yˆ )= 2
√
q 2 π 3 k 2 (cos( )ˆ x L 6
− sen( π6 )ˆy)
− π6
x ˆ
·
·
π
r L = π = sen( 6 ) sen( 2π ) 3
⇒ r = √ L3
√
q 2 qQ 3 k 2 = k L 2 = L ( √ 3 )
·
⇒ Q = √ 13 · q
2a
r = R (cos(θ)ˆ x + sen(θ)ˆ y) r 1 = a zˆ r 1 , r 2 r 2 = R2 + a2
· |r − r 1| = |r − | √
r
kqQ
= F 1 + F 2 = F
(R2 + a2 )
=
2
2kqQR 3 2
· (cos(θ)ˆx + sen(θ)ˆy)
|F | =
2kqQR (R2 + a2 )
3
·
1
df (R) (R2 + a2 ) 3R2 (R2 + a2 ) = =0 dR (R2 + a2 )3 2
⇒
.
= 2kqQ f (R)
2
3
=
−
2
⇒ (R2 + a2) · ((R2 + a2) − 3R2)) = 0 =⇒ R = √ 12 · a 1 · a R = √ 2
=
−aˆz
· (Rcos(θ)ˆx + Rsen(θ)ˆy − aˆz + Rcos(θ)ˆx + Rsen(θ)ˆy + aˆz)
3
(R2 + a2 )
r 2 =
1 2
E θ
α(C seg−1 ) θ
≪1
E F
·
q E = q E F
E
= E ˆi E
X
= T
−T sin(θ)ˆi + T cos(θ) jˆ
E = qE ˆi F
F X = qE
− T sin(θ) = 0
F Y = T cos(θ)
− mg = 0 T =
qE
mg cos(θ)
mg tan(θ) E q =
θ q (t) =
mg tan(θ) E
mgθ(t) E
dq mg dθ = dt E dt = α
(2)
mg − cos(θ) sin(θ) = 0 q =
dq dt
(1)
dθ dt
dθ αE = dt mg
θ
≪1
tan(θ)
≈
−
+Q q
d m
−q 1 F xˆi
1 F
r1 = d/2 jˆ
ˆ ˆ 1 = k( q )Q(xi + d/2 j) F l3
−
2 F
2 F
r2 =
ˆ 2 = k( q )Q(xi F l3
−
−d/2 jˆ
ˆ − d/2 j)
−q = F 1 + F 2 = −2kqQx ˆi F l3
(1)
ℓ
d/2
θ
d/2 = cos(θ) ℓ =
⇒ ℓ = (d/2) cos(θ) θ
θ
≪1
cos(θ)
= F
x
≈1
ℓ
−16kqQx
(2)
d3 = ma F
2
m a = m ddtx ˆi 2
−16kqQx = m d2x dt2
d3
2
⇒ ddtx2 + 16kqQx =0 md3
=
w = 4
T =
kqQ md3
2π π = w 2
≈ (d/2)
md3 kqQ
P P jˆ P 1 = 2kq jˆ E r2 P x y
y
E y = E cos(θ)
E E =
kq d2 + r2
cos(θ) = E y = (
√ r
d2 +r2
kq d2 + r 2
· √ d2r+ r2 ) jˆ = (d2 +kqrr2)3/2
y = E
− (d2 +kqrr2)3/2 jˆ
P = E y + E y + E 1 E
= ( 2kq E r2
− (d2 2kqr ) jˆ + r 2 )3/2
P
d = 2qk( 1 E r2
= 2qk( 1 E r2 =
2qk (1 r2
−
−
≪r
r (r 2
+
3
d2 ) 2
1 r2 (1 +
d2 32 ) r2
2
− (1 + dr2 )−
3 2
x (1 + x)α = 1 + αx
≪ r
d r
)
)
(1 + x)α
d
)
( dr )2
≈0 (1 +
d2 − ) r2
3 2
≈0 2
≈ (1 − 32 dr2 )
2 = 3qkd jˆ E r4
≈0
θ I
E F = qE
O
τ =
−F a sin(θ) + −F a sin(θ) = −2F a sin(θ) = −2qE sin(θ) I
α
O τ = I α
−2qEa sin(θ) = Iα sin(θ)
≈θ
α =
d2 θ dt2
θ
≪ 1
d2 θ 2qEaθ = I 2 dt
−
d2 θ 2qEaθ + =0 I dt2
w =
2qEa I
w 1 f = = 2π 2π
2qEa I
λ1 λ2
r
Ω = q F
x = L+d x = 2L+d
− r‘) |3 r − r‘ Ω 4πǫ 0 |
·
dq (r
r 1 = x 1 x ˆ r 2 = x 2 x ˆ
· ·
1
=
≤ x1 ≤ L
− r 1) | − r1|3 Ω L λ1 dx1 (x2 − x1 )ˆ x dq 2 4πǫ 0 |x2 − x1 |3 0 L λ1 dx1 · dq 2 · x ˆ· 4πǫ0 (x2 − x1 )2
21 = dq 2 dF =
0
dq 1 (r 2 4πǫ 0 r 2
0
L λ1 1 = dq 2 x ˆ 4πǫ0 x2 x1 0 λ1 x ˆ 1 1 = dq 2 ( ) 4πǫ 0 x2 L x2 λ1 λ2 x ˆ 1 1 = dx2 ( ) 4πǫ0 x2 L x2
·
· · − · · − − · · − −
x = 0 x = L L+d x2 2L+d
≤ ≤
=
⇒
F 21 = = =
=
⇒
2L+d λ1 λ2 x ˆ 4πǫ0 L+d λ1 λ2 x ˆ (ln(x2 4πǫ0
·
1
1 x2
− L − dx2 · − L) − ln(x2))|2L+d L+d λ1 λ2 x ˆ x2 − L 2L+d · ln( ) 4πǫ0
x2
λ1 λ2 F 21 = ln 4πǫ0
·
x2
L+d
(L + d)2 d(2L + d)
·
x ˆ
BC D AB = 2R q q q
DE = R
BC
−
DE
R CD AB O
O
2R
q λ1 =
dq
dq
q 2R
dq = λ 1 dx
ˆi
dx
kdq ˆ kλ 1 dx ˆ ) i = ( )i x2 x2
1 = ( dE x =
−3R
x =
1 = k(λ1 E
−R
1 ˆi = (kλ1 ( 1 dx) x2 x2
−
1 = 2kλ 1 ˆi E 3R
R
− −
))ˆi
3R
hati
BC CD x πR 2
λ2 = λ3 =
2q πR
2q − πR
dq
dE = λ2 ds = Rdθ
kdq R2 dq = λ 2 ds
ds
dE 2 = x
kλ2 Rdθ kλ 2 dθ = 2 R R
dE 2 dE x2 = dE 2 cos(θ) =
θ = 0
θ =
kλ 2 cos(θ)dθ R
π 2
kλ2 E x2 = R
π
2
cos(θ)dθ =
0
kλ2 R
x2 = kλ2 ˆi E R x x3 = kλ2 ˆi E R 2 + E 3 = E
2kλ 2 ˆ R i
q ′ O
R λ4 =
q ′ R AB
4 = (kλ4 E
2R
R
1 kλ4 ˆ dx)ˆi = ( )i 2 x 2R
−
1 + E 2 + E 3 + E 4 = 2kλ1 ˆi + 2kλ2 ˆi + ( kλ 4 )ˆi = 0 E 3R R 2R
−
λ4 2 (2λ1 + 6λ2 ) 3
λ4 = λ1 λ2
λ4
q ′ =
≈ 3
2 12 q (1 + ) 3 π
π
q ′ =
10 q 3
σ
dq (r r1 ) 4πǫ 0 r r 1 3
− | − |
(r) = E
Ω
r 1
Ω r = xˆ x + yyˆ + z zˆ r 1 = x 1 x ˆ + y1 yˆ
⇒ E (r) =
=
⇒
)2
)2
+ z2
|r − r1| = (x − x1 + (y − y1 +∞ +∞ σdx 1 dy1 ((x − x1 )ˆ x + (y − y1 )ˆ y + zˆ z) −∞ −∞ 4πǫ0 ((x − x1 )2 + (y − y1 )2 + z 2 ) w = x − x1 v = y − y1 =⇒ x1 = x − w, y1 = y − v
J =
=
r 2
3 2
(x1 )w (x1 )v (y1 )w (y1 )v +
+
=
−
1 0
−
∞ ∞ −∞ −∞ ∞ ∞
(r) = E
σ 4πǫ0
=
0 1
=1
σdwdv(wˆ x + vˆ y + z zˆ) 4πǫ0 (w2 + v 2 + z 2 ) + dwdv (z zˆ)
+
·
−∞ −∞ (w2 + v2 + z 2 )
3 2
3 2
w = rcos(θ) v = rsen(θ) J =
=
⇒
(r) = E = = = = =
wr wθ vr vθ
2π
+
∞ ∞ · ∞ · ∞ − ·
σ 4πǫ0
0
σ z zˆ 4ǫ0
+
du
(u)
1
3 2
z2
2
·| | ·
−
+
= r
3 2
(r2 + z 2 )
+
σ z zˆ 2u 4ǫ0 σ z zˆ 2ǫ0 z σ sign(z)ˆz 2ǫ0
·
(r2 + z2 ) rdr
0
z2
cos(θ) rsen(θ) sen(θ) rcos(θ)
rdrdθ (z zˆ)
0
σ 2πz zˆ 4πǫ0
=
3 2
pero
u = r 2 + z 2
→ du = 2rdr
⇒ E (r) = 2ǫσ0 · sign(z)ˆz
= + σ ǫ
0
σ
λ
σ
−σ 1 (r) = σ sign(x)ˆ E x 2ǫ0
·
2 (r) = −σ · x E ˆ 2ǫ0
=
⇒
− √
· √
sign(x)
= E 1 + E 2 = σ E 2ǫ0
x R2 + x2
x R2 + x2
x ˆ
dx
= dq E dF =
⇒
= F
d+a
con dq = λdx dq E
d
d+a
=
d
= = =
λσ 2ǫ0 λσ 2ǫ0 λσ 2ǫ0
σ λdx 2ǫ0 d+a
· √
x R2 + x2
xdx x ˆ R2 + x2
x ˆ
√ · · · · · − · d
R2 + x2
d+a d
R2 + (d + a)2
x ˆ
R2 + d2
xˆ
a
Q O
( E 0) =
− r ′) r − r ′3
kdq (r
dq = σdA
σ
dA θ γ
θ ds1 = R sin(γ )θ ds2 = Rdγ
γ
dA = ds 1 ds2 = R 2 sin(γ )dγdθ
·
dq = σR2 sin(γ )dγdθ
r ′
r ′ = 0
r
ˆ R cos(γ )kˆ r = R sin(γ ) cos(θ)ˆi + R sin(γ ) sin(θ) j +
r − r ′ = R ( E 0) =
3π
2π
2
π
0
kσ( R sin(γ ) cos(θ)ˆi
−
ˆ − R sin(γ )sin(θ) jˆ − R cos(γ )k) R2 sin(γ )dγdθ R3
2
kˆ
( E 0) =
2π
π
0
π 2
ˆ kσ( cos(γ )k)sin(γ )dγdθ = kσ
−
1 ( E 0) = πkσ( cos(2θ) 2
2π
π
dθ
0
π 2
π
1 ˆ )k = πkσ (cos(2π) 2 π 2
Q
σ =
Q 2πR 2
kQ ˆ ( E 0) = k 2R2
ˆ ( cos(γ )k)sin(γ )dγ = πkσ
−
π
π 2
ˆ ( sin(2θ)k))dγ
−
ˆ (πkσ)kˆ − cos(π))k = σ
AB
2L Q
L
Q
CD
jˆ
dE = kdq r r 2 ˆ d j r = x + d2 dq = λ 1 dx λ1
dq
2
xˆi
dq
dq λ1 =
Q 2L
dE =
kλ1 dx x2 + d2
jˆ dE y = dE cos(θ)
y cos(θ) =
√
d (x2 + d2 )
x
dE y = x =
−L
(x2 + d2 )
3 2
x = L
L
E y = kλ 1 d
kλ1 ddx
1
−L (x2 + d2 )
3
dx = kλ 1 d(
2
x x2 + d2
√ (d2
y
L
−
)= L
2kλ 1 L dF = dqE = dq d L2 + y2
Q L
λ2 = dq = λ 2 dy dF = λ 2 dy y = L
2kλ1 L y
y = 2L
L2 + y2
L
F = 2kλ1 λ2 L
L
y
1 F = 2kλ1 λ2 L( ln( L
λ1
L2 + y2
y
L2
+ y
y2
Q
√ − √ −
− L)
− L ) 2L)
√ 5 − 1 F = 2kλ1 λ2 ln( √ ) 2( 2 − 1) λ2
dy
L2 + y2 y
1 dy = ln( L L2 + y2
1
1
−
kQ2 5 1 F = 2 ln( ) L 2( 2 1)
L
2kλ1 L √ d L2 + d2 dq
b σ λ
d
±yˆ −∞ ≤ x ≤ +∞
r 2 = 0 r 1 = x xˆ + y yˆ
dq 1 r 1 , 4πǫ 0 r 1 3
d
≤ y ≤ b + d
· − | | ∞ ∞ · − − −∞ −∞ ∞ · − −∞ ∞ · −∞ ∞ − √ ·
21 = dq 2 dF
Ω +
=
b+d
dq 2
=
+
= =
=
+
dq 2 σ yˆ 4πǫ0
+
⇒
21 dF dx2
b+d
(x2 + y2 )
·
4πǫ0 (x2 + y2 )
3 2
3 2
d
1 x2 + d2
dx,
pero
dq2 = λdx2
∞ − √ · −∞ ∞ √ · − · → ∞ · ⇒ · +
= =
λσˆ y ln 2πǫ0
=
d
2
σdxdy yyˆ
dx
1 x2 + (b + d)2
λσˆ y 4πǫ0
=
dq 2
3
b+d
x2 + y2
λσˆ y 2πǫ0
b+d
dxdy y
1
−∞
+
σdxdy xˆ x
d
dq 2 σ yˆ 4πǫ0
dq1 = σdxdy
4πǫ0 (x2 + y2 )
d
dq 2 σˆ y 4πǫ0
con
1 x2 + (b + d)2
x + x2 + (b + d)2 x + x2 + d2
l´ım ln
x
λσˆ y ln 2πǫ0 =
+
1+
1+
0
2
1 + ( xd )
d b+d
dx
+
1 + ( b+d x )
d b+d
21 dF λσˆ y = ln dx2 2πǫ0
1 x2 + d2
2
ln
b+d d
R R λ
jˆ
d x
r ′ = xˆi
dq
= dE
r = d jˆ ˆ − r ′) = kdq (−xˆi + d j) |r − r ′|3 (x2 + d2 )
kdq (r
3 2
dq = λdx
= dE
−∞ = E
+
∞ −∞
ˆ kλdx( xˆi + d j)
−
(x2 + d2 )
3 2
+
∞
ˆ kλdx( xˆi + d j)
−
(x2 + d2 )
3 2
+
= kλ((
∞ −
x
−∞ (x2 + d2 )
3 2
)ˆi + (
+
∞
d
−∞ (x2 + d2 )
3 2
ˆ ) j)
f (x) =
x
3
2
(x +d2 ) 2
g(x) =
d 2
3
(x +d2 ) 2
+
−∞
+
∞
∞
+
∞
+
g(x)dx = 2
= 2kdλ E
+
∞ 0
1 2
2
(x +d )
3 2
=
x 2
2
g(x)dx
0
−∞
∞ 1 (x2
+
3
d2 ) 2
jˆ
1
d (x +d2 ) 2
= 2kdλ( E
x d2 (x2 + d2 )
3 2
∞ 0
ˆ 2kλ jˆ ) j = d
dq
dF = dqE
E dq
θ
dq dq
θ
d(θ) = R + R sin(θ) λds
dq
ds θ
ds = Rdθ
dq 2
dF = dqE = λRdθ θ = 0
θ = π
2kλ dθ · R + 2kλ = R sin(θ) 1 + sin(θ)
1 sin(θ) 1 = 1 + sin(θ) cos(θ)
−
F = 4kλ2
q > 0 H
R
q φ =
dS = q E ǫ0 S
(1)
·
S e φ = φ e + φc
S c =
⇒ φc = φ + φe
(2)
φe
φe =
φe
dS E
q
E =
S e
dS rˆ
·
1 q r 4πǫ0 R2 ˆ
rˆ = n dS ˆ dS
n ˆ
dS = E
·
φe =
S e
S e
1 q 1 q ˆ r n ˆ dS = dS 4πǫ0 R2 4πǫ0 R2
·
dS = E
·
S e
1 q 1 q dS = 2 4πǫ 0 R 4πǫ 0 R2
dS
dS
S e
S e
φe =
1 q q 2πR 2 = 2 4πǫ0 R 2ǫ0
·
φc =
dS = 2πR 2
(3)
q 2ǫ0
σ
1 ǫ0
Qint n E ˆ dS = ǫ0 Ω
Qint
·
Ω
∇ · E = ǫρ0 θ
(r) = E (r)ˆr E
z
r
rˆ
|E |
|E |
|E | r
Qint n E ˆ dS = =0 ǫ0 Ω
n E ˆ dS =
·
Ω
·
E (r)ˆ r n ˆ dS +
·
manto
= E (r)
E (r)ˆ r n ˆ dS
tapas
·
dS
manto
= E (r)(2πr)h
⇒ E (r)(2πr)h = 0 =⇒ E (r) = 0 =⇒ E = 0
=
R
≤r (2πR)hσ n E ˆ dS = E (r)(2πr)h = = ǫ0 Ω
·
⇒
(r) = E
+ σ ǫ
0
0
· σ ǫ0
R r
σ E (r) = ǫ0
r
≤
R r
Θ
ρ = αr
r
ǫ0 U = 2
(r) = E (r)ˆ E r
R
2 d3 x E
3
r
rˆ
r
Ω
Ω
n E ˆ dS =
·
E (r)ˆ r n ˆ dS =
·
E (r)4πr 2 = E (r)4πr 2 = E (r)4πr 2 = =
⇒
R
Qint ǫ0 1 ρd3 x ǫ0 Θ 1 2π π r 3 αr sen(ϕ)drdϕdθ ǫ0 0 0 0 4π r 3 αr dr ǫ0 0 παr 4 ǫ0 α 2 E (r) = r 4ǫ0
≤r
Θ
Ω
Qint ǫ0 4π R 3 = αr dr ǫ0 0 παR 4 = ǫ0 α R E (r) = 4ǫ0 r
n E ˆ dS =
·
E (r)4πr
2
E (r)4πr 2 =
⇒
2
R2
(r) = E
α 2 ˆ 4ǫ0 r r 2 α R R2 rˆ 4ǫ0 r
r
≤r
Θ
U =
ǫ0 2 d3 x E 2 R ǫ0 2π π +∞ E (r)2 r 2 sen(ϕ)drdϕdθ 2 0 0 0 +∞ 4πǫ0 E (r)2 r 2 dr 2 0 R +∞ 4πǫ0 α2 6 α2 R 8 dr 2 r dr + 2 16ǫ20 r2 0 16ǫ0 R
· − · 3
= = = = = =
πα 2 8ǫ0
r7 7
R
R
0
πα 2 R7 + R7 8ǫ0 7 2 πα 7 R 7ǫ0
81
r
∞ + R
R ρ 2R
−ρ d
ρ
(r) = E
ρr 3ǫ0
· rˆ
R
(r ) = E (r)ˆ E r
≤ r r>R
Ω
ndS E n ˆ dS =
·
E (r)4πr )4πr 2 =
3 (r) = R ρ ˆ E r 3ǫ0 r 2
=
⇒
(r ) = E
Qint ǫ0 4πR 3 ρ 3ǫ0
ρr ˆ 3ǫ0 r 3 R ρ ˆ r 3ǫ0 r2
·
r
≤
ρ
1 (r) = E 2 (r) = E
ρ r para r < R 3ǫ0 ρ (r d) para r 3ǫ0
·
−
||
| − d| < R
· −
|r − d | < R |r| < R (r) = E 1 (r ) + E 2 (r) = ρ r E 3ǫ0
· − 3ρǫ0 · (r − d) = 3ρǫ0 · d
ρ > 0
−Q < 0 m
0 = E 0 x E ˆ E 0 > 0 E 0
(r ) = E
ρr ˆ 3ǫ0 r R3 ρ ˆ r 3ǫ0 r2
·
r
≤
( E 0) = 0 = F
−Q ρr3ǫ · rˆ 0
−Q ρr · rˆ 3ǫ0 ρr · rˆ = 0 Q 3mǫ0 ρ · r = 0 Q 3mǫ0 w2 · r = 0
= m a = F a + = ¨r +
⇒
r¨ +
T =
2π w
= 2π 2π
3mǫ0 Qρ
(r ) = E
ρr ˆ + 3ǫ0 r 3 R ρ ˆ r + 3ǫ0 r2
·
E 0 x ˆ r
≤
(r) = E
( ρx ˆ R
·
0 3
(E 0
· · xˆ
2
0
−
−
R3 ρ ) 3ǫ0 x2
≤ x ≤ −R
−R < x < R ρx + E 0 = 0 = 3ǫ0
⇒ x1 = −3ǫρ0E 0
x1 x1 =
−3ǫ0E 0 > −R =⇒ E 0 < Rρ ρ
3ǫ0
E 0
∇2U U =
− −
−qV
(x) xdx E x ˆdx + c
V ( V (x) =
·
ρx + E 0 dx + c 3ǫ0 ρx 2 E 0 x +c 6ǫ0
= =
−
−
U (x) =
−QV QV ((x)
= QE 0 x + Q 2
⇒ ∇2U = ddxU 2
=
R
=
ρx2 6ǫ0
− Qc
Qρ > 0 3ǫ0
≤x R3 ρ + E 0 = 0 = 3ǫ0 x2
⇒ x2
imaginario R
≤x
x
≤ −R E 0
−
R3 ρ =0= 3ǫ0 x2
⇒ x3 = −R
ρR 3ǫ0 E 0
x1 x3 =
−R
ρR < 3ǫ0 E 0
−R =⇒
ρR > 1 = 3ǫ0 E 0
Rρ ⇒ E 0 < 3ǫ 0
E 0 < E 0 = 0 E 0 = x =
−R
E 0 >
Rρ 3ǫ0
Rρ 3ǫ0
Rρ 3ǫ0
R ρ = ρ 0 (1
−
r R)
ρ0
r
r
= E ˆ E r
S
n E ˆ dS =
·
n E ˆ dS =
·
tapas
q interior n E ˆ dS = ǫ0 manto
·
rˆ
rˆ n ˆ = 0
·
E rˆ n ˆ dS = 0
(2)
·
rˆ
rˆ n ˆ = 1
E rˆ n ˆ dS = E
·
manto
manto dS
· ·
n E ˆ dS = E 2πr L
manto
·
E 2πrL =
·
·
q interior ǫ0
dS
manto
manto dS =
(1)
·
tapas
n E ˆ dS =
manto
·
tapas
n ˆ
n ˆ
n E ˆ dS +
rˆ
2πrL
(3)
(4)
dq = ρ(r ′ ) dV
(5)
·
dV r′
r′ + dr ′
dV dV = 2πr ′ dr ′ L
(6)
dq = 2πr ′ dr′ L p(r′ )
· · · r′ ′ ′ dq = 2πr · dr · L · ρ0 (1 − ) R ′2
⇒ dq = 2πL · ρ0(r′ − rR )dr′
=
r
q = 2πLρ0
0
r ′ 2 ′ (r − )dr ′ = 2πLρ0 · (
r
R
0
r ′ dr ′ −
r
0
r′ 2 ′ dr ) R
2 r3 3Rr 2 − 2r3 ⇒ q = 2πLρ0 · ( r2 − 3R ) = 2πLρ0 · ( ) 6R
=
0 ⇒ q = πLρ (3Rr2 − 2r 3 ) 3R
=
E 2πrL =
·
πLρ0 (3Rr2 3Rǫ0
(7)
− 2r3)
E E =
ρ0 (3Rr 6Rǫ0
− 2r2)
(8)
r
E
dE ρ0 = (3R dr 6Rǫ0
− 4r)
r r =
3R 4
|E |max = 3ρ16ǫ0R0
2a ρ R
σ x > 0
xx>0 ρ σ E ρ y
E ρ
0 < x < a
•
x>a
0 < x < a A
2x
n E ˆ = 0
·
n ˆ dS = 0 E
·
(E ) n ˆ = E
·
S
φ = E
dS + E
S 1
S 1
S 2
n E ˆ dS = E
·
dS
S
dS = EA + EA = 2EA =
S 2
q int ρ q int = ρV = ρ2xA
ρ2xA ǫ0 ρx E = ǫ0
2EA =
ˆi ρ = ρxˆi, si 0 < x < a E ǫ0
q int ǫ0
(1)
•
x >a
φ = 2EA q int = ρ2aA
−a < x < a ρ2aA ǫ0 ρa E = ǫ0
φ = 2EA =
ˆi E ρ =
ρaˆ i, si x > a ǫ0
E σ
• r
σ r
φ= n E ˆ = E
·
n ˆ dS
S E
φ = E 4πr 2 =
n ˆ φ = E S dS = E 4πr 2 q int = 0
·
q int =0 ǫ0
E = 0, si r < R
• r>R
r > R
φ =
S
n E ˆ dS = E
·
dS = E 4πr 2
S
q int
q int = σ4πR 2 q int σ4πR 2 φ = E 4πr = = ǫ0 ǫ0 2
σR 2 E = ǫ0 r 2
2 σ = σR ˆ E r , si r > R ǫ0 r 2
rˆ
0 < x < a a < x < a + 2R x > a + 2R ρ = E
0 < x < a r
rˆ =
ˆi
rˆ a + R = x + r
−
σ = E
−σR2 ˆi ǫ0 (a + R − x)2
ρx ˆ ǫ0 i
σ = E
r = a + R
σR 2 ˆ r ǫ0 r 2
−x
= E ρ + E σ E = ( ρx E ǫ0
2
− ǫ0(a +σRR − x)2 )ˆi
a < x < a + 2R
E ρ = E ρ = ρaˆi, si a < x < a + 2R E ǫ0 ˆi
x > a + 2R σ E
rˆ = ˆi
x = a + R + r
σ = E
r = x
σR2 ˆi ǫ0 (x a R)2
− −
σR2 = ( ρa + E )ˆi, si x > a + 2R ǫ0 ǫ0 (x a R)2
− −
−a−R
= E
( ρx ǫ 0
( ρa ǫ + 0
2
σR − ǫ (a+R −x) )ˆi 2
0
ρaˆ ǫ0 i σR 2 ǫ0 (x a R)2
−−
0 < x < a a < x < a + 2R )ˆi x > a + 2R
R ρ
−ρ
d
2R
ρ
(r) = E
ρr 3ǫ0
· rˆ
R
≤ r
(r) = E (r)ˆ E r
r >R
Ω
n E ˆ dS =
·
E (r)4πr 2 =
R3 ρ E (r) = ˆ r 3ǫ0 r2
=
⇒
(r) = E
Qint ǫ0 4πR 3 ρ 3ǫ0
ρr ˆ 3ǫ0 r R3 ρ ˆ r 3ǫ0 r2
·
r
≤
ρ
1 (r) = E 2 (r) = E
ρ r para r < R 3ǫ0 ρ (r d) para r 3ǫ0
·
−
||
| − d| < R
· −
|r−d | < R |r| < R (r) = E 1 (r) + E 2 (r) = ρ r E 3ǫ0
· − 3ǫρ0 · (r − d) = 3ǫρ0 · d
σ>0 x = 0
aˆ x
−q < 0 −q m
−e < 0
< 0
−q
−q 1 = σ x E ˆ 2ǫ0 q 1 q 2 = 1 E xˆ = x ˆ 2 4πǫ0 (a x) 4πǫ0 (a x)2
− · − − −q
= E 1 + E 2 = E = E
−∇ V
− ·
σ 1 q + 2ǫ0 4πǫ 0 (a x)2
−
·
xˆ
Ω
∂V =0 ∂z =
⇒
∂V ∂y
=0
V (x) =
−
∂V σ 1 q = + ∂x 2ǫ0 4πǫ0 (a x)2 σx 1 q + + C 2ǫ0 4πǫ 0 (a x)
−
−
−e
−
a/2ˆ x
−eV (a/2) =
⇒
mv2 K = 2
= = =
mv 2 2
− eV (0) e (V (0) − V (a/2)) 1 q 1 2q σa − C e − + C + + 4πǫ0 a 4πǫ0 a 4ǫ0
= e
1 q σa + 4πǫ 0 a 4ǫ0
r1 , r2
q 1 , q 2
d >> r1 , r2
V (+ ) = 0
∞
V 1 = k
q 1 q 2 , V 2 = k r1 r2 d >> r1 , r2
(q 1 )f , (q 2 )f q 1 + q 2 = (q 1 )f + (q 2 )f
V 1 (q 1 )f k r1 4π(r1 )2 (σ1 )f r1 (σ2 )f = (σ1 )f
⇒
= V 2 (q 2 )f = k r2 4π(r2 )2 (σ2 )f = r2 r1 = r2
(q 1 )f + (q 2 )f = q 1 + q 2 2
4π(r1 ) (σ1 )f + 4π(r2 )2 (σ2 )f = q 1 + q 2 4πr 1 (σ1 )f (r1 + r2 ) = q 1 + q 2 1 q 1 + q 2 = (σ1 )f = 4πr 1 r1 + r2 1 q 1 + q 2 (σ2 )f = 4πr 2 r1 + r2
⇒
|E ⊥ (r)|
σ(r)
r
σ(r) ǫ0
r 0 < 2h << 1
Ω 0 < A << 1 Qint (r) n E ˆ dS = ǫ0 Ω
·
∼= σ(r)A ⇒ Qǫint ǫ0 0
∼
Qint = σ(r)A =
Ω
(r) n E ˆ dS =
·
(r) n E ˆ dS +
·
manto
1 (r) nˆ1 dS + E
carasup
·
2 (r) nˆ2 dS E
carainf
= E ⊥ + E // E
|E (r)|
nˆ1 =
carasup
1 (r) nˆ1 dS + E
·
−nˆ2
carainf
1 ) A 2 (r) nˆ2 dS = (E E ⊥
·
= σ(r)A ǫ0
h
manto
− (E 2)⊥A 1 ) − (E 2 ) )A ((E ⊥ ⊥
(r) n E ˆ dS
·
→ 0
cuando
h
→0
·
1 ) ((E ⊥
− (E 2)⊥ )A △E ⊥A =
σ(r)A ǫ0 σ(r)A = ǫ0 σ(r) E ⊥ (r) = + ǫ0 =
⇒ △
r (r) = + σ(r) n E ˆ ǫ0 n ˆ
r
q 0 > 0
V 0
q > 0 c>b
V = V (r)
rˆ
(r) = E (r)ˆr E q 1 , q 2 , q 3 , q 4 q 0 = q 1 + q 2
= E 0
q 1 n E ˆ dS = 0 = = ǫ0
·
⇒ q 1 = 0 =⇒ q 2 = q 0
q 1 + q 2 + q 3 n E ˆ dS = 0 = = ǫ0
⇒ q 1 + q 2 + q 3 = 0 =⇒ q 3 = −q 2 = −q 0
·
r>b
q 1 + q 2 + q 3 + q 4 n E ˆ dS = E (r)4πr 2 = = ǫ0
q 4 rˆ ⇒ E (r) = 4πǫ 2 0r
·
V 0 b
V (b)
− V (+∞) = V (b) = V 0
− ∞ −
=
+ b
V 0 =
+
∞
V 0 =
⇒ q 4
r
≤a
·
q 4 dr 4πǫ0 r2
q 4 1 = 4πǫ0 b = 4πǫ0 bV 0
r
d E r
≤a
q 1 n E ˆ dS = E (r)4πr 2 = =0= ǫ0
⇒ E = 0
·
a
r = b
q 0 n ˆ dS = E (r)4πr 2 = = E ǫ0
·
q 0 rˆ ⇒ E (r) = 4πǫ 2 0r
(b) = E 0
r>b
q 4 4πǫ0 bV 0 n E ˆ dS = E (r)4πr 2 = = = ǫ0 ǫ0
·
0 ⇒ E (r) = bV · rˆ 2 r
(r) = E
a
0
q0 4πǫ0
·
0
bV 0 r2
= =
C = V 0
· rˆ
≤
−
q0 4πǫ0
·
− −
·
·
V (b) = V 0
1 b
V (a)
≤a
dˆ E r + C
q 0 dr + C 4πǫ0 r2 q 0 1 + C 4πǫ0 r
q 0 V (r) = 4πǫ0
− − 1 r
1 b
q 0 − V (b) = 4πǫ 0
+ V 0
1 a
1 b
r = a q 0 V (a) = 4πǫ 0
V (r b
≤
≤r≤b V (r) =
r
r a a
rˆ r2
≤
− 1 a
q 0 a) = 4πǫ 0
1 b
+ V 0
− 1 a
1 b
+ V 0
≤r V (r) = = = =
− −
dˆ E r + C ‘
·
q 4 dr + C ‘ 4πǫ 0 r2 q 4 1 + C ‘ 4πǫ0 r bV 0 + C ‘ r
·
V (b) = V 0 =
⇒ C ‘ = 0 ⇒ V (r) = bV r 0
=
V (r) =
q0 4πǫ 0 q0 4πǫ 0
1 a 1 r
1 b 1 b
− − bV 0 r
+ V 0 r a + V 0 a < r < b b r
≤
≤
V 0 = 0 = E V 0 = 0
−∇ V
q 1 = 0 q 2 = q 0 q 3 =
−q 0
q 4 = 0
r
≤a
= : E 0
rˆ = q 0 a < r < b : E 4πǫ0 r2 = b r : E 0
·
≤
q 0 r a : V (r) = 4πǫ0 q 0 a < r < b : V (r) = 4πǫ0 b r : V (r) = 0
≤
≤
− − 1 a 1 r
V (a)
1 b 1 b
− V (b) = q > 0
q0 4πǫ0
1 a
1 b
−
q 1 , q 2 , q 3 , q 4
q 1 n E ˆ dS = 0 = = ǫ0
⇒ q 1 = 0 =⇒ q 2 = q 0
·
q 1 + q 2 + q 3 q 0 + q 3 n ˆ dS = 0 = = = E ǫ0 ǫ0
⇒ q 3 = −q 0 =⇒ q 4 = 0
·
q q
q > 0 q V (+ ) = 0
∞
V (b) =
−
b +
∞
dˆ E r=0
·
= 0 E
λ
V (r) =
Ω
r = x x ˆ + yyˆ + zˆ z
V (r) =
λdx1 (x1 x)2 + y2 + z 2
4πǫ0
−
√ √ − − − − − − · − − − · −
dx = x2 + 1
√
| − | 0 ≤ x1 ≤ L
0
dx x2 +1
dq 4πǫ0 r r 1
r 1 = x 1 x ˆ L
L
x = tan(θ)
sec2 (θ)dθ = tan2 (θ) + 1
sec(θ)dθ = ln(sec(θ)+tan(θ)) = ln(x+ x2 + 1) = arccosh(x)
dx1 (x1 x)2 +y 2 +z 2
(x1
dx1 = ln(x1 x)2 + y2 + z2
L
V (x,y,z) =
0
= = =
2d
x+
((x1
− x)2 + y2 + z2)
λdx1
4πǫ 0
(x1
x)2 + y2 + z 2
L λ dx1 4πǫ0 0 (x1 x)2 + y2 + z 2 λ ln(x1 x + (x1 x)2 + y2 + z 2 ) 4πǫ0
λ ln 4πǫ0
L
x+
x+
(L
x)2 + y 2 + z 2
x2 + y2 + z 2
L
0
q V = 4πǫ0
2 r+
1 r+
1 r−
−
− 2rd · cos π2 r2 + d2 − 2rd · sen(θ) 2
−
2
= r +d =
⇒ r1+
=
=
1 2rd sen (θ) 1
·
r 2 + d2
1 r
=
−
·
1 + ( dr )2
− 2 dr · sen (θ) d r
r >> d 1 r+
=
1 r
∼=
1 r
=
1 r
r−
·
θ
<< 1
1
− · · − − · · − · 1 + ( dr )2
1
1 2
1
1 2
2 dr sen (θ) 2
d r
d r
2
2
+
d sen(θ) r
d sen(θ) r
2
2
2
r− = r + d
− 2rd · cos
π + θ = r 2 + d2 + 2rd sen (θ) 2
·
∼ · − − ·
1 1 = r− r
1 2
1
d r
2
d sen(θ) r
r >> d
V (r, θ) =
∼= = = = V (r, θ) =
q 4πǫ0
q 4πǫ0 r
− · − · · 1 r+
1 r−
1
1 2
d r
2
− − − ·
d + sen(θ) r
·
1
1 2
d r
q 2d sen(θ) 4πǫ0 r r 2qd sen(θ) 4πǫ0 r2 p sen(θ) 4πǫ0 r2 1 p rˆ 4πǫ0 r2
· ·
· ·
p
= E
1 ∂V ˆ −∇ˆ V = − ∂V rˆ − θ ∂r r ∂θ
∂V = ∂r
− 2πǫ1 0 psen(θ) r3
1 ∂V 1 pcos(θ) = r ∂θ 4πǫ 0 r3
⇒ E (r, θ) = 4πǫp0r3
=
V (+ ) = 0
∞
2sen(θ)ˆr
− cos(θ)θˆ
2
d sen(θ) r
U = qV (x + dx,y + dy,z + dz) =
− qV (x,y,z) q (V (x + dx,y + dy,z + dz) − V (x,y,z))
= qdV ∂V ∂ V ∂ V = q dx + dy + dz ∂x ∂y ∂z ∂V ∂V ∂ V = px + p y + p z ∂x ∂y ∂z = p V
U =
·∇ − p · E
b a < r < b
ρ0
ρ(r) = r < a
p
V p =
−
d E ℓ
·
∞
= E ˆ E r
r
b
•
r
E
n ˆ φ =
·
S
n E ˆ dS = E
·
dS = E 4πr 2
S
q int = 0 E 4πr 2 = 0
•
n E ˆ = E
E = 0 a
·
rˆ
φ =
n E ˆ dS = E 4πr 2
·
S
q int q int = ρ 0 V V
r 4 q int = ρ 0 π(r3 3
E 4πr 2 = ρ 0
a
− a3 )
4 π(r3 3ǫ0
− a3)
ρ0 (r3 a3 ) E = 3r2 ǫ0
−
rˆ ρ0 (r3 a3 ) E = rˆ 3ǫ0 r2
−
•
r>b
q int =
4 πρ 0 (b3 3
− a3 )
ρ0 (b3 a3 ) E 4πr = 3rǫ0
−
2
E =
ρ0 (b3 a3 ) 3ǫ0 r 2
−
3 a3 ) (r) = ρ0 (b E rˆ 3ǫ0 r2
−
= E
0
r
ρ0 (r3 a3 ) rˆ 3ǫ0 r2 ρ0 (b3 a3 ) rˆ 3ǫ0 r2
− −
r>b
d ℓ r>b
d E ℓ = E ˆ r d ℓ = Edr
·
·
r
V (r) =
−
d E ℓ =
·
∞
r
r
−
Edr =
∞
V (r) =
ρ0 (b3 a3 ) )dr 3ǫ0 r 2
−
−
∞
ρ0 (b3 a3 ) , 3ǫ0 r
−
si r > b
r r
−
V (r) =
a < r < b
d E ℓ
·
∞
r
b
b b
V (r) =
−
d E ℓ+
·
∞
r
−
V (b) =
r
− b
·
r = b
r
− −
−
·
b
=
·
ρ0 (b3 a3 ) E dℓ = 3ǫ0 b ∞
−
d E ℓ =
d E ℓ
b
b
r
ρ0 (r3 a3 ) dr 3ǫ0 r 2
−
b ρ0 ( rdr 3ǫ0 ∞ r ρ0 r2 (( ) 3ǫ0 2 b
=
−
=
ρ0 3ǫ0
·
r
b
1 dr) r2
1 r a ( )) r b b2 r 2 (r b) ( + a3 ) 2 rb
·
·
−a
3
−
−
3
− −
r ρ0 b3 a3 b 2 r2 (r b) V (r) = ( + + a3 ), 3ǫ0 b 2 rb
−
−
−
si a < x < b
r
V (r) =
−
∞
d E ℓ =
·
b
−
d E ℓ+
·
∞
a
−
d E r +
b
·
r
− a
d E ℓ
·
r
−
d E ℓ = 0
·
a
ρ0 b3 a3 b 2 a2 (a b) V (r) = ( + + a3 ), 3ǫ0 b 2 ab
−
−
−
si r < a
r
σ
−σ
AB d
r
r =b
L
φ =
S
r
q int n E ˆ dS = ǫ0 S
·
n E ˆ dS = E
·
(1)
dS = E 2πrL
(2)
S
σ q int = σV = σ2πaL
(3)
E =
σa rǫ0
(4)
r = d E =
ˆi
σa dǫ0
y 2E y = 2Ecos( π3 ) E =
σa dǫ0
ˆi
0 < x < D
x
1 = 0, E
si 0 < x < a
1 = σa ˆi, E xǫ0
2 = 0, E
si x > a
si d
−a
0 < x < d
−a
x
2 = E
= E
d
∆V =
− − 0
a
=
0
= = =
−x
σa ˆ i (d x)ǫ0
−
σa ˆ (d x)ǫ0 i σa ˆ ( σa xǫ0 + (d x)ǫ0 )i σa ˆ xǫ0 i
0 < x < a a
−
−
−
−
δV x
d E ℓ
·
d E ℓ+
·
a
=
d
d a
− − · −
d E ℓ+
a
d
−
d a
−
d E ℓ
·
d a d σa σa σa σa ˆ ( + )dx + dx + i x)ǫ0 xǫ0 (d x)ǫ0 0 (d a d−a xǫ0 σa σa σa ( (ln(d x) a0 )) + ( (ln(x) ln(d x) da−a )) + ( ((ln(x) dd−a ))) ǫ0 ǫ0 ǫ0 σa d a 2σa d a σa d a ( ln( )) + ( ln( )) + ( ln( )) ǫ0 d ǫ0 a ǫ0 d 2σa d a d a (ln( ) ln( )) ǫ0 a d
− −
−
− | −
− −
−
−
−
−
− | −
|
−
∆V =
2σa d ln( ) ǫ0 a
R σ0 > 0
σ = σ0 (1 O
r P
− Rr )
OP = R q
P
a q
d
V = r r =
V =
kdq r
k a2 + d2
√
a2 + d2
dq =
√ a2kq + d2
dq
a = r dV =
√ r2kdq + R2
d = R
dq dq = σ(r) dA
·
r r dr
dA dA = 2πrdr dq = 2πσ 0 (r
2kπσ 0 (r R)dr r 2 + R2
√ −
dV = r = 0
r = R R
V = 2kπσ 0
− R)dr
(r
R)dr r 2 + R2 r dr r 2 + R2
√ − √ √ 0
R
= 2kπσ 0 (
0
= 2kπσ 0 ((
r2
+ R2
R
R
1 dr) r 2 + R2
√
−R
0
r2
)
+ R2
R
− (ln( + r) )) 0 0 √ = 2kπσ 0 (R 2 − R − R ln(R 2 + R) + R ln(R)) √ √ = 2kπσ 0 (R 2 − R − R ln(R( 2 + 1)) + R ln(R)) √ √ = 2kπσ 0 (R 2 − R − R ln(R) − R ln( 2 + 1)) + R ln(R)) √ √ = 2kπσ 0 (R 2 − R − R ln( 2 + 1)) √ √ = 2kπσ 0 R( 2 − 1 − ln( 2 + 1)) √ 2 − 1 − ln(√ 2 + 1) ≈ 1 2
V
≈ −kπσ 0R
Q dq = 2πσ 0 (r
R
Q =
0
2πσ0 (r
r2 = 2πσ0 ( ( 2 = V
2
−πσ0R
− R)dr
− Rr)
R
) 0
≈ −πσQ0R2 − kπσ 0R
− R)dr
V
R dq
≈ kQ R
m I
ℓ
q
II
θ θ
ℓ m
ℓ
U I = 2mgℓ +
kq 2 2ℓ 2
U II = 2mgℓ(1
− cos(θ)) + 2ℓ kq sin(θ) U I = U II
2mgℓ +
kq 2 = 2mgℓ(1 2ℓ
q =
±2ℓ
2
− cos(θ)) + 2ℓ kq sin(θ)
mg sin(θ) k(sec θ tan(θ))
−
θ
R
σ Z z > 0 z > 0
Z O
O
zkˆ
dq
dV = r
kdq r
zkˆ
dq
dV dq σ
dA dA = R 2 sin(γ )dγdθ
r r2 = R 2 + z 2
− 2zR cos(γ )
dq dV =
2π
V =
kσR 2 sin(γ )dγdθ R2 + z 2 2zR cos(γ )
π
−
kσR2 sin(γ )dγdθ R2 + z 2 2zR cos(γ )
0
V = 2kπσR2
π 2
π
π 2
−
sin(γ ) R2 + z2 2zR cos(γ )
−
sin γ dγ A + B cos(γ )
u2 = A + B cos(γ )
u =
sin γ dγ = A + B cos(γ )
−
2 B
sin γ dγ = A + B cos(γ )
− B2
dγ = 1du =
2udu
−B sin(γ )
− B2 u
A + B cos(γ )
A = R2 + z 2
sin γ 1 dγ = zR R2 + z 2 2zR cos(γ )
−
2kπσR2 V (z) = ( zR
R2
A + B cos(γ )
+
z2
− 2Rz cos(γ )
π
R2 + z 2
)= π 2
B =
−2zR
− 2Rz cos(γ )
2kπσR (z + R z
−
R2 + z 2 )
zkˆ = E
−∇V z
= E
− dV kˆ dz
V (z) dV dz
d 1 ( (z + R R2 + z 2 )) dz z 1 1 1 = 2kπσ( 2 (z + R R2 + z 2 ) + (1 + z z 2 = 2kπσR
−
= =
− −
√ R21+ z2 2z))
√ R2 + z2 1 − 1 R 1 √ − − 2kπσ( + + ) z z z2 z R2 + z 2 √ R2 + z2 − R −2kπσR2( z2√ R2 + z2 ) zkˆ
√ R2 + z2 − R (z) = 2kπσR2 ( √ E )kˆ 2 2 2 z R +z z = 0 O
2kπσR2 ( z
1
− √ z R+R 2
l´ım E (z) = 2kπσR l´ım (
→0
) E (z )kˆ
→ 0 2
z
2
z2
z
1
− √ z R+R 2
z2
→0
2
)
0 0
2
2kπσR l´ım ( z
→0
1
− √ z R+R 2
z2
2
2
) = 2kπσR l´ım ( z
→0
(1
− √ z R+R )′ 2
(z 2 )
2
′
3
)
R(z 2 + R2 )− 2 = 2kπσR l´ım ( ) z →0 2z 1 = 2kπσR2 2R2 = kπσ 2
·
(0) = k πσ kˆ E
r1 , r2
q 1 , q 2
d >> r1 , r2
V (+ ) = 0
∞
V 1 = k
q 1 q 2 , V 2 = k r1 r2 d >> r1 , r2
(q 1 )f , (q 2 )f q 1 + q 2 = (q 1 )f + (q 2 )f
V 1 (q 1 )f k r1 2 4π(r1 ) (σ1 )f r1 (σ2 )f = (σ1 )f
⇒
= V 2 (q 2 )f = k r2 4π(r2 )2 (σ2 )f = r2 r1 = r2
(q 1 )f + (q (q 2 )f = q 1 + q 2 4π (r1 )2 (σ1 )f + 4π 4π (r2 )2 (σ2 )f = q 1 + q 2 4πr 1 (σ1 )f (r1 + r2 ) = q 1 + q 2 1 q 1 + q 2 = (σ1 )f = 4πr 1 r1 + r2 1 q 1 + q 2 (σ2 )f = 4πr 2 r1 + r2
⇒
a
V 1
V 2
Q c > b
Q1
Q2
V 2
V 2
a < r < b a Q1 Q1 ndS E n ˆ dS = ǫ0 S
ndS n ˆ dS = E 4πr 2
S E
·
·
= E =
a
V 2
− V 1 =
− b
Q1 ˆ r 4πǫ0 r 2
Q1 Q1 1 ( dr = dr = 2 4πǫ 0 r 4πǫ 0 b
− a1 )
Q1
Q1 = 4πǫ0 (V 2
− V 1) b ab −a
r>b r>b Q1 + Q2
E 4πr 2 =
Q1 + Q2 ǫ0
= Q1 + Q2 E 4πǫ 0 r2 b b
V 2 =
−
∞
Q1 + Q2 Q1 + Q2 dr = dr = 4πǫ 0 r 2 4πǫ0 b
Q2 = 4πǫ 0 b(V 1 + (V ( V 2
− V 1) a ab− b )
c>b
c dq dW = dqV dq V ((c)
W = QV ( QV (c) r = c = c V ( V (c) =
W =
Q1 + Q2 4πǫ0 c
Q(Q1 + Q2 ) 4πǫ0
a b a
−Q
+Q
c
a
δ
≫ a
q
a
= K ( r )4 rˆ E a K
rˆ ρ(r)
φ =
q int n E ˆ dS = ǫ0 S
·
∇ · E = p(r) ǫ0 r
∇ · E
= = = =
1 ∂ 2 (r E (r)) r2 ∂r 1 ∂ 2 r 4 (r K ( ) ) r2 ∂r a 6 1 ∂ Kr ( ) r2 ∂r a4 6Kr 3 a4
6ǫ0 Kr 3 ρ(r) = a4
Q
K q
q =
a
6ǫ0 Kr 3 4πr 2 dr 4 a 0 a 24ǫ0 K r6 ) a4 6 0
= 4π =
ρ(r)dV
= 4πa2 ǫ0 K
q 4πǫ 0 a2
K =
ρ(r) = r = a + αδ
3 qr 3 2 πa6
0<α<1 q
− σint =
− 4πaq 2
σext =
Q + q 4πa2
Q +q
δ
int = E
q r 4 ( ) rˆ, 4πǫ0 a2 a
ext = Q + q ˆ E r, 4πǫ 0 r2
para r < a
si r > a
r>a V (r) =
−
ext d E r =
r
−
Q + q 4πǫ0
r
∞
1 Q + q dr = r2 4πǫ 0 r
V =
−
d E r =
·
b
− ∞ − −
V (r) =
ext d E r +
·
=
∞
V (r) =
·
b
si r > a
·
ext d E r +
·
Q + q 4πǫ0 a
Q + q 4πǫ 0 a
int dr , E
d E r
b
=
r
−
r
−
int d E r
·
b
− 20πaq 6ǫ0 (r5 − a5)
− 20πaq 6ǫ0 (r5 − a5),
si r < a
ρ(r) e−λr V (r) = q 4πr 2 ǫ0 ρ(r )
ρ(r) r = E
= E
−∇V
q − ∂V rˆ = e−λr (1 + λr)ˆr 2 ∂r 4πǫ0 r
∇ · E = ǫρ0 = E ˆ E r
∇ · E
= = =
1 ∂ 2 (r E (r)) r 2 ∂r 1 q ( λe−λr (1 + λr) + λe−λr ) 2 4πǫ0 r 1 q 2 −λr λ e 4πǫ0 r
−
−
ρ(r) =
− 4π1 q rλ2e−λr
