m q

R
R

= N cos(/3)i + N sin(/3)j
N
Fe
kq 2
Fe = 2
R
 ! kq 2
FX = N cos(/3) =0 (1)
R2
!
FY = N sin(/3) mg = 0 (2)

mg
N=
sin(/3)

mg kq 2
cos(/3) 2 = 0
sin(/3) R
q
"
mg cot(/3)
q = R
k
 q2
mg = k
x0 2
x0
#
k
= x0 = q (1)
mg

x0

d2 x kq 2
F =m = mg +
dt2 (x0 + x)2
|x| << |x0 | = | xx0 | << 1

d2 x kq 2 kq 2 x 2 kq 2 x
F =m = mg + = mg + (1 + ) = mg + (1 2 )
dt2 (x0 + x)2 x0 2 x0 x0 2 x0
kq 2
mg = x0 2

d2 x 2kq 2 x
= m =
dt2 x0 3
d2 x 2kq 2 x
= + =0
dt2 mx0 2 x0
kq 2
mx0 2 = g
d2 x 2g
= + x=0
dt2 x0

"
2g
w=
x0
 x
y

2 2

q q x 3
F1 = k 2 (cos( )x sen( )y) = k 2 ( y)
L 3 3 L 2 2
q 2
F2 = k 2 x
L

3 q 2 3 q 2 q 2 3 y q2
F = F1 + F2 = k 2 x k 2 y = 3 k 2 ( x ) = 3 k 2 (cos( )x sen( )y)
2 L 2 L L 2 2 L 6 6

6 x

r L L
= 2 = r =

sen( 6 ) sen( 3 ) 3

q2 qQ 1
3k 2
= k L 2 = Q = q
L ( 3 ) 3
 2a

r = R (cos()x + sen()y) r1 = az r2 = az
r r1 ,
r2
|r r1 | = |r r2 | = R2 + a2

kqQ
F = F1 + F2 = 3 (Rcos()x + Rsen()y az + Rcos()x + Rsen()y + az)
(R2 + a2 ) 2

2kqQR
= 3 (cos()x + sen()y) .
(R2 + a2 ) 2

2kqQR
|F | = 3 = 2kqQ f (R)
(R2 + a2 ) 2
3 1
df (R) (R2 + a2 ) 2 3R2 (R2 + a2 ) 2
= = =0
dR (R2 + a2 )3
1 1
= (R2 + a2 ) 2 ((R2 + a2 ) 3R2 )) = 0 = R = a
2
R= 1 a
2

E

(C seg 1 )
1

FE q

FE = q E

E X = E i
E FE = qE i

T = T sin()i + T cos()j
 !
FX = qE T sin() = 0 (1)

!
FY = T cos() mg = 0 (2)

mg
T =
cos()

mg
qE sin() = 0
cos()

mg tan()
q=
E
mg tan()
q= E 1 tan()

mg(t)
q(t) =
E

dq mg d
=
dt E dt
dq d
dt = dt

d E
=
dt mg
 +Q d
q m

F1
F1 r1 = d/2j
xi

k(q)Q(xi + d/2j)
F1 =
l3

F2
F2 r2 = d/2j

k(q)Q(xi d/2j)
F2 =
l3
q

2kqQx
F = F1 + F2 = i (1)
l3
 d/2

d/2
= cos()

= = (d/2) cos()

1 cos() 1 (d/2)

16kqQx
F = (2)
d3
F = ma
d2 x
x ma = m dt2 i

16kqQx d2 x
3
=m 2
d dt
d2 x 16kqQx
= + =0
dt2 md3

"
kqQ
w=4
md3

#
2 md3
T = =
w 2 kqQ
 P
P

j
P

1 = 2kq j
E
r2
P
x
y

y Ey = E cos()
E

kq
E=
d2 + r 2
 cos() = r
d2 +r2

kq r kqr
Ey = ( )j = 2
d2 +r 2 2
d +r 2 (d + r2 )3/2

y = kqr
E j
(d2 + r2 )3/2

P
=E
E y + E
y + E
1

= ( 2kq
E
2kqr
)j
r 2 (d + r2 )3/2
2

P dr

= 2qk( 1
E
r
)
r2 (r2 + d2 ) 32

= 2qk( 1
E
1
)
r 2 d2 32
r2 (1 + r2
)
2qk d2 3
= (1 (1 + ) 2)
r2 r2
(1 + x) x0

(1 + x) = 1 + x
d
dr r 0 ( dr )2 0

d2 3 3 d2
(1 + ) 2 (1 )
r2 2 r2

2
= 3qkd j
E
r4

I

E
F = qE

= F a sin() + F a sin() = 2F a sin() = 2qE sin()

I
O

= I

2qEa sin() = I

1
d2
sin() = dt2
 d2
2qEa = I
dt2

d2 2qEa
+ =0
dt2 I

"
2qEa
w=
I

"
w 1 2qEa
f= =
2 2 I
 1 2

$
dq(r r)
F = q
3
40 |r r|

x=0 x=L
x = L+d x = 2L+d r1 = x1 x r2 = x2 x 0 x1 L L+d x2 2L+d

dq1 (r2 r1 )
$
dF21 = dq2
1 40 |r2 r1 |3
L
1 dx1 (x2 x1 )x
$
= dq2
0 40 |x2 x1 |3
$ L
1 dx1
= dq2 x 2
40 0 (x2 x1 )
%L
1 1 %%
= dq2 x
40 x2 x1 % 0
1 x 1 1
= dq2 ( )
40 x2 L x2
1 2 x 1 1
= dx2 ( )
40 x2 L x2
 $ 2L+d & '
1 2 x 1 1
= F21 = dx2
40 L+d x2 L x2
1 2 x
= (ln(x2 L) ln(x2 ))|2L+d
L+d
40
%2L+d
1 2 x x2 L %%
= ln( )
40 x2 % L+d

(L + d)2
& '
1 2
= F21 = ln x
40 d(2L + d)
 BCD R
AB = 2R DE = R BC CD
q q AB
q
DE O

2R q
q
1 = 2R
dq dq = 1 dx dx

dq i
 1 = ( kdq k1 dx
dE 2
)i = ( )i
x x2
x = 3R x = R

1 %%R
%
1
$
1 = k(1
E dx)i = (k1 ( 2 % ))i
x2 x 3R

1 = 2k1 i
E
3R

hati

BC
CD

R
2

2q
2 =
R
2q
3 =
R

dq

kdq
dE =
R2
2 dq = 2 ds ds

ds = Rd
 k2 Rd k2 d
dE2 = =
R2 R
x dE2

k2 cos()d
dEx2 = dE2 cos() =
R

=0 = 2

k2 k2
$
2
Ex2 = cos()d =
R 0 R

x2 = k2 i
E
R
x

x3 = k2 i
E
R
2 + E
E 3 = 2k2
R i

q
O R

q
4 =
R
AB
2R
1 k4
$
4 = (k4
E dx)i = ( )i
x2 2R
R

1 + E
E 2 + E 4 = 2k1 i + 2k2 i + ( k4 )i = 0
3 + E
3R R 2R
4
2
4 = (21 + 62 )
3
1 2 4

2 12
q = q(1 + )
3
3
10
q = q
3


dq(r r1 )
$ $
r) =
E(
40 |r r1 |3
r1 r
r = xx + y y + z z r1 = x1 x + y1 y |r r1 |2 = (x x1 )2 + (y y1 )2 + z 2
$ + $ +
dx1 dy1 ((x x1 )x + (y y1 )y + z z)
= E(r) = 3
40 ((x x1 )2 + (y y1 )2 + z 2 ) 2
w = x x1 v = y y1 = x1 = x w, y1 = y v

% % % %
% (x ) (x1 )v % % 1 0
J = %% 1 w
%
%=% %=1
(y1 )w (y1 )v % % 0 1 %

+ $ +
dwdv(wx + v y + z z)
$
r) =
= E( 3
40 (w2 + v2 + z2) 2
+ $ +
dwdv (z z)
$
= 3
40 (w2 + v 2 + z 2 ) 2

w = rcos()
v = rsen()
% % % %
% wr w % % cos() rsen() %
J =%
% % = % %=r
vr v % % sen() rcos() %

2 +
rdrd (z z)
$ $
r) =
= E( 3
40 0 0 (r2 + z 2 ) 2
$ +
rdr
= 2z z 3 pero u = r2 + z 2 du = 2rdr
40 0 (r2 + z 2 ) 2
$ +
du
= z z 3
40 z2 (u) 2
( )% 2
1 %z
= z z 2u 2 %
40 +
z
= z
20 |z|

= sign(z)z
20
 r) = sign(z)z
= E(
20

+ 0


E1 (r) = sign(x)x
20

& '
x
E2 (r) = x sign(x)
20 R2 + x2

& '
= E1 + E2 =
= E
x
x
20 R + x2
2

dx
 dF = dq E

$ d+a
= F =
dq E
con dq = dx
d
$ d+a & '
x
= dx x
d 20 R2 + x2
$ d+a
xdx
= x
20 d R2 + x2
(* 2 )%d+a
= R + x2 % x
%
20 d
(* 2 * )
= R + (d + a)2 R2 + d2 x
20
 a Q
O

kdq(r r )
$
0) =
E(
r r 3

dq = dA

dA

ds1 = R sin()
ds2 = Rd

dA = ds1 ds2 = R2 sin()dd

dq = R2 sin()dd
 r r r = 0

r = R sin() cos()i + R sin() sin()j + R cos()k

r r = R

3
2
k(R sin() cos()i R sin() sin()j R cos()k) 2
$ $
2
0) =
E( R sin()dd
0 R3
2

$ 2 $ $ 2 $ $
0) =
E( k( cos()k) sin()dd = k d ( cos()k) sin()d = k ( sin(2)k))d

0 2
0 2 2

%
1 1
E(0) = k( cos(2)%% )k = k (cos(2) cos())k = (k)k
%
2 2
2

Q
=
2R2

0) = kQ k
E(
2R2
 AB 2L CD
L Q

Q
j

dq dE = kdq r
r2
dq xi dj r = x + d2
2

dq dq = 1 dx 1
Q
1 = 2L

k1 dx
dE =
x2 + d2
j x
y dEy = dE cos()
d
cos() = *
(x + d2 )
2
 k1 ddx
dEy = 3
(x2 + d2 ) 2
x = L x=L
L
%L
1 x % ) = 2k1 L
$
%
Ey = k1 d 3 dx = k1 d(
2 2 2 2 2
(d x + d %L d L2 + d2
L (x + d ) 2
dq
y
2k1 L
dF = dqE = dq *
d L2 + y 2

Q
2 =
L
dq = 2 dy
2k1 L
dF = 2 dy *
y L2 + y 2

y=L y = 2L
L
1
$
F = 2k1 2 L * dy
L y L2 + y 2

*
1 1 L2 + y 2 L
$
* dy = ln( )
2
y L +y 2 L y

* %2L
1 L2 + y 2 L %%
F = 2k1 2 L( ln( )% )
L y %
L

51
F = 2k1 2 ln( )
2( 2 1)
1 2 Q

kQ2 51
F = 2 ln( )
L 2( 2 1)
 b

d

r2 = 0 r1 = xx + y y x + d y b+d

dq1 r1
$ $
dF21 = dq2 3
, con dq1 = dxdy
40 |r1 |
$ + $ b+d + $ b+d
dxdy xx dxdy y y
$
= dq2 3 dq2 3
d 40 (x2 + y 2 ) 2 d 40 (x2 + y 2 ) 2
dq2 y + b+d dxdy y
$ $
= 3
40 d (x2 + y 2 ) 2
+ ,%b+d
dq2 y + 1
$ %
= dx
%
*
40
%
2 2
x + y %d
+ ,
dq2 y + 1 1
$
= * dx, pero dq2 = dx2
40 x2 + (b + d)2 x2 + d2

+ ,
dF21 y +
1 1
$
= = * dx
dx2 40 x2 + (b + d)2 x2 + d2
+ * ,%+
y x + x2 + (b + d)2 %%
= ln
20 x + x2 + d2
%
%
0
/
2
1 + 1 + ( b+d )
& '
y x b + d
= lm ln / ln
20 x+ d 2 d
1 + 1 + (x)
& '
y d
= ln
20 b+d

dF21
& '
y d
= = ln
dx2 20 b+d
 R
R

d j
x

dq r = xi r = dj

= kdq(r r ) kdq(xi + dj)

dE =
|r r |3
3
(x2 + d2 ) 2
dq = dx

= kdx(xi + dj)
dE 3
(x2 + d2 ) 2
+

+ + +
kdx(xi + dj) x d
$ $ $
=
E 3 = k(( 3 )i + ( 3 )j)
(x2 + d2 ) 2 (x2 + d2 ) 2 (x2 + d2 ) 2
 x
f (x) = 3
(x2 +d2 ) 2
d
g(x) = 3
(x2 +d2 ) 2
+ +
$ + $ +
g(x)dx = 2 g(x)dx
0

+
1
$
= 2kd
E 3 j
0 (x2 + d2 ) 2
1 x
2
3 = 1
(x2 +d2 ) 2 d2 (x2 +d2 ) 2
%
= 2kd( x % 2k
E )j = j
%
3 %
d2 (x2 + d2 ) 2 % d
0

dq dF = dqE
E
dq

dq
dq
d() = R + R sin() dq
ds ds
ds = Rd dq

2k 2k2 d
dF = dqE = Rd =
R + R sin() 1 + sin()
=0 =
 1 sin() 1
$
=
1 + sin() cos()

F = 4k2
 q>0
R H

= q
$
= dS
E (1)
S 0

Se
Sc = e + c
= c = + e (2)
e e

$
e = dS
E
Se
1 q
q E = 40 R2 r r

dS = ndS
dS n
r
 dS
= 1 q 1 q
E 2
r ndS = dS
40 R 40 R2

1 q 1 q
$ $ $
e = dS
E = dS = dS
40 R 2 40 R2
Se Se Se

dS = 2R2
2 2
Se dS Se

1 q q
e = 2
2R2 = (3)
40 R 20

q
c =
20

1
0
 ndS = Qint
3
E
0
Qint

=
E

0

z r) = E(r)r
E( r
r

|E|

|E|

|E|

r<R

ndS = Qint = 0
3
E
0

3 $ $
ndS =
E E(r)r ndS + E(r)r ndS
manto tapas
$
= E(r) dS
manto
= E(r)(2r)h
= 0
= E(r)(2r)h = 0 = E(r) = 0 = E

Rr

& '
ndS = E(r)(2r)h = (2R)h = E(r) = R
3
E
0 0 r

4

E(r) =
! R0 " r<R
0 r R r

+ 0
 = r
r<R

0
#
U= 2 d3 x
E
2 R3

r) = E(r)r
E( r r

r<R

Qint
$
ndS =
E
0
1
$ #
E(r)r ndS = d3 x
0
1 2 r 3
# # #
E(r)4r2 = r sen()drdd
0 0
# r 0 0
4
E(r)4r2 = r3 dr
0 0
r4
E(r)4r2 =
0
2
= E(r) = r
40

Rr

Qint
$
ndS =
E
0
4 R 3
#
2
E(r)4r = r dr
0 0
R4
E(r)4r2 =
0
R 2 2
% &
= E(r) = R
40 r
 '
2
r r r<R

E(r) = !4R0 "2 2

40 r R r R r

0
#
U = 2 d3 x
E
2 R3
0 2 +
# # #
= E(r)2 r2 sen()drdd
2 0 0 0
40 +
#
= E(r)2 r2 dr
2 0
%# R 2 # + 2 8 &
40 6 R
= 2 r dr + dr
2 0 160 R 1620 r2
( )R )+ *
2 r7 )) 1
8 )
)
= R
80 7 )0 r )R
2 R7
% &
= + R7
80 7
2 7
= R
70
 R

2R d

E(r) = r
r
30
R r
r) = E(r)r
E(

r>R

Qint
$
ndS =
E
0
4R3
E(r)4r2 =
30
3
= r) = R r
E(
30 r2

r
'
30 r r<R
E(r) = R3
30 r 2
r Rr

E1 (r) = r para |r| < R
30
para |r d|
<R
E2 (r) = (r d)
30

<R
|r d| |r| < R

r) = E1 (r) + E2 (r) = r (r d)
E( = d
30 30 30
 > 0

Q < 0

E0 = E0 x E0 > 0

E0

r
'
30 r r<R
E(r) = R3
30 r 2
r Rr

0) = 0
E(

F = Q 3
r
0
r

r
ma = F = Q r
30
r
a + Q r = 0
3m0

= r + Q r =0
3m0
r + w2 r = 0

+
2 3m0
T = w = 2 Q

r
'
30 r + E0 x r < R
E(r) = R3
30 r 2
r + E0 x R r
 x
( 330 + E0 ) x R < x < R


E(r) = ( 3R0 x2 + E0 ) x R x
3
(E0 3R0 x2 ) x x R

R < x < R

x 30 E0
+ E0 = 0 = x1 =
30

x1

30 E0 R
x1 = > R = E0 <
30

E0

2 U

U = qV

#
V (x) =
E(x) xdx + c
# % &
x
= + E0 dx + c
30
x2
= E0 x +c
60

U (x) = QV (x)
x2
= QE0 x + Q Qc
60
d2 U Q
= 2 U = = >0
dx2 30

Rx

R3
+ E0 = 0 = x2 imaginario
30 x2

Rx
 x R

R3
0
R
E0 = 0 = x3 = R
30 x2 30 E0
x1
0 0
R R R
x3 = R < R = > 1 = E0 <
30 E0 30 E0 30

R
E0 < 30
E0 = 0
R
E0 = 30
R
x = R E0 > 30
 R
r
= 0 (1 R) 0 r

r<R
R

= E r
E r

ndS = qinterior
# # #
ndS =
E ndS +
E E (1)
S tapas manto 0
n r r n = 0
# #
ndS =
E E r ndS = 0 (2)
tapas tapas
n r r n = 1
# # #
ndS =
E E r ndS = E dS
manto manto manto
1 1
manto dS manto dS = 2rL
#
ndS = E 2rL
E (3)
manto

qinterior
E 2rL = (4)
0
 dq = (r ) dV (5)

dV
r r + dr

dV

dV = 2r dr L (6)

dq = 2r dr L p(r )
r
dq = 2r dr L 0 (1 )
R
r 2
= dq = 2L 0 (r )dr
R

r 2
r # r # r 2
r
#

q = 2L0 (r )dr = 2L0 ( r dr dr )
0 R 0 0 R

r2 r3 3Rr2 2r3
= q = 2L0 ( ) = 2L0 ( )
2 3R 6R
L0
= q = (3Rr2 2r3 ) (7)
3R

L0
E 2rL = (3Rr2 2r3 )
3R0
E
0
E= (3Rr 2r2 ) (8)
6R0
r E

dE 0
= (3R 4r)
dr 6R0
r
3R
r=
4

30 R
|E|max =
160
 2a

R
x>0

xx>0

E
y E
 0<x<a x>a

0<x<a
A 2x

n = 0
E
#
ndS = 0
E

(E) n = E
# #
ndS = E
E dS
S S

qint
# #
=E dS + E dS = EA + EA = 2EA = (1)
S1 S2 0
S1 S2
qint

qint = V = 2xA

2xA
2EA =
0
x
E=
0
i

= x i, si 0 < x < a
E
0
 x>a

= 2EA
a < x < a qint = 2aA

2aA
= 2EA =
0
a
E=
0
i
a
E = i, si x > a
0

r<R

r<R
 ndS
1
= S E n
n = E = E S dS = E4r2
1
E
qint = 0
qint
= E4r2 = =0
0
E = 0, si r < R

r>R
r > R

# #
= ndS = E
E dS = E4r2
S S

qint
qint = 4R2

qint 4R2
= E4r2 = =
0 0

R2
E=
0 r2

2
= R r, si r > R
E
0 r2
r

0<x<a
a < x < a + 2R
x > a + 2R

= x = R2
0<x<a E 0 i E 0 r2
r
r r

a+R = x+r r = a+Rx

r = i

= R2
E i
0 (a + R x)2
 =E
E + E

= ( x R2
E )i
0 0 (a + R x)2
a < x < a + 2R E

= a i, si a < x < a + 2R
=E
E
0

x > a + 2R i
E

x = a+R+r r = xaR
r = i

= R2
E i
0 (x a R)2

= ( a + R2
E )i, si x > a + 2R
0 0 (x a R)2
 R2
x
( 0
0 (a+Rx)2
)i 0<x<a
= a
E 0 i a < x < a + 2R
( a + R2

0 0 (xaR)2
)i x > a + 2R
 R
2R d

E(r) = r
r
30
Rr
r) = E(r)r
E(

r >R

Qint
$
ndS =
E
0
4R3
E(r)4r2 =
30
3
= r) = R r
E(
30 r2

r
'
30 r r<R
E(r) = R3
30 r2
r Rr

E1 (r) = r para |r| < R
30
para |r d|
<R
E2 (r) = (r d)
30

<R
|r d| |r| < R

r) = E1 (r) + E2 (r) = r (r d)
E( = d
30 30 30
 >0
x=0 ax q < 0
q < 0

m e < 0 q

q

E1 = x
20
1 q 1 q
E2 = 2
x = x
40 (a x) 40 (a x)2
q
% &
= E1 + E2 = 1 q
E + x
20 40 (a x)2

= V
E
% &
V V V 1 q
=0 =0
y = +
z x 20 40 (a x)2
% &
x 1 q
= V (x) = + +C
20 40 (a x)

e a/2x
 mv 2
eV (a/2) = eV (0)
2
mv 2
= K = = e (V (0) V (a/2))
2 % &
1 q 1 2q a
= e +C + + C
40 a 40 a 40
% &
1 q a
= e +
40 a 40
 r1 , r2 q1 , q2
d >> r1 , r2

V (+) = 0

q1 q2
V1 = k , V2 = k
r1 r2

d >> r1 , r2

(q1 )f , (q2 )f

q1 + q2 = (q1 )f + (q2 )f

V1 = V2
(q1 )f (q2 )f
k = k
r1 r2
2
4(r1 ) (1 )f 4(r2 )2 (2 )f
=
r1 r2
(2 )f r1
= =
(1 )f r2

(q1 )f + (q2 )f = q1 + q2
2 2
4(r1 ) (1 )f + 4(r2 ) (2 )f = q1 + q2
4r1 (1 )f (r1 + r2 ) = q1 + q2
1 q1 + q2
= (1 )f =
4r1 r1 + r2
1 q1 + q2
(2 )f =
4r2 r1 + r2
 |E (r)|
(r) r

(r)
0

r
0 < 2h << 1
0 < A << 1

r) ndS = Qint
$
E(
0

Qint (r)A
Qint
= (r)A = =
0 0

$ # # #
r) ndS =
E( r) ndS +
E( E1 (r) n1 dS + E2 (r) n2 dS
manto carasup carainf

= E + E//
E

r)|
|E( n1 = n2

# #
E1 (r) n1 dS + E2 (r) n2 dS = (E1 ) A (E2 ) A
carasup carainf

= ((E1 ) (E2 ) )A
(
r)A
h 0
#
r) ndS 0
E( cuando h0
manto
 (r)A
((E1 ) (E2 ) )A =
0
(r)A
E A =
0
(r)
= E (r) = +
0

r) = + (r) n
E(
0
n r
q0 > 0

V0

q>0
c>b

r
V = V (r) r) = E(r)r
E(

q1 , q2 , q3 , q4
q0 = q1 + q2

ndS = 0 = q1 = q1 = 0 = q2 = q0
$
E
0
= 0
E
 ndS = 0 = q1 + q2 + q3 = q1 + q2 + q3 = 0 = q3 = q2 = q0
$
E
0

r>b

ndS = E(r)4r2 = q1 + q2 + q3 + q4 = E(r) q4 r

$
E =
0 40 r2

V0
# b
V (b) V (+) = V (b) = V0 = dr
E
+
# b
q4 dr
V0 =
+ 40 r2
q4 1
V0 =
40 b
= q4 = 40 bV0

ra
ra

ndS = E(r)4r2 = q1 = 0 = E
$
E = 0
0
a<r<b

ndS = E(r)4r2 = q0 = E(r) q0 r

$
E =
0 40 r2
r=b

E(b) = 0

r>b

ndS = E(r)4r2 = q4 = 40 bV0 = E(r) bV0

$
E = 2 r
0 0 r

0 ra
q0 r


40 r2
a<r<b
E(r) = 0

r=b
bV0
r2
r br

arb
#
V (r) = dr + C
E
q0 dr
#
= +C
40 r2
q0 1
= +C
40 r

V (b) = V0
q0 1
C = V0 40 b
% &
q0 1 1
V (r) = + V0
40 r b

% &
q0 1 1
V (a) V (b) =
40 a b
ra
r=a
% &
q0 1 1
V (a) = + V0
40 a b

% &
q0 1 1
V (r a) = + V0
40 a b
br
#
V (r) = dr + C
E
q4 dr
#
= + C
40 r2
q4 1
= + C
40 r
bV0
= + C
r
V (b) = V0 = C = 0

bV0
= V (r) =
r

q0 !1 1"
+ V0 r a

40 ! a1 1b "
q0
V (r) = 40 r b + V0 a < r < b
bV0
br

r

V0 = 0

= V
E
V0 = 0

q1 = 0
q2 = q0
q3 = q0
q4 = 0

= 0
ra : E
= q0 r
a<r<b : E
40 r2
= 0
br : E

% &
q0 1 1
r a : V (r) =
40 a b
% &
q0 1 1
a < r < b : V (r) =
40 r b
b r : V (r) = 0
q0 !1 1
"
V (a) V (b) = 40 a b

q>0
 q1 , q2 , q3 , q4

ndS = 0 = q1 = q1 = 0 = q2 = q0
$
E
0

ndS = 0 = q1 + q2 + q3 = q0 + q3 = q3 = q0 = q4 = 0
$
E
0 0

q>0
q

1b
V (+) = 0 V (b) = dr = 0 = 0
+ E E
 L

dq
#
V (r) =
40 |r r1 |

r = xx + y y + z z r1 = x1 x 0 x1 L

L
dx1
#
V (r) = 2
0 40 (x1 x)2 + y 2 + z 2

dx
1
x2 +1
x = tan()

dx sec2 ()d
# # # 2
= 2 = sec()d = ln(sec()+tan()) = ln(x+ x2 + 1) = arccosh(x)
x2 + 1 tan2 () + 1

dx1
1
(x1 x) +y +z 2
2 2

dx1
# 2
2 = ln(x1 x + ((x1 x)2 + y 2 + z 2 )
(x1 x)2 + y 2 + z 2

L
dx1
#
V (x, y, z) = 2
2 2 2
0 40 (x1 x) + y + z
# L
dx1
= 2
40 0 (x1 x)2 + y 2 + z 2
2 )L
= ln(x1 x + (x1 x)2 + y 2 + z 2 ))
)
40 0
( 2 *
2 2
L x + (L x) + y + z 2
= ln 2
40 x + x2 + y 2 + z 2

2d
 % &
q 1 1
V =
40 r+ r

3 4
2
r+ = r2 + d2 2rd cos
2
= r2 + d2 2rd sen()
1 1
= = 2
r+ 2 2
r + d 2rd sen ()
1 1
= +
r 1 + ( d )2 2 d sen ()
r r

d
r >> d r << 1

1 1 1
= +
r+ r 1 + ( dr )2 2 dr sen ()
( (% & **
1 1 d 2 d
= 1 2 sen()
r 2 r r
( % &2 *
1 1 d d
= 1 + sen()
r 2 r r

r
 3 4
2
r = r2 + d2 2rd cos + = r2 + d2 + 2rd sen ()
2

( % &2 *
1 1 1 d d
= 1 sen()
r r 2 r r

r >> d

% &
q 1 1
V (r, ) =
40 r + r
(( * ( **
1 d 2 d 1 d 2 d
% & % &
q
= 1 + sen() 1 sen()
40 r 2 r r 2 r r
% &
q 2d
= sen()
40 r r
2qd sen()
=
40 r2
p sen()
=
40 r2
1 p r
V (r, ) =
40 r2

= V r 1 V
= V
E
r r

V 1 psen()
=
r 20 r3

1 V 1 pcos()
=
r 40 r3

) = p 3 4
= E(r, 2sen()r cos()
40 r3

V (+) = 0
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 + py + pz
x y z

= p V
U =
pE
 b (r) =
0 a < r < b r < a

# p
Vp = d
E

= E r
E r

r<a
a<r<b
r>b

r<a
r<a
r

E n n = E
E
# #
= ndS = E
E dS = E4r2
S S

qint = 0

E4r2 = 0

E=0
a<r<b

n = E
E
 #
= ndS = E4r2
E
S
qint
qint = 0 V V r a

4
qint = 0 (r3 a3 )
3

4
E4r2 = 0 (r3 a3 )
30

0 (r3 a3 )
E=
3r2 0
r
3 3
= 0 (r a ) r
E
30 r2
r>b

4
qint = 0 (b3 a3 )
3

0 (b3 a3 )
E4r2 =
3r0

0 (b3 a3 )
E=
30 r2

0 (b3 a3 )
E(r) = r
30 r2

0 r<a
0 (r 3 a3 )

=
E r a<r<b
30 r 2
0 (b3 a3 )
r r>b

30 r 2

d
r>b
E d = E r d = Edr
# r r r
0 (b3 a3 )
# #
V (r) =
E d = Edr = )dr
30 r2

0 (b3 a3 )
V (r) = , si r > b
30 r

r a < r < b

# r
V (r) = d
E

r
b b r
# b # r
V (r) = d +
E d
E
b
r=b
b 3 3
d = 0 (b a )
#
V (b) = E
30 b

r r
0 (r3 a3 )
# #
d =
E dr
b b 30 r2
# b # r
0 3 1
= ( rdr a 2
dr)
30 b r
)r
r2 ) 1 )r
)
0
= (( )) ) a3 ( )) ))
30 2 b r b
0 b2 r 2 (r b)
= ( + a3 )
30 2 rb
 r

0 b3 a3 b2 r2 (r b)
V (r) = ( + + a3 ), si a < x < b
30 b 2 rb
r<a

# r # b # a # r
V (r) = d =
E d +
E dr +
E d
E
b a

r<a
# r
d = 0
E
a

0 b3 a3 b2 a2 (a b)
V (r) = ( + + a3 ), si r < a
30 b 2 ab
r

AB
d

r r = b

L r

ndS = qint
#
= E (1)
S 0

# #
ndS = E
E dS = E2rL (2)
S S

qint = V = 2aL (3)

 a
E= (4)
r0
r=d
a
E=
d0

y
i 2Ey = 2Ecos( 3 )

a
E=
d0

0<x<D i

x
 1 = 0,
E si 0 < x < a

1 = a i,
E si x > a
x0

2 = 0,
E si d a < x < d
x
0<x<da x dx

2 = a
E i
(d x)0

a


(dx)0 i 0<x<a
=
E a
( x a
+ (dx) )i a<x<da
0 0
a

x0 i da<x<d

V
x

# d
V = d
E
0
# a # da # d
= d +
E d +
E d
E
0 a da
a # da # d
a a a a
#
= dx + ( + )dx + i
0 (d x)0 a x0 (d x)0 da x0
a a a
= ( ( ln(d x)|a0 )) + ( ( ln(x) ln(d x)|da a )) + ( (( ln(x)|dda )))
0 0 0
a da 2a da a da
= ( ln( )) + ( ln( )) + ( ln( ))
0 d 0 a 0 d
2a da da
= (ln( ) ln( ))
0 a d
 2a d
V = ln( )
0 a
 R
R = 0 (1 r)
0 > 0 r O
P OP = R

q P

a
q d

kdq
#
V =
r
r
2
r= a2 + d2

k kq
#
V = dq =
a2 + d2 a2 + d2
dq d=R
a=r

kdq
dV =
r 2 + R2
dq

dq = (r) dA

r
r
dA dr

dA = 2rdr

dq = 20 (r R)dr

2k0 (r R)dr
dV =
r 2 + R2
r=0 r=R

R
(r R)dr
#
V = 2k0
0 r 2 + R2
# R # R
r 1
= 2k0 ( dr R dr)
2
r +R 2 r + R2
2
0 0
2 )R 2 )R
= 2k0 (( r2 + R2 ) ) ( ln( r2 + R2 + r)) ))
) )
0 0

= 2k0 (R 2 R R ln(R 2 + R) + R ln(R))

= 2k0 (R 2 R R ln(R( 2 + 1)) + R ln(R))

= 2k0 (R 2 R R ln(R) R ln( 2 + 1)) + R ln(R))

= 2k0 (R 2 R R ln( 2 + 1))

= 2k0 R( 2 1 ln( 2 + 1))

2 1 ln( 2 + 1) 21

V k0 R
Q
dq = 20 (r R)dr

# R
Q = 20 (r R)dr
0
)R
r2 )
= 20 ( ( Rr))) )
2 0
= 0 R2

Q
V k0 R
0 R2
 kQ
V
R

dq
 m q
I
II
m

kq 2
UI = 2mg +
2

kq 2
UII = 2mg(1 cos()) +
2 sin()

UI = UII

kq 2 kq 2
2mg + = 2mg(1 cos()) +
2 2 sin()

5
mg sin()
q = 2
k(sec tan())
 R
Z z>0
Z z>0
O

O
z k dq

kdq
dV =
r
r dq z k
dV
dq
dA

dA = R2 sin()dd

r2 = R2 + z 2 2zR cos()
dq

kR2 sin()dd
dV = 2
R2 + z 2 2zR cos()

2
kR2 sin()dd
# #
V = 2
0
2
R2 + z 2 2zR cos()

sin()
#
V = 2kR2 2

2
R + z 2 2zR cos()
2
 sin
#
2 d
A + B cos()

2udu
u2 = A + B cos() d = B sin()

sin 2 2
# #
2 d = 1du = u
A + B cos() B B
2
u= A + B cos()

sin 22
#
2 d = A + B cos()
A + B cos() B

A = R2 + z 2 B = 2zR

sin 1 2 2
#
2 d = R + z 2 2Rz cos()
R2 + z 2 2zR cos() zR

2kR2 2 2 ) 2kR 2
V (z) = ( R + z 2 2Rz cos()) ) = (z + R R2 + z 2 )
)
zR 2
z

z k

= V
E
z

= dV k
E
dz
V (z)

dV d 1 2
= 2kR ( (z + R R2 + z 2 ))
dz dz z
1 2 1 1 1
= 2k( 2 (z + R R2 + z 2 ) + (1 + 2z))
z z 2 R + z2
2

1 R R2 + z 2 1 1
= 2k( 2
+ + )
z z z z R + z2
2

2 R2 + z 2 R
= 2kR ( )
z 2 R2 + z 2

z k

2 R2 + z 2 R
E(z) = 2kR ( )k
z 2 R2 + z 2
z=0
O

1 R
2 z 2 +R2
2kR ( )
z2
z0 E(z)k

1 R
2 z 2 +R2
lm E(z) = 2kR lm ( )
z0 z0 z2
0
0

1 R (1 R )
z 2 +R2 z 2 +R2
2kR2 lm ( ) = 2kR2 lm ( )
z0 z2 z0 (z 2 )
3
2R(z 2 + R2 ) 2
= 2kR lm ( )
z0 2z
1
= 2kR2
2R2
= k

E(0) = k k
 r1 , r2 q1 , q2
d >> r1 , r2

V (+) = 0

q1 q2
V1 = k , V2 = k
r1 r2

d >> r1 , r2

(q1 )f , (q2 )f

q1 + q2 = (q1 )f + (q2 )f

V1 = V2
(q1 )f (q2 )f
k = k
r1 r2
2
4(r1 ) (1 )f 4(r2 )2 (2 )f
=
r1 r2
(2 )f r1
= =
(1 )f r2
 (q1 )f + (q2 )f = q1 + q2
2 2
4(r1 ) (1 )f + 4(r2 ) (2 )f = q1 + q2
4r1 (1 )f (r1 + r2 ) = q1 + q2
1 q1 + q2
= (1 )f =
4r1 r1 + r2
1 q1 + q2
(2 )f =
4r2 r1 + r2
 a < b V1 V2

Q
c > b

Q1 Q2 V2 V2

a < r < b
a Q1

ndS = Q1
#
E
S 0
ndS = E4r2
1
S E

= Q1
E r
40 r2

a
Q1 Q1 1 1
#
V2 V1 = 2
dr = ( )
b 40 r 40 b a
Q1

ab
Q1 = 40 (V2 V1 )
ba
r>b
r >b
Q1 + Q2

Q1 + Q2
E4r2 =
0
 = Q1 + Q2
E
40 r2
b
b
Q1 + Q2 Q1 + Q2
#
V2 = dr =
40 r2 40 b

ab
Q2 = 40 b(V1 + (V2 V1 ) )
ab

c>b c
dq

dW = dqV (c)

W = QV (c)

r=c
Q1 + Q2
V (c) =
40 c

Q(Q1 + Q2 )
W =
40
 a b c
a<b<c
Q +Q
 a a Q
q a

= K( r )4 r
E
a
K r
(r)

ndS = qint
#
= E
S 0

= p(r)
E

0
r

E
= 1 2
(r E(r))
r2 r
1 2 r 4
= (r K( ) )
r2 r a
1 Kr 6
= ( )
r2 r a4
6Kr3
=
a4

60 Kr3
(r) =
a4
 K
q

#
q = (r)dV
# a
60 Kr3
= 4 4
4r2 dr
0 a
)a
240 K r6 ))
= )
a4 6) 0
= 4a2 0 K

q
K=
40 a2

3 qr3
(r) =
2 a6

r = a + 0<<1
q

q
int =
4a2

Q+q
Q+q
ext =
4a2

int = q r
E ( )4 r, para r < a
40 a2 a

ext = Q + q r,
E si r > a
40 r2
r>a
r
ext dr = Q + q 1 Q+q
# #
V (r) = E 2
dr =
40 r 40 r
r<a
 # # b # r
V = dr =
E ext dr +
E int dr,
E si r > a
b

#
V (r) = dr
E
# b # r
= ext dr +
E int dr
E
b
Q+q q
= (r5 a5 )
40 a 20a6 0

Q+q q
V (r) = (r5 a5 ), si r < a
40 a 20a6 0
 (r)

er
V (r) = q
4r2 0
(r)

(r)
r

= V
E

= V r =
E
q
er (1 + r)r
r 40 r2

=
E

0
= E r
E

E
= 1 2
(r E(r))
r2 r
1 q
= (er (1 + r) + er )
40 r2
1 q 2 r
= e
40 r

1 q 2 r
(r) = e
4 r