P 3.1

a) Q = 8πρ0R
3
15
b) ~E(r) =

{
ρ0

15R2ε0
(5R2r − 3r3)r̂ r ≤ R

2ρ0R
3
15ε0r2
r̂ r > R
c) V (r) =

{
2ρ0R
2
15ε0
− ρ0(r2−R2)
15ε0R2
(7R2−3r2)
4
r ≤ R

2ρ0R
3
15ε0r
r > R
P 3.2 ~E(r) =

{
0 0 < r < R1

ρ2(r2−R2
1)
2ε0r
r̂ R1 ≤ r ≤ R2

ρ2(R2
2−R
2
1)
2ε0r
r̂ R2 ≤ r ≤ R3

ρ2(R2
2−R
2
1)+ρ4(r2−R2
3)
2ε0r
r̂ R3 ≤ r ≤ R4

ρ2(R2
2−R
2
1)+ρ4(R2
4−R
2
3)
2ε0r
r̂ r ≥ R4
P 3.4

a) ~E = − σ

2ε0
ẑ + ρD

2ε0
ẑ
b) ~E = + σ

2ε0
ẑ + ρ(D−2d)

2ε0
ẑ
c) ~E = + σ

2ε0
ẑ − ρD

2ε0
ẑ
P 3.3 ~E =

{
−nρr

3ε0
r̂ 0 < r < a

ρ
3ε0
(
r − (n+ 1)a
3
r2
)
r̂ a ≤ r ≤ b

ρ
3ε0r2
(
b3 − (n+ 1)a3
)
r̂ r ≥ b
P 3.5 y(x) =
qσ

2ε0mv2
x
x2 (trayectoria), tan θ =
qσd

mε0v2
x
(ángulo).
P 3.6 ΦE = Q
24ε0
P 3.7 ~E = ρ0 ~d

3ε0
P 3.8
a) ~E = − ρx0

ε0
x̂
b) σ(θ) = 2x0ρ cos θ
P 3.9

a) ρ(r) = 4E0r
a2
b) ~E(r > a) = E0a
2
ε0r2
r̂, V (r > a) = E0a
2
ε0r
c) V (r ≤ a) = E0
3ε0
(
4a− r3
a2
)
P 3.10

a) Q(r) =

{
2πhr3
3a
r ≤ R

2πhR3
3a
r > R
b) ~E(r) =

{
r2
3aε0
r̂ r ≤ R

R3
3aε0r
r̂ r > R
c) V (r) =

{
− r3
9aε0
r ≤ R

− R3
3aε0
[
ln
(
r
R
)
+ 1
3
]
r > R
P 3.11

a) ~E(r) =

{
λ

2πε0r
r̂ r < R(

λ

2πε0r
+ Rσ
ε0r
)
r̂ r > R
El campo eléctrico es discontinuo en r = R.
b) V (r) =

{
λ

2πε0
ln
(
R
r
)
r ≤ R
1
ε0
(
λ

2π
+Rσ
)
ln
(
R
r
)
r > R
El potencial eléctrico es siempre continuo.
P 3.12 b) ~E(z) = ρ
kε0
(1− e−kz)ẑ
P 3.13 a) ~ET (h) = λ sin (2α)
2πhε0
x̂+
(
σ

2ε0
− λ cos (2α)
2πhε0
)
ŷ b) W = qσh

4ε0
+ qλ ln(2)
2πε0
(
sin 2α
tanα
− cos 2α
)