Ch. 21
Coulomb’s law:
~
F
=
k
q
1
q
2
r
2
ˆ
r
,
k
=
1
4
π±
0
E
=
F
q
, so
dE
=
k
ˆ
rdq
r
2
Given uniform charge density:
dq
=
Q
V
dV
=
ρdV
for a volume,
σdA
for a sheet,
λdl
for a line
Dipoles
For
q
at
~
d
and

q
at 0:
~
P
≡
q
~
d
.
~
F
net
= 0
~
τ
=
~
r
×
~
F
=
~
P
×
~
E
=
PE
sin
θ
U
=

~
P
·
~
E
Ch. 22
Φ =
H
~
E
·
d
~
A
=
q
encl
±
0
. For a closed surface,
~
A
points
outward.
H
is the integral over a closed surface.
E
(
X
) =
1
±
0
R
X
0
ρ
(
x
)(
x
X
)
D

1
dx
, where D is the dimen
sion of the symmetry (1 slab, 2 cylindrical, 3 spheri
cal).
ring:
E
=
kQx
(
x
2
+
a
2
)
3
/
2
(a radius, x distance)
line:
1
2
π±
0
λ
x
√
x
2
/a
2
+1
(2a length, x distance)
∞
line:
E
=
λ
2
π±
0
r
∞
sheet:
E
=
σ
2
±
0
. (Think one end (half) of a Gaus
sian cylinder.)
disk:
E
=
σ
2
±
0
(1

1
√
R
2
/x
2
+1
)
parallel plates:
E
outside
= 0;
E
between
=
σ
±
0
insulating sphere:
E
inside
=
kQr
R
3
A
sphere
= 4
πr
2
Conductors: A conductor is enclosed by an equipo
tential surface, so
E
=
E
⊥
at surface.
E
inside
=
q
inside
= 0.
Ch. 23
~
F
is a conservative force if
W
a
→
a
=
H
a
a
~
F
·
d
~
l
= 0.
≡
W
a
→
b
is pathindependent.
≡ ∃
U
:
~
F
(
~
r
) =

~
∇
U
(
~
r
)
≡
U
+
K
is constant
≡
Δ
U
=

W
for a slow displacement from rest to
rest.
≡
U
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 Fall '07
 Evrard
 Force, Magnetic Field, Trigraph, Dq Dt, dt dt surf, AC w/, dt mag dipole

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