Physics 8.02
Exam One Conflict Exam Solutions
Fall 2007
Please Remove this Tear Sheet from Your Exam
2
3
ˆ
4
4
o
o
r
r
πε
πε
=
=
E
r
q
q
r
G
G
ˆ
points
source q
observer
r
r =
from
to
r
G
insidefree
o
closed surface
κ
ε
⋅
=
∫∫
E dA
w
Q
G
G
points from inside to outside
dA
G
q
=
F
E
b
G
G
G
∫
⋅
−
=
−
=
Δ
a
a
b
b
to
a
from
moving
V
V
V
s
d
E
G
0
=
⋅
∫
path
closed
s
d
E
G
G
W
U
q V
= Δ
=
Δ
point charge
4
o
V
r
q
πε
=
∑
−
=
pairs
all
j
i
o
j
i
U
r
r
G
G
πε
4
q
q
2
1
2
o
v
U
E
ε
⎡
⎤
=
⎢
⎥
⎣
⎦
∫∫∫
x
ol
dV
E
x
=

∂
V
∂
x
,
E
y
=

∂
V
∂
y
,
E
z
=

∂
V
∂
z
r
V
E
r
∂
= −
∂
for spherical symmetry
Q
C
V
=
Δ
U =
1
2
C
Δ
V
2
=
Q
2
2C
1
parallel
C
C
=
+
Circumferences, Areas, Volumes:
Circle of radius
r
:
Area =
π
r
2
Circumference =
2
π
r
Sphere of radius
r
Surface Area =
4
π
r
2
Volume =
3
4
3
r
π
Cylinder of radius
r
and height
h
Side surface area =
2
π
r h
End cap surface area =
2
π
r
2
Volume =
π
r
2
h
Definition of trig functions
sin is opposite/hypotenuse;
cos is adjacent/hypotenuse;
tangent is opposite over adjacent;
Properties of
30, 45,
and 60 degrees
(
π
/6,
π
/4, and
π
/3 radians)
:
sin(
π
/6) = cos(
π
/3) = 1/2,
sin(
π
/3) = cos(
π
/6) =
2
/
3
;
sin(
π
/4) = cos(
π
/4) =
2
/
1
;
Integrals that may be useful
a
b
dr
b
a
−
=
∫
)
/
ln(
1
a
b
dr
r
b
a
=
∫
⎟
⎠
⎞
⎜
⎝
⎛
−
=
∫
b
a
dr
r
b
a
1
1
1
2
Some potentially useful numbers
2
9
2
0
1
N
9
10
4
C
e
k
πε
=
=
×
m
Breakdown of air
E ~ 3 x 10
6
V/m
2
C
1
2
1
2
series
C C
C
C
C
=
+
Speed of light
c = 3 x 10
8
m/s
Electron charge
e = 1.6 x 10
19
C
Avagadro’s number
N
A
= 6.02 x 10
23
mol
1
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