Physics 8.02
Exam One Solutions
Spring 2004
PLEASE DETACH THIS SHEET
AND USE AT YOUR CONVENIENCE
Some (possibly useful) Relations:
F
=
1
4
πε
o
q
1
q
2
r
2
2
3
ˆ
4
4
o
o
q
q
r
r
πε
πε
=
=
E
r
G
r
G
ˆ
points
source q
observer
r
r
r =
from
to
G
2
0
1
ˆ
4
V
dq
r
πε
=
∫
E
r
G
inside
o
closed surface
Q
κ
ε
⋅
=
∫∫
E dA
G
G
w
outside
to
inside
from
points
A
d
G
∫
⋅
−
=
−
=
∆
b
a
a
b
b
to
a
from
moving
V
V
V
s
d
E
G
G
0
=
⋅
∫
path
closed
s
d
E
G
G
point charge
4
o
q
V
r
πε
=
many point charges
1
4
N
i
i
o
i
q
V
πε
=
=
−
∑
r
r
G
K
∑
−
=
pairs
all
j
i
o
j
i
q
q
U
r
r
G
G
πε
4
2
1
2
o
v
U
E
ε
⎡
⎤
=
⎢
⎥
⎣
⎦
∫∫∫
x
ol
dV
E
x
=

∂
V
∂
x
,
E
y
=

∂
V
∂
y
,
E
z
=

∂
V
∂
z
Q
C
V
=
∆
U =
1
2
C
∆
V
2
=
Q
2
2C
o
parallel plate
A
C
d
κ ε
=
1
parallel
C
C
2
C
=
+
1
2
1
2
series
C C
C
C
C
=
+
Circumferences, Areas, Volumes:
1) The area of a circle of radius
r
is
π
r
2
Its circumference is
2
π
r
2) The surface area of a sphere of radius
r
is
4
π
r
2
Its volume is
3
4
3
r
π
3) The area of the sides of a cylinder of
radius
r
and height
h
is
2
π
r h
Its volume is
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
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