1
Physics 8.02
Equation Sheet Exam One
Please Remove this Tear Sheet from Your Exam
2
3
ˆ
4
4
o
o
q
q
r
r
πε
πε
=
=
E
r
r
G
G
ˆ
points
source q
observer
r
r
r =
from
to
G
many point charges
3
1
(
)
4
N
i
i
i
o
i
q
πε
=
=
−
−
∑
E
r
r
r
r
G
G
K
G
K
2
0
1
ˆ
( )
4
source
dq
πε
=
′
−
∫
E r
r
r
r
G
G
G
G
q
source
q
=
F
E
G
G
enc
o
closed
surface
Q
ε
⋅
=
∫∫
E dA
G
G
w
points from inside to outside
dA
G
0
closed
path
⋅
=
∫
E ds
G
G
v
b
moving from a to b
b
a
a
V
V
V
Δ
=
−
= −
⋅
∫
E ds
G
G
U
q V
Δ
=
Δ
point charge
;
(
)
0
4
o
q
V
V
r
πε
=
∞ =
many point charges
1
;
(
)
0
4
N
i
i
o
i
q
V
V
πε
=
=
∞ =
−
∑
r
r
G
K
0
1
( )
;
(
)
0
4
source
dq
V
V
πε
=
∞ =
′
−
∫
r
r
r
G
G
G
;
(
)
0
4
i
j
all pairs
o
i
j
q q
U
U
πε
=
∞ =
∑
−
r
r
G
G
2
1
2
o
vol
all space
U
E dV
ε
=
∫∫∫
E
r
= −
∂
V
∂
r
for spherical symmetry,
V
= −∇
E
G
G
x
V
E
x
∂
= −
∂
y
V
E
y
∂
= −
∂
z
V
E
z
∂
= −
∂
C
=
Q
Δ
V
U
=
1
2
C
Δ
V
2
=
Q
2
2
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
(4 / 3)
π
r
3
3)
The area of the sides of a cylinder
of radius
r
and height
h
is
2
π
rh
.
Its volume is
π
r
2
h
Integrals that may be useful
dr
a
b
∫
=
b
−
a
ln( /
)
b
a
dr
b a
r
=
∫
1
r
2
dr
a
b
∫
=
1
a
−
1
b
⎛
⎝
⎜
⎞
⎠
⎟
Some potentially useful numbers
k
e
=
1
4
πε
0
=
9
×
10
9
N m
2
C
2

This
** preview**
has intentionally

**sections.**

*blurred***to view the full version.**

*Sign up*
2