Physics 8.02
Exam Two Mashup Solutions
Fall 2003
Some (possibly useful) Relations:
F
=
1
4
πε
o
q
1
q
2
r
2
E
=
q
4
πε
o
r
2
ˆ
r
observer
to
source
from
points
r
ˆ
∫
=
⋅
surface
closed
o
inside
Q
ε
dA
E
outside
to
inside
from
points
dA
∫
⋅
−
=
−
=
∆
b
a
a
b
b
to
a
from
moving
V
V
V
ds
E
0
=
⋅
−
∫
ds
E
V
point charge
=
q
4
πε
o
r
∑
=
−
=
N
i
i
o
i
q
V
1
4
r
r
πε
charges
point
many
V
Q
C
∆
=
C
Q
V
=
∆
U =
1
2
C
∆
V
2
=
Q
2
2C
I
=
q n A
wire
v
drift
drift
wire
I
qn
A
=
=
J
v
where
is the resistivity
ρ
ρ
=
E
J
1/
where
the conductivity
σ
σ
ρ
=
=
=
J
E
∆
V
=
i R
L
L
R
A
A
ρ
σ
=
=
1
R
parallel
=
1
R
1
+
1
R
2
R
series
= R
1
+ R
2
R
V
R
i
V
i
P
heating
ohmic
2
2
∆
=
=
∆
=
(joules/sec)
2
ˆ
4
o
q
r
µ
π
×
=
v
r
B
2
ˆ
4
o
I
r
µ
π
×
=
ds
r
dB
G
G
Ampere’s Law:
through
o
contour
I
µ
∫
=
⋅
ds
B
where I
through
is the current flowing through
the open surface bounded by the contour
∫
⋅
=
surface
open
through
I
dA
J
ds
right-handed wrt
dA
F
=
q
v
×
B
ext
ext
I
=
×
dF
ds
B
The length of an arc of a circle of radius
R
that
subtends an angle
θ
radians is
R
θ
, right handed with respect to
IA
I
=
μ
G
The area of the sides of a cylinder of radius
r
and height
h
is
2
π
rh
Circular motion:
For a particle of mass
m
to
move in a circle of radius
R
at speed
v
,
there must be an inward force of
magnitude
R
mv
2

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