Physics 8.02
Final Exam
Spring 2004
Some (possibly useful) Relations:
enclosed
o
closed
surface
Q
d
ε
⋅
=
∫∫
E
A
G
G
w
total
closed
open
path
surface
external
self
open
surface
d
d
d
dt
d
d
dt
⋅
= −
⋅
⎡
⎤
= −
+
⋅
⎣
⎦
∫
∫∫
∫∫
E
s
B
A
B
B
G
G
G
G
G
G
G
v
A
0
closed
surface
d
⋅
=
∫∫
B
A
G
G
w
closed
path
thru
o
E
E
open
surface
d
d
I
dt
where
d
µ
µ ε
⋅
=
+
Φ
Φ
=
⋅
∫
∫∫
o
o
B
s
E
A
G
G
G
G
v
2
0
1
2
elec
u
E
=
ε
2
0
1
2
mag
u
B
=
µ
0
2
ˆ
4
µ
π
×
=
=
∫
∫
s
r
B
B
source
source
Id
d
r
G
G
G
q
q
q
=
+
×
F
E
v
G
G
G
B
G
B
d
I d
=
×
F
s
G
G
G
0
1
ˆ
4
2
dq
r
πε
=
dE
r
G
2
0
1
ˆ
4
V
dq
r
πε
=
∫
E
r
G
2
1
2
1
P
2
1
P
The electric potential at point P
minus that at point P is given by
V( P )
V( P )
d
−
= −
⋅
∫
E
s
G
G
i
N
i
i 1
0
i,
q
1
V
4
r
πε
=
=
∆
=
P
∑
V
IR
∆
=
2
2
Joule Heating
P
I R
V
=
= ∆
/ R
Q
C
V
=
∆
2
2
capacitor
1
2
2
Q
U
C
V
C
=
∆
=
self field
self
one turn
N
d
⋅
=
∫∫
B
A
L
I
G
G
2
inductor
1
2
U
L
=
I
0
1
LC
=
ω
0
0
1
c
=
µ ε
1
f
T
=
2
2
f
T
ω =
π
=
π
2
k
=
π λ
c
T
f
k
= λ
= λ
= ω
0
1
=
×
µ
S
E
G
B
G
G
DOUBLE SLIT:
constructive:
sin
,
0,
1,
2,
3, ...
d
m
m
θ =
λ
=
±
±
±
destructive:
1
sin
,
0,
1,
2,
3, ..
2
d
m
m
⎛
⎞
θ =
+
λ
=
±
±
±
⎜
⎟
⎝
⎠
SINGLE SLIT:
destructive:
sin
,
1,
2,
3, ...
a
n
n
θ =
λ
= ±
±
±
Areas, Volumes, etc.:
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
3
r
4
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
USEFUL INTEGRALS:
d
c
dx
d
c
=
−
∫
(
)
2
2
1
2
d
c
r dr
d
c
=
−
∫
1
ln
d
c
d
dr
r
c
⎡
⎤
=
⎢
⎥
⎣
⎦
∫
2
1
1
d
c
dr
r
c
1
d
⎡
⎤
⎛
=
−
⎜
⎟
⎢
⎥
⎞
⎣
⎦
⎝
∫
⎠

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