Physics 8.02
Final Exam
Spring 2003
Some (possibly useful) Relations:
enclosed
o
closed
surface
Q
d
ε
⋅
=
∫∫
E
A
r
r
±
closed
open
path
surface
d
d
d
dt
⋅
= −
⋅
∫
∫∫
E
s
B
A
r
r
r
r
±
one turn
N
d
⋅
=
∫∫
B
A
r
r
LI
closed
path
dI
d
L
dt
⋅
= −
∫
E
s
r
r
±
0
closed
surface
d
⋅
=
∫∫
B
A
r
r
±
thru
o
E
closed
path
E
open
surface
d
d
I
dt
where
d
µ
µ ε
⋅
=
+
Φ
Φ
=
⋅
∫
∫∫
o
0
B
s
E
A
r
r
r
r
±
0
2
ˆ
4
µ
π
×
=
=
∫
∫
s
r
B
B
source
source
Id
d
r
r
r
r
q
q
q
=
+
×
F
E
v
B
B
r
r
r
r
d
I d
=
×
F
s
r
r
r
0
ˆ
1
4
2
dq
r
πε
=
r
dE
r
2
0
1
ˆ
4
V
dq
r
πε
=
∫
E
r
r
b
a
V
V(b )
V( a )
d
∆
≡
−
= −
⋅
∫
E
s
r
r
i
N
i
i 1
0
i,
q
1
V
4
r
πε
=
=
∆
=
∑
P
/ R
2
2
Joule Heating
P
I R
V
=
= ∆
Q
C
V
=
∆
2
2
1
2
2
C
Q
U
C
V
C
=
∆
=
0
1
LC
=
ω
2
0
1
2
elec
u
E
=
ε
2
0
1
2
mag
u
B
=
µ
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
r
B
r
r
0
1
( , )
( , )
T
x t
x
T
=
∫
S
S
r
r
t dt
2
0
1
2
sin
(
)
2
T
x
ct
dt
T
π
⎛
⎞
1
−
=
⎜
⎟
λ
⎝
⎠
∫
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
=
±
±
±
⎜
⎟
⎝
⎠
intensity:
2
max
cos
sin
I
I
d
π
⎛
⎞
=
θ
⎜
⎟
λ
⎝
⎠
SINGLE SLIT:
destructive:
sin
,
1,
2,
3, ...
a
n
n
θ =
λ
= ±
±
±
intensity:
2
max
sin
sin
sin
a
I
I
a
⎛
π
⎛
⎞
θ
⎜
⎟
⎜
⎟
λ
⎝
⎠
⎜
⎟
=
π
⎜
⎟
θ
⎜
⎟
λ
⎝
⎠
⎞
USEFUL INTEGRALS:
b
a
dx
b
a
=
−
∫
(
)
2
2
1
2
b
a
x dx
b
a
=
−
∫
1
ln
b
a
b
dx
x
a
⎡
⎤
=
⎢
⎥
⎣
⎦
∫
2
1
1
b
a
dx
1
x
a
b
⎡
⎤
⎛
=
−
⎜
⎟
⎢
⎥
⎞
⎣
⎦
⎝
∫
⎠

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