repeated vector operations curl (grad a) = 0 divcurl a = 0 div (grad a) = laplacian (a)
TRANSCRIPT
repeated vector operations
curl (grad a) = 0
div•curl A = 0
div (grad a) = Laplacian (a)
a a aa
x y z
x y zu u u
ax y z
a a a
x y z
x y zu u uxux
y xz z
y x
A AA A
x y z y z x
A A
z x y
A
x y z
x y z
A A A
x y zu u u
A
a a aa
x y z
x y zu u u
aa ayx z
ax y z
x y z
x y zu u u
2 2 22
2 2 2
a a aa
x y z
Phasors v = V cos (t + ø) v = Re [V ejt] where V = V ejø
vjdt
dv
vj
1'vdt
LjRI
Vo
o
53j
120j
e10
e5
o120je5 V
o30t4sin5)t(v
oo 9030t4cos5
o173t4cos5.0)t(i
24j6
e5o120j
cb
ay
0.010.01
1 100
1
100
y
d
0.010.01
1 100
1
100
y
a0.01
0.011 100
1
100
y
b
0.010.01
1 100
1
100
y
c
Physics of Plasmas September, 2008
Electric charge force electric field
ruF2
o
21
r4
12o 10854.8
Coulombs10602.1Q 19
+Q +Q- Q+Q
meter
farads10
36
1 9
+Q - Q
Yes Senator, Electrical and Computer Engineers have particles that have
charges with different signs – positive and negative! This was not invented at
Microsoft in order to limit competition
10 to 20 Coulombs
ruF2
o
21
r4
2
2
o LF
Q 3
22
ML
TQ
22
2
LTL
M
Q
2o
21
r4
QQF
Charged dust grains
Coulombs10602.110Q 194
23
9
215
103610
4
10602.1
m10mm1r 3
Newtons10602.19 152
history 900 BC - Magnus, a Greek shepherd,
walks across a field of black stones which pull the iron nails out of his sandals and the iron tip from his shepherd's staff (authenticity not guaranteed). This region becomes known as Magnesia.
Thales of Miletus 624-547 BC amber rod picked up “fluff and stuff” “elektron” in Greek “elektrum” in Hebrew Ezekiel 1:27
history
history 1269 - Petrus Peregrinus of Picardy,
Italy, discovers that natural spherical magnets (lodestones) align needles with lines of longitude pointing between two pole positions on the stone.
x
y
3
3
00 6
6
Superposition of forces
vectors
5
43
54
5
43
54
yx
2o
gr
yx
2o
gr
uu
uuF
ruF2
o
21
r4
5
8
54
QQ y
2o
gr u
x
y
3
3
00 6
6
22
yx
2
o
gr
52
52
294
uuF
x
y
3
3
00 6
6
ruF2
o
21
r4
22
yx
2
o
gr
54
54
414
uu
21 Q
FE ru
2o
1
r4
Q
Coulomb
Newtons
21 Q
FE ru
2o
1
r4
Q
Coulomb
Newtons
Fundamental units
mass M, length L, time T, charge Q
QTL
M 2
2
2L
M1T
Q L
meter
CoulombJoules
Superposition of electric fields
x
y
3
3
00 6
6vectors
5
43
54
Q yx
2o
uuE
x
y
3
3
00 6
6ruE
2or4
Q
5
6
54
Q x2
o
u
5
43
54
Q yx
2o
uu
Distributed charge density
volume charge density
surface charge density
linear charge density
3v )meter/(Coulombs
2S )meter/(Coulombs
)meter/(CoulombsL
cosr4
dydE
2o
L
r
x
r4
dydE
2o
L
2o
L
r4
dydE
22 yxr
x
y
dyr
r
x
r4
dydE
2o
L
22 yxr x
y
dyr
a
a 22222
o
L
yx
x
yx4
dyE
22o
L
ax
a
x2
x2
Ea
lim
o
L
y
x
Z
dE
cosr2
dydE
o
s
22
o
s
yz2
zdyE
22o
s
yz
z
r2
dy
o
s
2
o
s
2E
r2E
o
L
2or4
QE
100
1
0.01
1 100 r
E
aa 2/3222
o
saa
zyx4
zdydxE
y
x
Z
-a-a
a
a
finite size square sheetuniform charge density
E
“Lately I've been wondering where the slide rules went. Back in the 1960’s, slide rules were a prime accessory for those with a quantitative bent. They were suspended from belts long before calculators. In fact, they were the personal calculation engine of choice for almost three centuries.”
The principle of the slide rule was first enumerated by E. Gunter in 1623, and in 1671, S. Partridge constructed an instrument similar to the modern slide rule. The slide rule was an indispensable tool for scientists and engineers through the 1960’s.
In 50 years, the computer you are using to view this will be landfill, but your slide rule will just be nicely broken in.
It happens elsewhere other than under the Iowa Avenue bridge
