the fundamental theorem of calculus (integral of a derivative over a region is related to values at...
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The Fundamental Theorem of Calculus
b
a
afbfdxdxdf
(Integral of a derivative over a region is related to values at the boundary)
Cross Product: determinant of matrix with unit vector
zyx
zyx
BBB
AAA
kji
BA
ˆˆˆ
det
zzyyxx BABABABA
Dot Product: multiply components and add
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Scalar Field : a scalar quantity defined at every point of a 2D or 3D space.
),(),( yxfyxS )sin(),( xyxyxS Ex:
0
50
100
0
50
100-10
-5
0
5
10
20 40 60 80 100
10
20
30
40
50
60
70
80
90
100
-4
-2
0
2
4
6
20 40 60 80 100
10
20
30
40
50
60
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80
90
100
-6
-4
-2
0
2
4
6
EM Fields
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0
10
20 05
1015
200
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15
20
0
0.1
0.2
0.3
0.4
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0.9
3D scalar field
3D scatter plot with color giving the field value:
510210310 222
1),,( zyx eeezyxS
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Vector Field: a vector quantity defined at every point of a 2D or 3D space.
jyxixyyx ˆˆ)sin(),( S
kVjViVzyx zyxˆˆˆ),,( V
Functions of (x,y,z)
NOT constantsNOT partial derivatives
2D Ex:
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
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Temperature Map: a scalar field
Wind Map: a vector field
Two Fields
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1. Gradient
“the derivative of a scalar field”
y
S
x-2 -1 0 1 2
-2
0
20
5
10
15
20
22 yxS
![Page 7: The Fundamental Theorem of Calculus (Integral of a derivative over a region is related to values at the boundary) Cross Product: determinant of matrix](https://reader035.vdocuments.us/reader035/viewer/2022062714/56649d565503460f94a34547/html5/thumbnails/7.jpg)
Derivative (slope) depends on direction!
dyyS
dxxS
dS
kz
jy
ix
ˆˆˆ
Total Differential:
Looks like a dot product: jdyidxjyS
ixS
dS ˆˆˆˆ
ldSdS
“del”
“nabla”
Del is not a vector and it does not multiply a field – it is an operator!
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1. The Fundamental Theorem of Gradients
b
a
aSbSldS
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
0
2
4
6
8
10
12
0
50
100
0
50
100-10
-5
0
5
10
a
b
(Integral of a derivative over a region is related to values at the boundary)
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V
“the creation or destruction of a vector field”
kVjViVkz
jy
ix zyx
ˆˆˆˆˆˆ
zyx Vz
Vy
Vx
(a scalar field!)
jyxixyyx ˆˆ)cos(),( 22 V
yxyy 2)sin( V
2. Divergence
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kjyix ˆ0ˆˆ V
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
X
Y
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-2 -1 0 1 2-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
X
Y
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
kjyix ˆ0ˆˆ V
2 V
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-2 -1.5 -1 -0.5 0 0.5 1 1.5 2-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
jyix ˆˆ 22 V
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jyix ˆˆ 22 V
-2 -1 0 1 2-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
-6
-4
-2
0
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yx 22 V
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2. The Fundamental Theorem of Divergence
SV
add
VV
volume integral surface integral
(The Divergence Theorem)
(Integral of a derivative over a region is related to values at the boundary)
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+ -
+ -
I. Gauss’ Law: relation between a charge distribution and the electric field
point charge
E field lines
0q
dS
aE
0
1 E Gauss’ Law
(differential form)
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First Results from a Superconductive Detector for Moving Magnetic Monopoles
Blas Cabrera Physics Department, Stanford University, Stanford, California
94305 Received 5 April 1982
A velocity- and mass-independent search for moving magnetic monopoles is being performed by continuously monitoring the current in a 20-cm2-area superconducting loop. A single candidate event, consistent with one Dirac unit of magnetic charge, has been detected during five runs totaling 151 days. These data set an upper limit of 6.1×10-10 cm-2 sec-1 sr-1 for magnetically charged particles moving through the earth's surface.
PRL 48, p1378 (1982)
The Valentine’s Day Monopole
II. Gauss’ Law for Magnetism: relation between magnetic monopole distribution and the magnetic field
0 B
0S
daB
Cabrera
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V
kVjViVkz
jy
ix zyx
ˆˆˆˆˆˆ
“How much a vector field causes something to twist”
zyx
zyx
VVV
kji ˆˆˆ
det
3. Curl
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-2 -1 0 1 2-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
kjyix ˆ0ˆˆ V
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-2 -1 0 1 2-2
-1.5
-1
-0.5
0
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-0.8
-0.6
-0.4
-0.2
0
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011 V
colorplot = z component of curl(V)
kjyix ˆ0ˆˆ V
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-2 -1.5 -1 -0.5 0 0.5 1 1.5 2-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
kjxiyy ˆ0ˆˆsin 22 V
![Page 21: The Fundamental Theorem of Calculus (Integral of a derivative over a region is related to values at the boundary) Cross Product: determinant of matrix](https://reader035.vdocuments.us/reader035/viewer/2022062714/56649d565503460f94a34547/html5/thumbnails/21.jpg)
-2 -1 0 1 2-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
-7
-6
-5
-4
-3
-2
-1
0
1
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kyyyyx ˆ cossin22 2 V
colorplot = z component of curl(V)
kjxiyy ˆ0ˆˆsin 22 V
![Page 22: The Fundamental Theorem of Calculus (Integral of a derivative over a region is related to values at the boundary) Cross Product: determinant of matrix](https://reader035.vdocuments.us/reader035/viewer/2022062714/56649d565503460f94a34547/html5/thumbnails/22.jpg)
3. The Fundamental Theorem of Curl
PS
dd lVaV
open surface integral closed perimeter line integral
(Really called Stokes’ Theorem)
V
V
(Integral of a derivative over a region is related to values at the boundary)
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Faraday
III. Faraday’s Law: A changing magnetic field induces an electric field.
B
emf
temf B
0
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Moving coil in a varying B field.Force on electrons:
BvEF
q
Forces don’t cancel: 0emf
FF
v
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v
EE
BvEF
q
0v
EFq
Electric field must be created!
Only left with:
Stationary coil with moving B source:
But we still get an emf …
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EE
i
In general:
temf B
SCd
td aBlE
Faraday’s Law(integral form)
Stationary coil and B source, but increasing B strength:
0emf
t B
E
Faraday’s Law(differential form)
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IV. Ampere’s Law
iB
enclosedCid 0 lB
More general:
SCdd aJlB
0
J = free current density
MaxwellAmpere
“Something is missing..”
![Page 28: The Fundamental Theorem of Calculus (Integral of a derivative over a region is related to values at the boundary) Cross Product: determinant of matrix](https://reader035.vdocuments.us/reader035/viewer/2022062714/56649d565503460f94a34547/html5/thumbnails/28.jpg)
Cd lB
SdaJ
Charging a capacitor
i- +- +
- +- +
- +
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Cd lB 0S daJ
i- +- +
- +- +
- +
Charging a capacitor
Maxwell: “…the changing electric field in the capacitor is also a current.”
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SCd
td aEJlB
00
“Displacement current”
Ampere-Maxwell Eqn.(Integral Form)
SSd
td aEJaB
00
Get Stoked:
EJB
00
t
Ampere-Maxwell Eqn.(differential form)
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Maxwell’s Equations in Free Space with no free charges or currents
0 E
0 B
t B
E
EB
t 00
Faraday
MaxwellAmpere
GaussYour Name Here!