B r A r
''
4 '
j rB r
r rO dv
''
4 '
j rB r
r rO dv
Slide through the integral!
j(r’)
r’
ur’ - r
'2' '
4 'r ru
B r j rr r
O dv
+ B(r)
r
3/ 22 2
''
4 '
u uu z rOz
z rI dz
z r
2 22uOI L
r L r
R
z’
r
dz’2 L
I
B
z
'2' '
4 'r ru
B r j rr r
O dv
2 22
B r uOI L
r L r
lim
2B r uoI
L r
R
z’
r
dz’2 L
I
B
z
3/ 22 2
'4
u uu r zO
a zI ad
a z
2
3/ 22 22O
z
I aB
a z
'2' '
4 'r ru
B r j rr r
O dv
summary
js
Three techniques to find B
1] Ampere’s circuital law - lots of symmetry
2] find vector potential A, then B = x A
3] Biot - Savart law
B
A
BB
magnetic forces
BvF Q
x
yz
BV
F+
F-Lorentz force
BvEF Q
jjaMF
vBQF j
j
2
jj r
vMvBQ
j
jj BQ
vMr
B
v
Ma
Mb
j
2
j r
vM
Please Note: Rocket Sets contain rocket motors that are classified as hazardous materials for shipping purposes. Given this, we must ship Rocket Sets via surface mail through the USPS.
MHD plasma drift F = Q (E + v x B) = 0 0 = E x + v B x V v B x
C 60
+ -B
B
BvEF Q
LBvLEV
hours2
miles25000v
Tesla105B 5
miles12L volts5000V
electric power in space!
12 miles
plasma physics
N
S
N
S
N
S
plasma physics
plasma physics
magnetic field
particle loss
Example: Charged Particle Orbits
in a Tokamak
finite element method
FEMLAB using MATLAB
poloidal coils
transformer core
BIp
Icoil
primary winding
Tokamak principleTokamak – Problem Definition
Introduction A tokamak is a “magnetic bottle” for the
confinement of high energy plasmas in Controlled Nuclear Fusion.
The ions and electrons in the plasma travel far between collisions.• collisionless model• particle motion is governed by the EM field• losses are mainly due to collisions with the
walls
Tokamak
Modeling approach
Coils and plasma current are modeled in FEMLAB and the magnetic field is computed.
• plasma current = 2.2 MA• peak toroidal magnetic field = 2 T
Particle orbits are computed using an ODE solver.
Tokamak
Results – particle simulation
Proton orbit - Results
banana orbits
work
b
aW dlF
xxQW uBv
xyx xBvQ uuu 0
x
yz
BV
F
x
VIS Dr. Lou Frank & Dr. John Sigwarth
1
I1 dl1
B2
I2 dl2
B1
BvdF dQ
dvv BvdF dsdlBjdF
BIdldF
lightning arrestor or STAR WARS
Magnetic resonance imaging?
electric machines
rotate about an axis
no translation
torque
(forces)
z
x
yI
y B
x
BIdldF zoB uB
looponforces
zox BxI uu zoy ByI uu zoy ByI uu zox BxI uu
Torque
cos 2
yFcos
2
yFT
zx uuF Bx I
+
•
y
zB
y
x
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Euro-Wash Laundry Ball
MagneForce The first shoe with its own power supply. Comfortable, quality footwear constructedwith a lightweight, flexible magnetic insole to generate a deep penetrating magnetic field which increases circulation; reduces foot, leg, and back fatigue; provides natural pain relief and improved energy level. The James Randi Educational Foundation offers $1 million to Florsheim if it can demonstrate its claims for MagneForce Footwear in a formal double-blind scientific test.
Variable Specific Impulse Magnetoplasma Rocket
The VASMIR engine has three magnetic cells.
1) Hydrogen is ionized.
2) Microwaves heat the plasma.
3) Magnetic nozzle converts energy into a directed flow.
Mars, here we come!
cnn 6/2000
1) Hydrogen is ionized.
2) Microwaves heat the plasma.
3) Magnetic nozzle converts energy into a directed flow.
1) Hydrogen is ionized.
2) Microwaves heat the plasma.
3) Magnetic nozzle converts energy into a directed flow.
1) Hydrogen is ionized.
2) Microwaves heat the plasma.
3) Magnetic nozzle converts energy into a directed flow.