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.