Compare to principle of superposition applied to determine the electric field inside and outside an infinite parallel plate capacitor

Compare to principle of superposition applied to determine the electric field inside and outside an infinite parallel plate capacitor

ELEC 3105 BASIC EM AND POWER ENGINEERING

Force and torque on Magnetic Dipole

Magnetic dipole = product of current in loop with surface area of loop

DEFINITION OF MAGNETIC DIPOLE

I = current in loopA “S” = surface area of the loop = unit vector normal to loop surface - RHR

vector in the direction of Magnetic moment

|��|= 𝐼𝐴=𝐼 𝑆

Units of Am2

FORCE ON A MAGNETIC DIPOLE B

Consider a circular ring of current I placed at the end of a

solenoid as shown in the figure. The current in the

solenoid produces a magnetic field in which the current loop is placed into. By postulates 1

and 2 of magnetic fields, the current ring will be subjected

to a magnetic force.

out of page into page

I

z


I

z

outF

outF

downF

downF

Circular ring

Cancel in pairs around the ringoutF

outF

downF

downF

Will add in same direction on ring giving a net force.


I

z

outF

outF

downF

downF

Circular ring

downF

downF

Will add in same direction on ring giving a net force.

Using postulate 1: dBIFd

r2

B

zB

rB

downF

We need for find Br

Gives:rdown rIBF 2

FORCE ON A MAGNETIC DIPOLE

z

z

Gaussian cylinder

We will relate Br to z

Bz

Total magnetic flux through Gaussian cylindrical surface must be zero. As many magnetic field lines that enter the surface, leave the surface. No magnetic charges or monopoles. 0 B

11

Another important property of B

0 B

Recall everywhere

No net magnetic flux through any closed surface.

Closed surface S

S

adB 0

vol

voldvB 0

Using divergence theorem

0 3-D view


z

z

Flux through side:

3-D view

siderzBr 2

Flux through top:

topz

zzBr 2

Flux through bottom:

bottomz

zBr 2

0bottomtopside


z

z

3-D view

02 22 zBrzzBrzBrzzr

0bottomtopside

z

zBzzBrB zz

r

2

z

BrB z

r

2

We can now use this in our force on current ring expression


I

z

outF

outF

downF

downF

Circular ring

rdownrIBF 2

r2

B

zB

rB

downF

We have found Br

z

BrB z

r

2

z

BrrIF z

down

22


I

z

outF

outF

downF

downF

Circular ring

r2

B

zB

rB

downF

z

BrrIF z

down

2

2

z

BIrF z

down

2

z

BIAF z

down

z

BmF z

down

z

BmF z

z

Force pulls dipole into region of stronger magnetic field


3-D view

z

In general

x

BmF x

x

z

BmF z

z

y

BmF y

y

Solenoid with axis along z

Solenoid with axis along x

Solenoid with axis along y

Form suggests some sort of dot product with the del

operator


3-D view

z

In general

xxBmF

yyBmF

zzBmF

BmF

TORQUE ON A MAGNETIC DIPOLE

We will consider a dipole in a uniform magnetic field. We can use any shape we want for the dipole. Here we will select a square loop of wire.

I out of page

I into page

m

B

a

a

I

Side view

Topview

Wire loop

a

a

I

Topview

Wire loop

a

2

a


m

B

a

a

I

Side view

Topview

Wire loop

F

F

Torque attempts to align dipole

moment with .m

B

Pivot point

Pivot line


m

B

Side view

F

F

Torque attempts to align dipole

moment with .m

B

Pivot point

sin2

2a

F

Fr

2

a

Total torque

F => Magnetic force on wire of length a


m

B

Side view

F

F

Pivot point

sin2

2a

F

2

a

F => Magnetic force on wire of length a

IBaF Through postulate 1 for magnetic fields

sin2IBaThen


m

B

Side view

F

F

Pivot point

2

a

sin2IBa

a

a

I

Wire loopIam 2

sinBm

Bm


21

Boundary conditions

Inductance

Magnetic energy

Principle of virtual work


22

NORMAL COMPONENT OF B FIELD AT BOUNDARY

Gaussian surfaceArea A

Area A

0TVery thin

1B

2B

Interface

S

AdB 0

1

2

Net flux through a closed surface is zero.

)(12 normalBnormalB

The normal components of B are continuous across the interface

normalB1

normalB2

23

TANGENTIAL COMPONENT OF H FIELD AT BOUNDARY

Square closed pathLength L

0TVery thin

1H

2H

Interface

0 IdHS

1

2

Integral of H around closed path is equal to the current enclosed (I = 0)

)tangential(tangential 12 HH

The tangential components of H are continuous across the interface

tangential1

H

tangential2

H

Length L

HB

THIS BOUNDARY CONDITION ASSUMES NO SURFACE CURRENT AT THE INTERFACE.

24

TANGENTIAL COMPONENT OF H FIELD AT BOUNDARY

Square closed pathLength L

0TVery thin

1H

2H

Interface

1

2

)tangential(tangential21

HH

The tangential components of H are discontinuous across the interface

tangential1

H

tangential2

H

Length L

HB

THIS BOUNDARY CONDITION ASSUMES A SURFACE CURRENT AT THE INTERFACE.

Surface Current K

X XX X X XX X X XX X X XX X

KHH )tangential(tangential 12

IdHS

25

SUMMARY OF BOUNDARY CONDITIONS (GENERAL)

26

SUMMARY OF BOUNDARY CONDITIONS (CONDUCTORS)

27

SUMMARY OF BOUNDARY CONDITIONS (CONDUCTORS)

28

SELF INDUCTANCEA transformer is a device in which the current in one circuit induces an EMF in a second circuit through the changing magnetic field.Introduction

17

B, H, and M relationship

To understand how current in one circuit induced EMF in another, we will first examine how a current in a circuit can induce an EMF in the same circuit.

29

SELF INDUCTANCEConsider a single wire loop

v

i

iB

Enclosed surface S

Current in loop produces a magnetic field , giving a flux through the loop.B

From Biot-Savard Law

13

Biot-Savard Law

The Biot-Savard law applied to the small segment gives an element of magnetic field at the point P.

Consider a small segment of wire of overall length

I

d

P

21r

This expression isThe Biot-Savard Law

26

MagnetostaticsPOSTULATE POSTULATE 22 FOR THE MAGNETIC FIELDFOR THE MAGNETIC FIELD

A current element produces a magnetic field which at a distance R is given by:

d

R

RIBd o

2

ˆ

4

dI

B

Bd

Units of {T,G,Wb/m2}

d

Bd

Same result as postulate 2 for the magnetic field

Lecture 15 slid 26

Bd

2

21

211

ˆ

4 r

rdIrBd o

iB

Thus: i

WRITE: Li

SELF INDUCTANCE

30

Current in loop produces a magnetic field , giving a flux through the loop.

Consider a single wire loop

v

i

iB

Enclosed surface S

B

Li

L is the self inductance of the loop

dt

diL

dt

dv

temf

SELF INDUCTANCE

31

Consider a single wire loop

v

i

iB

Enclosed surface S

Current in loop produces a magnetic field , giving a flux through the loop.B

Li

dt

diL

dt

dv

It is difficult to compute L for a simple wire loop since the magnetic field produced by the loop is not constant across the surface of the loop.

A possible solution is to find B at center of loop and then approximate:

SBcenter

SELF INDUCTANCE

32

A simple example for the calculation of a self inductance is the long solenoid.

41

Magnetic field of a long solenoidCurrent out of page

Current into page

Infinite coil of wire carrying a current I

Axis of solenoid

P

Evaluate B field here

42

Magnetic field of a long solenoidIn the vicinity of the point P

P 3

12

45Axis of solenoid

resultant

Expect B to lie along axis of the solenoid

1

Current out of page

0B Implies that B field has no radial component. I.e. no component pointing towards

or away from the solenoid axis.

B

50

M ag n etic fie ld o f a lo n g so len o id

C u rre n t o u t o f p a g e

C u rre n t in to p a g e

0bB

P

L

NIB o

N : n u m b e r o f tu rn s e n c lo se d b y le n g th L

•B is in d e p e n d e n t o f d is ta n ce fro m th e a x is o f th e lo n g so le n o id a s w e a re in s id e th e so le n o id !•B is u n ifo rm in s id e th e lo n g so le n o id .fin ite so len o id sta rt

in fin ite so lenoid (36 )

28

Magnetic field of a FINITEFINITEsolenoidCurrent out of page

Current into page

finitefinitecoil of wire carrying a current I

Axis of solenoid

P


a

Radius of solenoid is a. Cross-section cut through solenoid axis

L

3 4

M a g n e t i c f i e l d o f a F I N I T EF I N I T E s o l e n o i d

zdB

d

r

d

a

3

2

2 r

adIdB o

z

dL

NIrdI

sin

r

asin

s u b i n

d

LNI

ra

dLa

NIrra

dB oo

z 22

2

3

2

d

LNI

dB o

zsin

2

3 5


L

d

2

1

sin2

d

L

NIB o

z

12

W e c a n n o w s u m ( i n t e g r a t e ) t h e e x p r e s s i o n f o r o v e r t h e a n g u l a r e x t e n t o f t h e c o i l . I . e . s u m o v e r a l l t h e r i n g s o f t h e f i n i t e l e n g t h s o l e n o i d .

zdB

z

21

coscos2

L

NIB o

z

zL

NIB o ˆcoscos

2 21

SELF INDUCTANCE

33

Current out of page

N turns of wire carrying current I

is uniform over the cross-section of the solenoid

B

B

AREAA

Long solenoid of length

NIB o

SELF INDUCTANCE

34

B AREA

A


NIB o

Flux through one loop of area A

NIA

o

1

SELF INDUCTANCE

35

AREAA


NIB o

Flux through all N loops of solenoid

IAN

N o

N

2

1

B

From LIThen

AN

L o

2

SELF INDUCTANCE

36

AREAA


NIB o

LI

AN

L o

2

Self inductance of a long solenoid of N turns with a

current I in the windings. The solenoid has cross-sectional

area A.

SELF INDUCTANCE

37

Calculate the “self inductance” per unit length for a segment of a coax cable. Inner radius (a),

outer radius (b).

EXAMPLE: SELF INDUCTANCE

Example completed in class

𝐼

38

Consider a long solenoid in order to develop a general expression for the energy stored in a magnetic field.

41

Magnetic field of a long solenoidCurrent out of page

Current into page

Infinite coil of wire carrying a current I

Axis of solenoid

P


42

Magnetic field of a long solenoidIn the vicinity of the point P

P 3

12

45Axis of solenoid

resultant

Expect B to lie along axis of the solenoid

1

Current out of page

0B Implies that B field has no radial component. I.e. no component pointing towards

or away from the solenoid axis.

B

50

M ag n etic fie ld o f a lo n g so len o id

C u rre n t o u t o f p a g e

C u rre n t in to p a g e

0bB

P

L

NIB o

N : n u m b e r o f tu rn s e n c lo se d b y le n g th L

•B is in d e p e n d e n t o f d is ta n ce fro m th e a x is o f th e lo n g so le n o id a s w e a re in s id e th e so le n o id !•B is u n ifo rm in s id e th e lo n g so le n o id .fin ite so len o id sta rt

in fin ite so lenoid (36 )

28

Magnetic field of a FINITEFINITEsolenoidCurrent out of page

Current into page

finitefinitecoil of wire carrying a current I

Axis of solenoid

P


a

Radius of solenoid is a. Cross-section cut through solenoid axis

L

3 4


zdB

d

r

d

a

3

2

2 r

adIdB o

z

dL

NIrdI

sin

r

asin

s u b i n

d

LNI

ra

dLa

NIrra

dB oo

z 22

2

3

2

d

LNI

dB o

zsin

2

3 5


L

d

2

1

sin2

d

L

NIB o

z

12

W e c a n n o w s u m ( i n t e g r a t e ) t h e e x p r e s s i o n f o r o v e r t h e a n g u l a r e x t e n t o f t h e c o i l . I . e . s u m o v e r a l l t h e r i n g s o f t h e f i n i t e l e n g t h s o l e n o i d .

zdB

z

21

coscos2

L

NIB o

z

zL

NIB o ˆcoscos

2 21

Energy in Magnetic Field

39

Current out of page

AREAA


NIB

May have core with constant permeability

Find work done by current source in building up magnetic field:

N turns of wire carrying current I

IVPower dt

dIL

dt

dV


40

ENERGY IN MAGNETIC FIELD

dt

dIL

dt

dV

IVPower

dt

dW

THEN

dtIdt

ddW

THEN

tdItd

ddW

I

IdILW0

THEN

2

2LIW

Energy stored


41


2

2LIW

Energy stored AN

L2

NI

B

For core solenoid2

22AINW

AIN

W

2

222

2

1

ABW 2

2

1

enclosed volume


42


2

2B

VOLUME

W

Energy density VOLUME

W

ABW 2

2

1

Total magnetic energy stored in solenoid

EXPRESSION VALID FOR ALL

vol

dvBW 2

2

1


43


4 0

E n e r g y s t o r e d i n e l e c t r i c fi e l d

+ -

+ Q - Q

A

QE

oo

s

V

C o n s i d e r a c a p a c i t o r a t p o t e n t i a l d i f f e r e n c e V a n d o f c h a r g e + Q , - Q o n t h e p l a t e s . A r e a o f p l a t e s ( A ) a n d s p a c i n g ( D )

E n e r g y s t o r e d i n t h e c a p a c i t o r :22

2CVQVU

B u t :

ADE

D

VAD

D

AVCVU ooo

2222

2222

D

A

platesbetween volume2

2EU o

41

Energy stored in electric field

In general for any volume where electric field exists:

Energy stored is:

Volume

o dVEU 2

2

Potential energy stored in electrostatic field

Gauss’s Law Uniform spherical charge Uniform spherical charge distribution distribution All points P (inside and outside)All points P (inside and outside)

P

2

3

3 r

RE V

PrR

3

rE V

Plot of E versus R

R > r R < r

Lecture 4 slide 16

42

Energy stored in electric fieldalternate derivation of general expression

Reference (9) page 172

E n e r g y i n M a g n e t i c F i e l d

C u r r e n t o u t o f p a g e

A R E AA

L o n g s o l e n o i d o f l e n g t h

NIB

M a y h a v e c o r e w i t h c o n s t a n t p e r m e a b i l i t y

F i n d w o r k d o n e b y c u r r e n t s o u r c e i n b u i l d i n g u p m a g n e t i c f i e l d :

N t u r n s o f w i r e c a r r y i n g c u r r e n t I

IVPower dt

dIL

dt

dV


dt

dIL

dt

dV IVPower

dt

dW

THEN

dtIdt

dLdW

THEN

tdItd

dLdW

I

IdILW0

THEN

2

2LIW

Energy stored

E n e r g y i n M a g n e t i c F i e l d

2

2B

VOLUME

W

E n e r g y d e n s i t y VOLUME

W

ABW 2

2

1

T o t a l m a g n e t i c e n e r g y s t o r e d i n s o l e n o i d

E x p r e s s i o n v a l i d

f o r a l l

Energy in Electric Field


44

For electric fields, we argued that the energy was really stored in the potential energy of the charged particle’s

positions, since it would require that much energy to take separate charges and form that distribution from a universe

with equally distributed charges.

This is harder to do for magnetic fields since there are no magnetic charges. But one possible approach is to take

current loops enclosing zero area, and consider the forces on the wires as we expand the loops so as to form the

current distributions which generate the magnetic field.


Energy in Electric Field

45

PRINCIPLE OF VIRTUAL WORK (MAGNETIC)

We can use the principle of virtual work to determine forces as we did for electric forces.

Forces in ElectrostaticsConductor caries a surface charge of density Find force on plates of a parallel plate capacitor. Plate area A

Principle of virtual work: Find the work W required to increase plate separation by S

S

U

S

WF

oE

sF

F Field between plates

+Q

-Q

Recall WSF 42

Energy stored in electric fieldalternate derivation of general expression

Reference (9) page 172 Lecture 6 slide 41

F o r c e s i n E l e c t r o s t a t i c s W = c h a n g e i n e n e r g y s t o r e d i n t h e s y s t e m = U

222

2o

oo

QE

A

QQQ

S

UF

4 0

E n e rg y s to red in e le ctr ic fi e ld

+ -

+ Q - Q

A

QE

oo

s

V

C o n s i d e r a c a p a c i t o r a t p o t e n t i a l d i f f e r e n c e V a n d o f c h a r g e + Q , - Q o n t h e p la te s . A r e a o f p l a t e s ( A ) a n d s p a c i n g ( D )

E n e r g y s t o r e d i n t h e c a p a c i t o r :22

2CVQVU

B u t :

ADE

D

VAD

D

AVCVU ooo

2222

2222

D

A

p latesb etween v o lu me2

2EU o

L e c t u r e 6 s l i d e 4 0

22

22

A

QASAS

EU

o

oo

R e a s o n f o r 2

F o r c e s i n E l e c t r o s t a t i c sF o r c es i n E le c tr o s t a t ic s

nEEo

o t h e rd q u pˆ

c o n du c t or

+ + + + + +

u pd qE

d o w nd qE

o th e rsE

o th e rsE

S a m e m a g n it u deB y G a u s s ’ s la w

T h e s e ad d to g iv e:

T he s e c a n c e l e x a c t ly o th e rd o w nd q EE

T w o e q ua tio n sa n d tw o u n k n o w n s

S o l v i n g t w o e q u a t i o n s g i v e s :

nEo

ot her ˆ2

T h u s t h e s e g m e n t d a c o n t a i n i n g c h a r g e d q p r o d u c e s 1 / 2 “ h a l f ” o f t h e e l e c t r i c f i e l d c l o s e t o i t s c h a r g e d i s t r i b u t i o n .

A l l o t h e r c h a r g e s p r o d u c e t h e o t h e r 1 / 2 “ h a l f ” o f t h e e l e c t r i c fi e l d .

L e c t u r e 9s l i d e 1 0

S = D

F o r c e s i n E l e c t r o s t a t i c sC o n d u c t o r c a r i e s a s u r f a c e c h a r g e o f d e n s i t y F i n d f o r c e o n p l a t e s o f a p a r a l l e l p la t e c a p a c i t o r . P la t e a r e a A

x d

yL

E

sF

d

L

ydLSxE

U o 2

2

2

2 xDE

y

UF o

F o r c e p u l l i n g m e ta l i n s e r t i n t o c a p a c i t o r2

2 xDEF o

Be very careful using the principle of virtual work

s

UF mag

mag

Energy stored in magnetic field

Position variableGives correct magnitude

46


Magnetic Relays

47


Magnetic Relays

Movable contact

I

VMetal spring provides restoring force when current is zero

GAP

Use principle of virtual work to obtain expression for the magnetic force on the movable contact.

Example completed in class

48

MUTUAL INDUCTANCE

2v

i2i

Enclosed surface S2

1v

1i

B

Loop 1 Loop 2

Enclosed surface S1

1 2

We shall consider two current loops close together.

49

MUTUAL INDUCTANCE

1v

1i

B

Loop 1 Loop 2

1

2

Suppose current i1 flows in loop 1, creating a flux in the loop and a flux in loop 2. We will set the source current i2 zero for now.

1

12

2

2112S

adB

Magnetic field of loop 1 in the region of loop 2

Integral over loop 2 surface

Flux of loop 2 produced by current in loop 1

Now some math!!!!

1S

2S

50

MUTUAL INDUCTANCE

2

2112S

adB

2

2112S

adA Using magnetic vector potential

Using Stoke’s theorem

2

2112dA

Using definition of magnetic vector potential

2 1

2

21

11

12 4

d

r

dio

2 1

21

21

112 4 r

ddi o

Rearrange terms

51

MUTUAL INDUCTANCE

2 1

21

21

112 4 r

ddi o

12112Mi

Constant that depends on loop geometry

FLUX IN LOOP 2 DUE TO CURRENT IN LOOP 1

MUTUAL INDUCTANCE

1v

2i

B

Loop 1 Loop 2

1

2

Suppose current i2 flows in loop 2, creating a flux in the loop and a flux in loop 1. We will set the source current i1 zero for now.

2

21

1

1221S

adB

Magnetic field of loop 2 in the region of loop 1

Integral over loop 1 surface

Flux of loop 1 produced by current in loop 2

Now some math!!!!

1S

2S

53

MUTUAL INDUCTANCE

1

1221S

adB

1

1221S

adA Using magnetic vector potential

Using Stoke’s theorem

1

1221dA Using definition of magnetic vector potential

1 2

1

12

22

21 4

d

r

dio

1 212

12

221 4 r

ddi o

Rearrange terms

54

MUTUAL INDUCTANCE

1 212

12

221 4 r

ddi o

21221Mi

Constant that depends on loop geometry

FLUX IN LOOP 1 DUE TO CURRENT IN LOOP 2

55

MUTUAL INDUCTANCE

2 1

21

21

112 4 r

ddi o

12112Mi

1 2

12

12

221 4 r

ddi o

21221MiConclusion

M’s are geometrical factors

MMM 2112

MUTUAL INDUCTANCE BETWEEN LOOPS

56

MUTUAL INDUCTANCE

Mutual Inductance

2v

i2i

Enclosed surface S2

1v

1i

B

Loop 1 Loop 2

Enclosed surface S1

1 2

We shall consider two current loops close together.We shall consider two current loops close together.

General result

dt

diM

dt

diL

dt

d

dt

dv 21

1

211

1

dt

diL

dt

diM

dt

d

dt

dv 2

2

1122

2

Sign convention

1v

1i

primary

2v

2i

Indicates v2 positive when v1 is positive

57

START Extra


58

EXTRA ON INDUCTANCE

59

EXTRA ON INDUCTANCE

60

EXTRA ON INDUCTANCE

61

EXTRA ON INDUCTANCE

62

EXTRA ON INDUCTANCE

63

EXTRA ON INDUCTANCE

64

EXTRA ON INDUCTANCE

65

Particle accelerators


VAN DER GRAAFF GENERATORA Van de Graaff generator is an electrostatic generator which uses a moving belt to accumulate very high amounts of electrical potential on a hollow metal globe on the top of the stand. It was invented by American physicist Robert J. Van de Graaff in 1929. The potential difference achieved in modern Van de Graaff generators can reach 5 megavolts. A tabletop version can produce on the order of 100,000 volts and can store enough energy to produce a visible spark.

A Van de Graaff generator operates by transferring electric charge from a moving belt to a terminal. The high voltages generated by the Van de Graaff generator can be used for accelerating subatomic particles to high speeds, making the generator a useful tool for fundamental physics research.

This Van de Graaff generator of the first Hungarian linear particle accelerator achieved 700 kV in 1951 and 1000 kV in 1952. (Constructor: Simonyi Károly; Sopron, 1951)

http://en.wikipedia.org/wiki/Van_de_Graaff_generator



PARTICLE ACCELERATOR LINIACA linear particle accelerator (often shortened to linac) is a type of particle accelerator that greatly increases the velocity of charged subatomic particles or ions by subjecting the charged particles to a series of oscillating electric potentials along a linear beamline; this method of particle acceleration was invented by Leó Szilárd. It was patented in 1928 by Rolf Widerøe,[1] who also built the first operational device and was influenced by a publication of Gustav Ising.[2]

Linacs have many applications: they generate X-rays and high energy electrons for medicinal purposes in radiation therapy, serve as particle injectors for higher-energy accelerators, and are used directly to achieve the highest kinetic energy for light particles (electrons and positrons) for particle physics.

The design of a linac depends on the type of particle that is being accelerated: electrons, protons or ions. Linac range in size from a cathode ray tube (which is a type of linac) to the 3.2-kilometre-long (2.0 mi) linac at the SLAC National Accelerator Laboratory in Menlo Park, California.

http://en.wikipedia.org/wiki/Linear_particle_accelerator

http://en.wikipedia.org/wiki/Linear_particle_accelerator

PARTICLE ACCELERATOR CYCLOTRON The American physicist Ernest O.

Lawrence won the 1939 Nobel Prize in physics for a breakthrough inaccelerator design in the early 1930s. He developed the cyclotron, the first circular accelerator. A cyclotron is somewhat like a linac wrapped into a tight spiral. Instead of many tubes, the machine has only two hollow vacuum chambers, called dees, that are shaped like capital letter Ds back to back .A magnetic field, produced by a powerful electromagnet, keeps the particles moving in a circle. Each time the charged particles pass through the gap between the dees, they are accelerated. As the particles gain energy, they spiral out toward the edge of the accelerator until they gain enough energy to exit the accelerator.

http://en.wikipedia.org/wiki/Cyclotron

http://en.wikipedia.org/wiki/Cyclotron

PARTICLE ACCELERATOR SYNCHROCYCLOTRON AND ISOCHRONOUS CYCLOTRON

A classic cyclotron can be modified to increase its energy limit. The historically first approach was the synchrocyclotron, which accelerates the particles in bunches. It uses a constant magnetic field B, but reduces the accelerating field's frequency so as to keep the particles in step as they spiral outward, matching their mass-dependent cyclotron resonance frequency. This approach suffers from low average beam intensity due to the bunching, and again from the need for a huge magnet of large radius and constant field over the larger orbit demanded by high energy.

The second approach to the problem of accelerating relativistic particles is the isochronous cyclotron. In such a structure, the accelerating field's frequency (and the cyclotron resonance frequency) is kept constant for all energies by shaping the magnet poles so to increase magnetic field with radius. Thus, all particles get accelerated in isochronous time intervals. Higher energy particles travel a shorter distance in each orbit than they would in a classical cyclotron, thus remaining in phase with the accelerating field. The advantage of the isochronous cyclotron is that it can deliver continuous beams of higher average intensity, which is useful for some applications. The main disadvantages are the size and cost of the large magnet needed, and the difficulty in achieving the high magnetic field values required at the outer edge of the structure.

Synchrocyclotrons have not been built since the isochronous cyclotron was developed.

http://en.wikipedia.org/wiki/Synchrocyclotron

http://en.wikipedia.org/wiki/Cyclotron#Isochronous_cyclotron

PARTICLE ACCELERATOR BETATRONA betatron is a cyclic particle accelerator developed by Donald Kerst at the University of Illinois in 1940 to accelerate electrons,[1][2][3] but the concepts ultimately originate from Rolf Widerøe,[4][5] whose development of an induction accelerator failed due to the lack of transverse focusing.[6] Previous development in Germany also occurred through Max Steenbeck in the 40s.[7]

The betatron is essentially a transformer with a torus-shaped vacuum tube as its secondary coil. An alternating current in the primary coils accelerates electrons in the vacuum around a circular path. The betatron was the first important machine for producing high energy electrons.

http://en.wikipedia.org/wiki/Betatron

http://en.wikipedia.org/wiki/Betatron

TRIUMF is Canada's national laboratory for particle and nuclear physics. Its headquarters are located on the south campus of the University of British Columbia in Vancouver, British Columbia. TRIUMF houses the world's largest cyclotron,[1] a source of 500 MeV protons, which was named an IEEE Milestone in 2010.[2] TRIUMF's activities involve particle physics, nuclear physics, nuclear medicine, and materials science.

There are over 450 scientists, engineers, and staff on the TRIUMF site, as well as 150 students and postdoctoral fellows. The lab attracts over 1000 national and international researchers every year TRIUMF has generated over $1B in economic impact activity over the last decade.

TRIUMF scientists and university-based physicists develop and implement Natural Sciences and Engineering Research Council’s (NSERC) long-range plan for subatomic physics. TRIUMF uses these plans to develop its own priorities. TRIUMF supports only those projects that have been independently peer reviewed and endorsed by the international scientific community. TRIUMF has over 50 international agreements for collaborative scientific research.

Asteroid 14959 TRIUMF is named in honour of the laboratory

PARTICLE ACCELERATOR TRIUMPH

http://en.wikipedia.org/wiki/TRIUMF

PARTICLE ACCELERATOR FUSORA fusor is a device that uses an electric field to heat ions to conditions suitable for nuclear fusion. The machine has a voltage between two metal cages inside a vacuum. Positive ions fall down this voltage drop, building up speed. If they collide in the center, they can fuse. This is a type of Inertial electrostatic confinement device.

A Farnsworth–Hirsch fusor is the most common type of fusor.[2] This design came from work by Philo T. Farnsworth in (1964) and Robert L. Hirsch in 1967.[3][4] A variant of fusor had been proposed previously by: William Elmore, James L. Tuck, and Ken Watson at the Los Alamos National Laboratory[5] though they never built the machine.

Fusors have been built by various institutions. These include academic institutions such as the University of Wisconsin–Madison,[6] the Massachusetts Institute of Technology[7] and government entities, such as the Atomic Energy Organization of Iran and the Turkish Atomic Energy Authority.[8][9] Fusors have also been developed commercially, as sources for neutrons by DaimlerChrysler Aerospace[10] and as a method for generating medical isotopes.[11][12][13] Fusors have also become very popular for hobbyists and amateurs. A growing number of amateurs have performed nuclear fusion using simple fusor machines.

http://en.wikipedia.org/wiki/Fusor

http://en.wikipedia.org/wiki/Fusor

Documents

Compare to principle of superposition applied to determine the electric field inside and outside an infinite parallel plate capacitor