eric prebys, fnal. in terms of total charge and current in terms of free charge an current uspas,...

E&M and RelativityEric Prebys, FNAL

Maxwell’s Equations In terms of total charge and current

In terms of free charge an current

USPAS, Knoxville, TN, January 20-31, 2013Lecture 2 - Basic E&M and Relativity 2

BHAdD

tIldH

t

DJH

EDQAdDD

SenclosedfCf

encfSf

;

;

0,

,

Law sAmpere'

Law sFaraday'

00

Law Gauss'

000000

00

SenclosedC

SC

S

enc

S

AdEt

IldBt

EJB

AdBt

ldEt

BE

AdBB

QAdEE

Local effects of media

Example: Field in a permeable dipole Cross section of dipole magnet


g

Integration loop

enclosedgap

gap

steelC

IgB

gBldBldH

0

0steelin path

1

g

INB turns

gap0

Electrodynamics and Electrodynamic Potentials We can write the electric and magnetic fields in terms of

Vector and Scalar potentials

Particle dynamics are governed by the Lorentz force law


t

AE

trAB

,

correctically relativist ;

for ;

dt

pd

cvdt

vdm

dt

pdBvEeF

Cyclotron (1930’s) A charged particle in a

uniform magnetic field will follow a circular path of radius

side view

B

top view

B

m

qBfm

qB

vf

cvqB

mv

s

2

)!(constant! 2

2

)(

MHz ][2.15 TBfC

“Cyclotron Frequency”

For a proton:

Accelerating “DEES”5

USPAS, Knoxville, TN, January 20-31, 2013 Lecture 2 - Basic E&M and Relativity

Red box = remember!

Relativity Basics

A word about units For the most part, we will use SI units, except

Energy: eV (keV, MeV, etc) [1 eV = 1.6x10-19 J] Mass: eV/c2 [proton = 1.67x10-27 kg = 938 MeV/c2] Momentum: eV/c [proton @ b=.9 = 1.94 GeV/c]


2222

2

2

2

energy kinetic

energy total

momentum

1

1

pcmcE

mcEK

mcE

mvp

c

v

Some Handy Relationships (homework)

4-Vectors and Lorentz Transformations We’ll use the conventions

Note that for a system of particles

We’ll worry about field transformations later, as needed


222222

2

222222

axis) z along(velocity

00

00

0010

0001

,,,

,,,

mcpppc

E

czyxct

c

Eppp

ctzyx

zyx

zyx

P

X

AΛAA

P

X

scM effi 222P

Some Handy Relationships Know all of these by heart because you’re going to use them over

and over!


)2cos(12

1sin

)2cos(12

1cos

)cos()cos(2

1sinsin

)cos()cos(2

1coscos

)sin()sin(2

1sincos

)sin()sin(2

1cossin

1cos22cos

cossin22sin

sinsincoscos)cos(

sinsincoscos)cos(

sincoscossin)sin(

sincoscossin)sin(

2

2

2

AA

AA

BABABA

BABABA

BABABA

BABABA

AA

AAA

BABABA

BABABA

BABABA

BABABA

Synchrotrons and beam “rigidity” The relativistic form of Newton’s Laws for a particle in a

magneticfield is:

A particle in a uniform magnetic field will move in a circle of radius

In a “synchrotron”, the magnetic fields are varied as the beam accelerates such that at all points , and beam motion can be analyzed in a momentum independent way.

It is usual to talk about he beam “rigidity” in T-m

Bvqdt

pdF

)(),( tptxB

300

]MeV/c[]Tm)[()(

pB

q

pB

]T[

300/]MeV/c[]m[

B

p

qB

p

9

Booster: (Br)~30 TmLHC : (Br)~23000 Tm

USPAS, Knoxville, TN, January 20-31, 2013 Lecture 2 - Basic E&M and Relativity

Thin lens approximation and magnetic “kick”

If the path length through a transverse magnetic field is short compared to the bend radius of the particle, then we can think ofthe particle receiving a transverse “kick”

and it will be bent through small angle

In this “thin lens approximation”, a dipole is the equivalent of a prism in classical optics.

l

B p

)(

B

Bl

p

p

qBlvlqvBqvBtp )/(

USPAS, Knoxville, TN, January 20-31, 2013 10Lecture 2 - Basic E&M and Relativity

Field multipole expansion Formally, in a current free region

The general solution in two dimensions


00

02

B

BB

Magnetic field is the gradient of a scalar…

…which satisfies Laplace’s equation

02

2

2

2

Re),(0m

mm iyxCyx

yx

Solving for B components

Combining


1

1

1

1

Re

Re

m

mmy

m

mmx

iyximCy

B

iyxmCx

B

0n

nnxy iyxKiBB

Symmetry properties of mulitpoles

The phase angle δm represents a rotation of each component about the axis. Set all δm =0 for the moment


in

n

nin

in

n

nn

n

nnxy

ereK

erKiyxKiBB

n

0

00

),()2/,(

0)4/,(;)4/,(

sextupole)0,(;0)0,(2

),(),(

0)2/,(;)2/,(

quadrupole)0,(;0)0,(1

dipole;00

,,

22

22

,,

1

1

0

rBrB

rBKrrB

KrrBrBn

rBrB

rBKrrB

KrrBrBn

KBBn

yxyx

yx

yx

yxyx

yx

yx

yx

Back to Cartesian Coordinates. Differentiate both sides n times wrt x

And we can rewrite this as

“Normal” terms always have Bx=0 on x axis.

“Skew” terms always have By=0 on x axis. Generally define


n

yx

nx

n

yx

n

yn

n

nnxy

Knx

Bi

x

B

iyxKiBB

!00

0

0

00

~

;~

!

1

yx

xn

n

n

yx

yn

n

nn

nnnxy

Bx

B

Bx

BiyxBiBn

iBB“normal”

“skew”

etc ,~~

,~~

,, 2121 BBBBBBBB

Expand first few terms…

Note: in the absence of skew terms, on the x axis


...2

~~~

...~

2

~

220

220

xyByxB

yBxBBB

xyByxB

yBxBBB

x

y

dipole

quadrupole

sextupole

nny x

n

Bx

Bx

BxBBB

!...

6232

0

dipole

quadrupole

sextupole octupole

Application of Multipoles Dipoles: bend Quadrupoles: focus or defocus


A positive particle coming out of the page off center in the horizontal plane will experience a restoring kick

xB

y

yB

x

)()(

)(

B

lxB

B

lxBx

lB

Bf

'

)(

Sextupoles Octupoles Sextupole magnets have a

field(on the principle axis) given by

One common application of this is to provide an effective position-dependent gradient.

In a similar way, octupoles have a field given by

So high amplitude particles will see a different average gradiant

2

2

1)( xBxBy

x

yB

x

BxBeff

USPAS, Knoxville, TN, January 20-31, 2013 17Lecture 2 - Basic E&M and Relativity

3

6

1)( xBxBy

x

yB

maxx

Bx

Beff 2

2max

eric prebys, fnal. in terms of total charge and current in terms of free charge an current uspas,...

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