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

BHAdDt

IldHtDJH

EDQAdDD

SenclosedfCf

encfSf

;

;

0,

,

Law sAmpere'

Law sFaraday'

00

Law Gauss'

000000

00

SenclosedC

SC

S

encS

AdEt

IldBtEJB

AdBt

ldEtBE

AdBB

QAdEE

Local effects of media

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

USPAS, Knoxville, TN, January 20-31, 2013Lecture 2 - Basic E&M and Relativity 3

g

Integration loop

enclosedgap

gap

steelC

IgB

gBldBldH

0

0steelin path

1

gINB 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

USPAS, Knoxville, TN, January 20-31, 2013Lecture 2 - Basic E&M and Relativity 4

tAE

trAB

,

correctically relativist ;

for ;

dtpd

cvdtvdm

dtpdBvEeF

Cyclotron (1930’s) A charged particle in a

uniform magnetic field will follow a circular path of radius

side view

B

top view

B

mqBf

mqB

vf

cvqBmv

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]

USPAS, Knoxville, TN, January 20-31, 2013Lecture 2 - Basic E&M and Relativity 6

2222

2

2

2

energy kinetic

energy total

momentum1

1

pcmcE

mcEK

mcE

mvp

cv

b

b

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

USPAS, Knoxville, TN, January 20-31, 2013Lecture 2 - Basic E&M and Relativity 7

222222

2

222222

axis) z along(velocity

0000

00100001

,,,

,,,

mcpppcE

czyxct

cEppp

ctzyx

zyx

zyx

P

X

AΛAA

P

X

bb

scM effi 222P

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

and over!

USPAS, Knoxville, TN, January 20-31, 2013Lecture 2 - Basic E&M and Relativity 8

)2cos(121sin

)2cos(121cos

)cos()cos(21sinsin

)cos()cos(21coscos

)sin()sin(21sincos

)sin()sin(21cossin

1cos22cos

cossin22sinsinsincoscos)cos(sinsincoscos)cos(sincoscossin)sin(sincoscossin)sin(

2

2

2

AA

AA

BABABA

BABABA

BABABA

BABABA

AA

AAABABABABABABABABABABABABA

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

BvqdtpdF

)(),( tptxB

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

qpB

]T[300/]MeV/c[]m[

Bp

qBp

9

Booster: (B)~30 TmLHC : (B)~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

)(

BBl

pp

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

USPAS, Knoxville, TN, January 20-31, 2013Lecture 2 - Basic E&M and Relativity 11

000

2

BBB

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

USPAS, Knoxville, TN, January 20-31, 2013Lecture 2 - Basic E&M and Relativity 12

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

USPAS, Knoxville, TN, January 20-31, 2013Lecture 2 - Basic E&M and Relativity 13

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,(1dipole;00

,,

22

22

,,

1

1

0

rBrBrBKrrB

KrrBrBnrBrB

rBKrrBKrrBrBn

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

USPAS, Knoxville, TN, January 20-31, 2013Lecture 2 - Basic E&M and Relativity 14

nyx

nx

n

yx

ny

n

n

nnxy

KnxBi

xB

iyxKiBB

!00

0

0

00

~

;~!

1

yxxn

n

n

yxyn

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

USPAS, Knoxville, TN, January 20-31, 2013Lecture 2 - Basic E&M and Relativity 15

...2

~~~

...~2

~

220

220

xyByxByBxBBB

xyByxByBxBBB

x

y

dipole

quadrupole

sextupole

nny x

nBxBxBxBBB

!...

6232

0

dipole

quadrupole

sextupole octupole

Application of Multipoles Dipoles: bend Quadrupoles: focus or defocus

USPAS, Knoxville, TN, January 20-31, 2013Lecture 2 - Basic E&M and Relativity 16

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

xB

y

yB

x

)()()(

BlxB

BlxBx

lBBf

')(

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

21)( xBxBy

x

yB

x

BxBeff

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

3

61)( xBxBy

x

yB

maxx

Bx

Beff 2

2max