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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