Eric Prebys, FNAL

Ø  Math Refresher (Expectations) Ø  Maxwell’s Equations Ø  Special Relativity Ø  Multipole Expansion of Magnetic Fields

Ft. Collins, CO, June 13-24, 2016 E. Prebys, Accelerator Fundamentals: Basic EM and Relativity 2

Ø  Matrix Operations


a bc d

!

"#

$

%&

V1V2

!

"

##

$

%

&&=

aV1 +bV2cV1 +dV2

!

"

##

$

%

&&

a bc d

!

"#

$

%&

−1

=1

ad −bcd −b−c a

!

"#

$

%&

a bc d

≡ det a bc d

"

#$

%

&'= ad −bc( )

a b cd e fg h i

= a e fh i

−bd fg i

+ c d eg h

Ø  Vector Operations u Dot product

u Cross product


!A ⋅!B = (AxBx + AyBy + AzBz )

!A×!B =

i j kAx Ay AzBx By Bz

= (AyBz − AzBy )i + (AzBx − AxBz ) j + (AxBy − AyBx )k

Ø  Vector differential operations u Grad operator

u Gradient

u Divergence

u Curl


∇!"≡

∂∂xi + ∂

∂yj + ∂

∂zk

$

%&

'

()

∇!"φ ≡

∂φ∂xi + ∂φ

∂yj + ∂φ

∂zk

%

&'

(

)*

∇!"×"A ≡

i j k∂∂x

∂∂y

∂∂z

Ax Ay Az

=∂Az∂y

−∂Ay

∂z&

'(

)

*+ i +

∂Ax

∂z−∂Az∂x

&

'(

)

*+ j +

∂Ax

∂y−∂Ay

∂x&

'(

)

*+ k

∇!"⋅"A ≡ ∂Ax

∂x+∂Ay

∂y+∂Az∂z

%

&'

(

)*

Ø  You should be very comfortable with the complex plane

Ø  Also remember the Taylor expansions of trig functions


eiθ = cosθ + isinθ

cosθ = eiθ + e−iθ

2

sinθ = eiθ − e−iθ

2i

eθ ≈ 1+θ +θ2

2!+θ 3

3!+ ...

sinθ ≈θ −θ3

3!+θ 5

5!− ...

cosθ ≈ 1−θ2

2!+θ 4

4!− ...

Ø  Memorize these because we’ll use them a lot!


( )

( )

( )

( )

( )

( ))2cos(121sin

)2cos(121cos

)cos()cos(21sinsin

)cos()cos(21coscos

)sin()sin(21sincos

)sin()sin(21cossin

1cos22coscossin22sin

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

2

2

2

AA

AA

BABABA

BABABA

BABABA

BABABA

AAAAA

BABABABABABABABABABABABA

−=

+=

+−−=

−++=

−−+=

−++=

−=

=

+=−

−=+

−=−

+=+

Ø  In 1861, James Maxwell began his attempt to find a self-consistent set of equations consistent with all of the E&M experiments which had been done up until that point. u Because vector calculus hadn’t been invented yet, his final paper is

55 pages long and completely incomprehensible.

Ø  In in modern notation, it reduces to the following four equations:


!∇•!E =

ρε0

⇒!E •d!A

S"∫ =Qenc

ε0

Gauss' Law

!∇•!B = 0 ⇒

!B•d!A

S"∫ = 0 No Name Law

!∇×!E = −

∂!B∂t

⇒!E •d!l

C"∫ = −∂∂t

!B•d!A

S"∫ Faraday's Law

!∇×!B = µ0

!J +µ0ε0

∂!E∂t

⇒!B•d!l

C"∫ = µ0Ienclosed +µ0ε0∂∂t

!E •d!A

S"∫ Ampere's Law

Ø  The electric field passing through a surface depends only on the charge contained within the surface

Ø  Example: deriving Coulomb’s Law


!E •d!A

S"∫ =Qenc

ε0

!E •d!A

S"∫ = E •A

= 4πr2E

=qε0

→ E = q4πr2ε0

!B•d!A

S"∫ = 0→No magnetic monopoles

Ø  The integrated electric field around any closed loop is proportional to the rate of change of the magnetic flux passing through the loop

Ø  Example: magnetic induction


!E •d!l

C"∫ = −∂∂t

!B•d!A

S"∫

V =!E •d!l

C"∫= −

∂∂t

!B•d!A

S"∫

= −B dAdt

= −Bwv

w

Ø  The integrated magnetic field around any closed loop is proportional to the total current passing through the loop.

Ø  Example: Magnetic field of a wire


!B•d!l


!E •d!A

S"∫Set to 0 for a minute

!B•d!l

C"∫ = 2πrB

= µ0Ienclosed = µ0I

→ B = µ0I2πr

Ø  Maxwell’s first version of Ampere’s Law did not have the second term

Ø  However, you should be able to draw the surface anywhere, and you get in trouble if you draw it through a break in the current

Ø  Maxwell added the second term just so he would get the same answer in both cases!


!B•d!l

C"∫ = µ0Ienclosed

Current flowing here

Field changing here

However, anywhere there’s a break in the current, you’ll get a changing electric field.

!B•d!l


!E •d!A

S"∫

Ø  The “displacement current” was added for purely mathematical reasons u  It would not be proven experimentally for many years

Ø  However, the implications were profound Ø  Previously, it was believed you could not have electric or magnetic

fields without electric charges, but now, even in a complete vacuum, you can have u (changing electric field)è(changing magnetic field)è�

(changing electric field)è“Electromagnetic Wave”! Ø  Moreover, Maxwell could calculate the velocity,

and he found it was the speed of light! Ø  He wrote (with trembling hands, maybe?)

"we can scarcely avoid the inference that light consists in the transverse undulations of the same medium which is the cause of electric and magnetic phenomena"


Ø  In one fell swoop, Maxwell not only unified electricity and magnetism, but his results would eventually show that light, heat, radio waves, x-rays, gamma rays, etc., are all really the same thing – differing only in wavelength!


The entire visible spectrum.

Ø  As often happens science, one answer raised a lot more questions.

Ø  All (other) known waves require a “medium” (air, water, earth, “the wave”) to travel through.

Ø  Light at least appears to travel through a vacuum. Ø  In science, always try the simplest answer first:

u Maybe vacuum isn’t really empty?

Ø  Scientists hypothesized the existence of “luminiferous aether”, and started to look for it…


Ø  If aether exists, then it must fill space and the earth must be passing through it.

Ø  Light traveling along the direction of the Earth’s motion should have a slightly different wavelength than light traveling transverse to it.

Ø  In 1887, Albert Michelson and Edward Morley performed a sensitive experiment to measure this difference.

Ø  Their result: u No difference è no aether!

Ø  Biggest mystery in science for almost 20 years.


Ø  In 1905, Albert Einstein postulated that perhaps the equations meant exactly what they appeared to mean: u The speed of light was the same in any frame in which is was

measured.

Ø He showed that this could “work”, but only if you gave up the notion of fixed time. u è “Special Theory of Relativity”

Ø Profound implications…


Ø  Einstein said, “The speed of light must be the same in any reference frame”. For example, the time it takes light to bounce off a mirror in a spaceship must be the same whether it’s measured by someone in the spaceship, or someone outside of the spaceship.

Ø  This seems weird, but it applies to everything we do at the lab u Example: the faster pions and muons move, the longer they live.


• These two people have to measure the same speed for light, even though light is traveling a different distance for the two of them.

• The only solution? More time passes for the stationary observer than the guy in the spaceship!

• “Twin Paradox”

Ø  Generally, relativity treats time more or less like one more spatial dimension. Both time and space transform between two frames


Ø  Classically:

Ø  Relativistically:


momentum: !p = m!v

kinetic energy: K =12mv2

momentum: !p = m!v1− (v / c)2

total energy: E2 = (mc2 )2 + (pc)2

kinetic energy: K = E-mc2

Emc2

pc

0"

1"

2"

3"

4"

5"

6"

7"

8"

0" 0.1" 0.2" 0.3" 0.4" 0.5" 0.6" 0.7" 0.8" 0.9" 1"

Rela%v

is%c*"gamma"*fa

ctor*

(velocity)/(speed*of*light)*

Rest Energy

Kinetic

Energy

For v<<c (speed of light),

Kinetic energy ~ ½mv2

γ = 1

1− vc

⎛⎝⎜

⎞⎠⎟2

c = (speed of light) = 300,000 km/s!

Ø  Basics

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

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


β ≡vc

γ ≡1

1−β 2

momentum p = γmvtotal energy E = γmc2

kinetic energy K = E −mc2

E = mc2( )2+ pc( )2

β = pcE

dγ = βγ 3dβdββ

= 1γ 2

dpp

dpp

= 1β 2

dEE

Some Handy Relationships (homework)

Ø  We’ll use the conventions

Ø  Note that for a system of particles

Ø  We’ll worry about field transformations later, as needed


( )

( ) ( )

( )222222

2

222222

axis) z along(velocity

0000

00100001

,,,

,,,

mcpppcE

czyxct

cEppp

ctzyx

zyx

zyx

≡−−−⎟⎠

⎞⎜⎝

⎛=

≡−−−=

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

−

−==ʹ′

⎟⎠

⎞⎜⎝

⎛≡

≡

P

X

AΛAA

P

X

τ

γβ

βγ

( ) scMeffi ≡=∑222

P

Ø  The equations we’ve talked about so far are correct if you account for all electric charges in the system; however, in real life situation, much, or even most, of the charge is a system is contained in matter, and it’s behavior can generally be parameterized in a more convenient way. In terms of just the free electric charge, Gauss’ Law and Ampere’s Law become: where


!∇•!D = ρ f ⇒

!D•d

!A

S"∫ =Qf ,enc ;!D ≡ε

!E

!∇×!H =

!J f +

∂!D∂t

⇒!H •d

!l

C"∫ = I f ,enclosed +0∂∂t

!D•d

!A

S"∫ ;!H ≡

!Bµ

Local effects of media

ε = "electric permitivity"µ = "magnetic permiability"

Ø  The “electric permittivity” comes from the tendency of charge in matter to form electric dipoles in the presence of an external field, reducing the the true field

Ø  The “magnetic permeability” comes from the tendency of magnetic dipoles in some materials to align with the external magnetic field, increasing the true field.


Ø  Cross section of dipole magnet


g

Integration loop

!H •d

!l

C"∫ =

1µsteel

!B•d!l

path in steel∫ +

Bgapgµ0

≈Bgapgµ0

= Ienclosed

gINB turns

gap0µ≈⇒

µsteel µgap

Ø  The relativistially correct form for the motion of charged particles in electric and magnetic fields is given by the Lorentz equation:


F = d

!pdt

= q(!E + !v ×

!B)

radius of curvature r = pqB

Ø  A charged particle in a uniform magnetic field will follow a circular path of radius

side view

B

ρ

top view

Bρ =

mvqB

(v << c)

f =v

2πρ

=qB

2πm (constant!!)

Ωs = 2π f = qBm

MHz ][2.15 TBfC ×=

“Cyclotron Frequency”

For a proton:

Accelerating “DEES” 27

Ft. Collins, CO, June 13-24, 2016

E. Prebys, Accelerator Fundamentals: Basic EM and Relativity

Ø  The relativistically correct form of Newton’s Laws for a particle in an electromagnetic field is:

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

!F = d

!pdt

= q!E + !v ×

!B( ); !p = γm!v

ρ = peB

Bρ( ) = pe

Bρ( )c = pce

side view

B

ρ

top view

Bconstant for fixed energy!

T-m2/s=V units of eV in our usual convention

Bρ( )[T-m]= p[eV/c]c[m/s]

≈ p[MeV/c]300

Beam “rigidity” = constant at a given momentum (even when B=0!)

Remember forever!

If all magnetic fields are scaled with the momentum as particles accelerate, the trajectories remain the same è“synchrotron” [E. McMillan, 1945]


Ø  Compare Fermilab LINAC (K=400 MeV) to LHC (K=7000 GeV)


Parameter Symbol Equa0on Injec0on Extrac0on proton mass m [GeV/c2] 0.938 kine9c energy K [GeV] .4 7000 total energy E [GeV] 1.3382 7000.938 momentum p [GeV/c] 0.95426 7000.938 rel. beta β 0.713 0.999999991

rel. gamma γ 1.426 7461.5 beta-‐gamma βγ 1.017 7461.5

rigidity (Bρ) [T-‐m] 3.18 23353.

K +mc2

E2 − mc2( )2

pc( ) / EE / (mc2 )

p[GeV]/(.2997)pc( ) / (mc2 )

This would be the radius of curvature in a 1 T magnetic field or the field in Tesla

needed to give a 1 m radius of curvature.

Ø  If the path length through a transverse magnetic field is short compared to the bend radius of the particle, then we can think of

the particle receiving a transverse “kick” and it will be bent through small angle

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

lB θΔ

p

)( ρθ

BBl

pp

=≈Δ ⊥

qBlvlqvBqvBtp ==≈⊥ )/(

θΔ

Ft. Collins, CO, June 13-24, 2016 30 E. Prebys, Accelerator Fundamentals: Basic EM and Relativity

Ø  Define the “gradient” operator


∇!"≡∂∂xi + ∂

∂yj + ∂

∂zk

∇!"⋅"A = ∂Ax

∂x+∂Ay

∂y+∂Az∂z

∇!"×"A = ∂Az

∂y−∂Ay

∂z'

()

*

+, i +

∂Az∂x

−∂Ax

∂z'

()

*

+, j +

∂Ay

∂x−∂Ax

∂y'

()

*

+, k

=

i j k∂∂x

∂∂y

∂∂z

Ax Ay Az

Ø  Formally, in a current free region, the curl of the magnetic field is:

Ø  This means that the magnetic field can be expressed as the gradient of a scalar:

Ø  The zero divergence then gives us:

Ø  If the field is uniform in z, then δφ/δz=0, so


!∇×!B = µ0

!J = 0

Laplace Equation !B = −∇

"!φ

∇!"⋅"B = −∇2φ =

∂2φ∂x2

+∂2φ∂y2

+∂2φ∂z2

&

'(

)

*+= 0

∂2ϕ∂x2

+∂2ϕ∂y2

= 0

Ø  The general solution is

Ø  Solving for B components

Ø  Combining and redefining the constants


Bx = −∂ϕ∂x

= −Re mCm x+ iy( )m−1m=1

∞

∑ = −Im imCm x+ iy( )m−1m=1

∞

∑

By = −∂ϕ∂y

= −Re imCm x+ iy( )m−1m=1

∞

∑

By + iBx = Kn x+ iy( )nn=0

∞

∑ ;Kn = i(n+1)Cn+1

∂2ϕ∂x2

+∂2ϕ∂y2

= 0⇒ϕ (x, y)=Re Cm x+ iy( )mm=0

∞

∑

Note order!

Ø  We can express the complex numbers in notation


By + iBx = Kn x+ iy( )nn=0

∞

∑ = Knrn

n=0

∞

∑ einθ

= Kn eiδn rn

n=0

∞

∑ einθ

Amplitude rotation

r is real Kn is complex

Ø  In our general expression the phase angle δm represents a rotation of each component about the z axis. Set all δm =0 for the moment, and we see the following symmetry properties for the first few multipoles


),()2/,(0)4/,(;)4/,(

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

quadrupole)0,(;0)0,(1dipole;00

,,

22

22

,,

1

1

0

θπθ

ππ

θπθ

ππ

rBrBrBKrrB

KrrBrBnrBrB

rBKrrBKrrBrBn

KBBn

yxyx

yx

yx

yxyx

yx

yx

yx

−=+

==

≡==⇒=

−=+

==

≡==⇒=

≡==⇒=

By + iBx = Kn eiδn rn

n=0

∞

∑ einθ

Ø  Back to Cartesian Coordinates. Expand by differentiating both sides n times wrt x

Ø  And we can rewrite this as

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


( )

nyx

nx

n

yxny

nn

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 ≡ʹ′ʹ′≡ʹ′≡ʹ′ʹ′≡ʹ′

Ø  Expand first few terms…

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


( )

( ) ...2

~~~

...~2

~

220

220

+ʹ′ʹ′+−ʹ′ʹ′

+ʹ′+ʹ′+=

+ʹ′ʹ′−−ʹ′ʹ′

+ʹ′−ʹ′+=

xyByxByBxBBB

xyByxByBxBBB

x

y

dipole quadrupole sextupole

nny x

nBxBxBxBBB!

...62

320 +

ʹ′ʹ′ʹ′+

ʹ′ʹ′+ʹ′+=

dipole quadrupole sextupole octupole

Ø  Dipoles: bend Ø  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

)()()(

ρρθ

BlxB

BlxBx ʹ′

−=−≈Δ

lBBf')( ρ

=

∇!"×"B = 0

→∂By

∂x=∂Bx

∂y

Ø  Sextupole magnets have a field (on the principle axis) given by

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

Ø  In a similar way, octupoles have a field given by

Ø  So high amplitude particles will see a different average gradiant

2

21)( xBxBy ʹ′ʹ′=

x

yB

x

BxBeff ʹ′ʹ′=ʹ′

Ft. Collins, CO, June 13-24, 2016 39 E. Prebys, Accelerator Fundamentals: Basic EM and

Relativity

3

61)( xBxBy ʹ′ʹ′ʹ′=

x

yB

maxx

Bx

Beff ʹ′ʹ′ʹ′=ʹ′2

2max

Eric Prebys, FNAL

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