Electrodynamics and Relativity Explained: Maxwell's Equations, Charged Particle Behavior, and More

E&M and Relativity Eric Prebys, FNAL

Maxwells Equations In terms of total charge and current Q = = Gauss' Law enc E E d A S 0 0 = = 0 0 B B d A S B = = Faraday' Law s E E l d B d A t t C S E = + = + Ampere' Law s B J B l d I E d A 0 0 0 0 0 0 enclosed t t C S In terms of free charge an current = = ; D D S d A Q D E , f f enc D B H C D S = + l d = + ; H J I d A H , 0 f f enclosed t t Local effects of media USPAS, Knoxville, TN, January 20-31, 2013 Lecture 2 - Basic E&M and Relativity 2

Example: Field in a permeable dipole Cross section of dipole magnet Integration loop g B g 1 in gap l d = l d + H B 0 steel path steel C B g gap = I enclosed 0 msteel mgap N I 0 turns B gap g USPAS, Knoxville, TN, January 20-31, 2013 Lecture 2 - Basic E&M and Relativity 3

Electrodynamics and Electrodynamic Potentials We can write the electric and magnetic fields in terms of Vector and Scalar potentials B = ( ) , A r t A = E t Particle dynamics are governed by the Lorentz force law ( dt ) d p d v = + = = ; for F e E v B m v c dt d p = relativist ; ically correct dt USPAS, Knoxville, TN, January 20-31, 2013 Lecture 2 - Basic E&M and Relativity 4

Cyclotron (1930s) A charged particle in a uniform magnetic field will follow a circular path of radius side view top view B B mv = ( ) v c qB v 2 = f Cyclotron Frequency qB 2 = (constant! ) ! m f 2 qB = = s m Red box = remember! For a proton: = 15 2 . [ MHz ] fC B T Accelerating DEES USPAS, Knoxville, TN, January 20-31, 2013 5 Lecture 2 - Basic E&M and Relativity

Relativity Some Handy Relationships (homework) Basics v b =pc c E 1 dg = bg db b dp p 3db dp p dE E 2 1 =1 g2 =1 b2 = momentum p mv = 2 total energy E mc = 2 kinetic energy K E mc ( ) ( )2 2 = + 2 2 E mc pc 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 @ =.9 = 1.94 GeV/c] USPAS, Knoxville, TN, January 20-31, 2013 Lecture 2 - Basic E&M and Relativity 6

4-Vectors and Lorentz Transformations We ll use the conventions ( , , , ct z y x P ) X E , , , p p p x y z c 1 0 0 0 0 1 0 0 = = A (velocity A along z axis) A 0 0 0 0 ( ) ct ( ) ( 2 2 2 = 2 2 2 X x y z c 2 ) E 2 2 = 2 x 2 y 2 z 2 P p p p mc c ( ) 2 2 = 2 P Meff c s Note that for a system of particles i We ll worry about field transformations later, as needed USPAS, Knoxville, TN, January 20-31, 2013 Lecture 2 - Basic E&M and Relativity 7

Some Handy Relationships Know all of these by heart because you re going to use them over and over! cos sin ) sin( B A B A = + + = + cos sin A B A B A B = sin( ) sin cos A cos sin A A B cos( ) cos cos sin sin A B B B = + cos( ) cos A cos A sin sin A B A B A B = sin 2 2 sin cos A = 2 cos 2 2 cos 1 A A 1 ( ) = + + sin cos sin( ) sin( ) A B A B A B 2 1 ( ) = + cos sin sin( ) sin( ) A B A B A B 2 1 ( ) = + + cos cos cos( ) cos( ) A B A B A B 2 1 ( ) = + sin sin cos( ) cos( ) A B A B A B 2 1 ( ) = + 2 cos 1 cos( 2 ) A A 2 1 ( ) ) A = 2 sin 1 cos( 2 A 2 USPAS, Knoxville, TN, January 20-31, 2013 Lecture 2 - Basic E&M and Relativity 8

Synchrotrons and beam rigidity The relativistic form of Newton s Laws for a particle in a magnetic field is: ( v q dt ) d p = = F B A particle in a uniform magnetic field will move in a circle of radius [ MeV/c B / ] 300 p p = [ m ] [ T ] qB 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 ( , ) ( ) B x t p t [ MeV/c ] p p Booster: (B )~30 Tm LHC : (B )~23000 Tm = ( ) ( )[ Tm ] B B 300 q USPAS, Knoxville, TN, January 20-31, 2013 9 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 of the particle receiving a transverse kick p p B l = = ( / ) qvBt qvB l v qBl and it will be bent through small angle p Bl = ( B ) p In this thin lens approximation , a dipole is the equivalent of a prism in classical optics. USPAS, Knoxville, TN, January 20-31, 2013 Lecture 2 - Basic E&M and Relativity 10

Field multipole expansion Formally, in a current free region Magnetic field is the gradient of a scalar = = 0 B B which satisfies Laplace s equation = = 2 0 0 B The general solution in two dimensions 2 2 ( ) = m m + = = + 0 ( , ) Re x y C x iy m 2 2 x y 0 USPAS, Knoxville, TN, January 20-31, 2013 Lecture 2 - Basic E&M and Relativity 11

Solving for B components ( ) = m 1 m = = + Re B mC x iy x m x 1 ( ) = m 1 m = = + Re B imC x iy y m y 1 Combining ( ) = n n + = + B iB K x iy y x n 0 USPAS, Knoxville, TN, January 20-31, 2013 Lecture 2 - Basic E&M and Relativity 12

Symmetry properties of mulitpoles ( ) = = n + = + = n in B iB K x iy K r e y x n n 0 0 n n = i = n in K e r e n n 0 n The phase angle m represents a rotation of each component about the axis. Set all m =0 for the moment = = = 0 0 ; dipole n B B K 0 x y = = = 1 ( ) 0 , r 0 ; ( ) 0 , r quadrupole n B r B r r K = 1 x y = ( , r ) 2 / = ; = ( B , ) 2 / r = 0 B r K B 1 x y + ( , ) ( , ) B , , x y x y = 2 2 ( ) 0 , r 0 ; ( ) 0 , r sextupole n B r B r r K 2 x y = = 2 ( , r ) 4 / ; = ( B , ) 4 / r 0 B r K B 2 x y + ( , ) 2 / ( , ) B , , x y x y USPAS, Knoxville, TN, January 20-31, 2013 Lecture 2 - Basic E&M and Relativity 13

Back to Cartesian Coordinates. Differentiate both sides n times wrt x x y iB B = + ( ) = n + K x iy n 0 n n B n B y + = ! x i n K n n n x x = = 0 x y = = 0 x y And we can rewrite this as = + n normal ( )( n 1 n ~ ) n + + ; B iB B B i x iy B B y x n n n y n ! x = 0 = = 0 x y n ~ B B n x n x skew = = 0 x y Normal terms always have Bx=0 on x axis. Skew terms always have By=0 on x axis. Generally define , 1 B B B B ~ B ~ B ~ B ~ B , , etc , 2 1 2 USPAS, Knoxville, TN, January 20-31, 2013 Lecture 2 - Basic E&M and Relativity 14

Expand first few terms ( ) B ~ B ~ B = + + + 2 2 ... B B B x y x y xy 0 y 2 ~ B ( ) ~ B ~ B = + + + + + 2 2 ... B x B y x y B xy 0 x 2 quadrupole sextupole dipole Note: in the absence of skew terms, on the x axis + + = B B 2 B 6 + + 2 3 n ... n ! B B B x x x x 0 y n sextupole quadrupole octupole dipole USPAS, Knoxville, TN, January 20-31, 2013 Lecture 2 - Basic E&M and Relativity 15

Application of Multipoles Dipoles: bend Quadrupoles: focus or defocus B B x y y x A positive particle coming out of the page off center in the horizontal plane will experience a restoring kick lx ( ) Bx x l B = ( ) ( ) B B ( ) B = f ' B l USPAS, Knoxville, TN, January 20-31, 2013 Lecture 2 - Basic E&M and Relativity 16

Sextupoles Octupoles In a similar way, octupoles have a field given by 1 ) ( B x By = Sextupole magnets have a field (on the principle axis) given by 1 = 2 3 ( ) By x B x x 2 6 So high amplitude particles will see a different average gradiant One common application of this is to provide an effective position- dependent gradient. B B y y x x x x max 2 max x = Beff x B = Beff B 2 17 Lecture 2 - Basic E&M and Relativity USPAS, Knoxville, TN, January 20-31, 2013