Appendix A Maxwell’s Equations, Equations of Motion, and Energy Balance in an Electromagnetic Field

A.1 Maxwell’s Equations

Classical electrodynamics in vacuum is governed by the Maxwell equations. In the SI system of units, the Maxwell equations are

∇·D = ρ , (A.1a) ∇·B = 0 , (A.1b) ∂ B ∇×E =− , (A.1c) ∂t ∂ D ∇×H = j + , (A.1d) ∂t where ρ is the charge density, j is the current density, D = 0 E and H = B/μ0. Traditionally B is called the magnetic induction, and H is called the magnetic ﬁeld, but in this book we refer to B as the magnetic ﬁeld. The Maxwell equations are linear: a sum of two solutions, E1, B1 and E2, B2, is also a solution corresponding ρ + ρ + to the sum of densities 1 2, j 1 j 2. For a point charge q moving along a trajectory r = r0(t) the charge density and the current density are

ρ(r, t) = qδ(r − r0(t)) , j(r, t) = qv(t)δ(r − r0(t)) , (A.2) with v(t) = dr0(t)/dt. To ﬁnd a particular solution of the Maxwell equations in a volume, proper bound- ary conditions should be speciﬁed at the volume boundary. On a surface of a good conducting metal the boundary condition requires the tangential component of the electric ﬁeld to be equal to zero, Et |S = 0.

© Springer International Publishing AG, part of Springer Nature 2018 269 G. Stupakov and G. Penn, Classical Mechanics and Electromagnetism in Accelerator Physics, Graduate Texts in Physics, https://doi.org/10.1007/978-3-319-90188-6 270 AppendixA: Maxwell’s Equations, Equations of Motion, and Energy Balance …

A.2 Wave Equations

In free space with no local charges and currents the electric ﬁeld satisﬁes the wave equation,

1 ∂2 E ∂2 E ∂2 E ∂2 E − − − = 0 . (A.3) c2 ∂t2 ∂x2 ∂y2 ∂z2

The same equation is valid for the magnetic ﬁeld B. A particular solution of Eq. (A.3) is a sinusoidal wave characterized by the frequency ω and the wave vector k,

E = E0 sin(ωt − k · r), (A.4) where E0 is a constant vector perpendicular to k, E0 · k = 0, and ω = ck.

A.3 Vector and Scalar Potentials

It is often convenient to express the ﬁelds in terms of the vector potential A(r, t) and the scalar potential φ(r, t):

∂ A E =−∇φ − , ∂t B =∇×A . (A.5)

Substituting these equations into Maxwell’s equations, we ﬁnd that the second and third equations are satisﬁed identically. We only need to take care of the ﬁrst and fourth equations.

A.4 Energy Balance and the Poynting Theorem

The electromagnetic ﬁeld has an energy and momentum associated with it. The energy density of the ﬁeld (energy per unit volume) is

1 u = (E · D + H · B) = 0 (E2 + c2 B2). (A.6) 2 2 The Poynting vector,

S = E × H , (A.7) AppendixA: Maxwell’s Equations, Equations of Motion, and Energy Balance … 271

Fig. A.1 Point charges shown by red dots move inside volume V and interact through the electromagnetic ﬁeld

gives the energy ﬂow (energy per unit area per unit time) in the electromagnetic ﬁeld. Consider charges that move inside a volume V enclosed by a surface A, see Fig. A.1. The Poynting theorem states ∂ udV =− j · E dV − n · S dA, (A.8) ∂t V V A where n is the unit vector normal to the surface and directed outward. The left-hand side of this equation is the rate of change of the electromagnetic energy due to the interaction with moving charges. The ﬁrst term on the right-hand side is the work done by the electric ﬁeld on the moving charges. The second term describes the electromagnetic energy ﬂow from the volume through the enclosing surface.

A.5 Photons

The quantum view on electromagnetic radiation is that the electromagnetic ﬁeld is represented by photons. Each photon carries the energy ω and the momentum k, where the vector k is the wavenumber which points to the direction of propagation of the radiation, = 1.05 × 10−34 J· sec is the Planck constant divided by 2π, and k = ω/c. Appendix B Lorentz Transformations and the Relativistic Doppler Effect

B.1 Lorentz Transformation and Matrices

Consider two coordinate systems, K and K . The system K is moving with velocity v in the z direction relative to the system K (see Fig. B.1). The coordinates of an event in both systems are related by the Lorentz transformation

x = x , y = y , z = γ(z + βct), t = γ(t + βz/c), (B.1) where β = v/c, and γ = 1/ 1 − β2. The vector (ct, r) = (ct, x, y, z) is called a 4-vector, and the above transformation is valid for any 4-vector quantity. The transformation from K to K is also a Lorentz transformation, but the original frame K has a velocity −v relative to the system K , so the inverse transformation is obtained from Eq. (B.1) by changing the sign of β:

Fig. B.1 Laboratory frame K and a moving frame K

© Springer International Publishing AG, part of Springer Nature 2018 273 G. Stupakov and G. Penn, Classical Mechanics and Electromagnetism in Accelerator Physics, Graduate Texts in Physics, https://doi.org/10.1007/978-3-319-90188-6 274 Appendix B: Lorentz Transformations and the Relativistic Doppler Effect

x = x , y = y , z = γ(z − βct), t = γ(t − βz/c). (B.2)

The Lorentz transformation (B.1) can also be written in the matrix notation ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ x 10 0 0 x x ⎜ y ⎟ ⎜ 01 0 0 ⎟ ⎜ y ⎟ ⎜ y ⎟ ⎜ ⎟ = ⎜ ⎟ ⎜ ⎟ = L ⎜ ⎟ , (B.3) ⎝ z ⎠ ⎝ 00 γ cβγ ⎠ ⎝ z ⎠ ⎝ z ⎠ βγ γ t 00 c t t where L denotes the 4 × 4 matrix in the middle of the equation. The advantage of using matrices is that consecutive transformations reduce to matrix multiplication.

B.2 Lorentz Contraction and Time Dilation

Two events occurring in the moving frame at the same point and separated by the time interval t will be measured by the lab observers as separated by t,

t = γt . (B.4)

This is the effect of relativistic time dilation. An object of length l aligned in the moving frame with the z axis will have the length l in the lab frame:

l l = . γ (B.5)

This is the effect of relativistic contraction. The length in the direction transverse to the velocity is not changed.

B.3 Relativistic Doppler Effect

Consider a wave propagating in a moving frame K . It has the time-space dependence:

∝ cos(ωt − k · r), (B.6) where ω is the frequency and k is the wavenumber of the wave in K . An observer that measures this wave in the reference frame K will see the time-space dependence Appendix B: Lorentz Transformations and the Relativistic Doppler Effect 275 that is obtained from the Lorentz transformation of coordinates and time in Eq. (B.6):

cos(ω t − k · r ) = cos(ω γ(t − βz/c) − kx x − ky y − kz γ(z − βct)) = cos(γ(ω + kz βc)t − kx x − ky y − γ(kz + ω β/c)z). (B.7)

We see that in the K frame this process is also a wave

∝ cos(ωt − k · r), (B.8) with the frequency and wavenumber

kx = kx , ky = ky , kz = γ(kz + βω /c), ω = γ(ω + βckz ). (B.9)

Hence, the combination (ω, ck) is a 4-vector. The above transformation is valid for any type of waves (electromagnetic, acoustic, plasma waves, etc.) Let us now apply it to electromagnetic waves in vacuum. For these waves we know that

ω = ck . (B.10)

Assume that an electromagnetic wave propagates at angle θ in the frame K ,

k cos θ = z , (B.11) k and has a frequency ω in that frame. What is the angle θ and the frequency ω of this wave in the lab frame? We can always choose the coordinate system such that k = (0, ky, kz), then

k θ θ = ky = y = sin . tan (B.12) kz γ(kz + βω /c) γ(cos θ + β)

In the limit γ 1 almost all angles θ (except for the angles very close to π)are transformed to angles θ ∼ 1/γ. This explains why radiation of an ultrarelativistic beam goes mostly in the forward direction, within an angle of the order of 1/γ. A general expression for how the Lorentz transformation changes the angle θ of an electromagnetic wave with respect to the direction of the velocity is:

cos θ − β sin θ cos θ = , sin θ = . (B.13) 1 − β cos θ γ(1 − β cos θ) 276 Appendix B: Lorentz Transformations and the Relativistic Doppler Effect

For the frequency, a convenient formula relates ω with ω and θ (not θ). To derive it, we use the inverse Lorentz transformation

ω = γ(ω − βckz) = γ(ω − βck cos θ), (B.14) which gives

ω ω = . (B.15) γ(1 − β cos θ)

Assuming a large γ and a small angle θ and using β ≈ 1−1/2γ2 and cos θ = 1−θ2/2, we obtain

2γω ω = . (B.16) 1 + γ2θ2

The radiation in the forward direction (θ = 0) gets a large factor 2γ in the frequency transformation.

B.4 Lorentz Transformation of Fields

The electromagnetic ﬁeld (E, B) is transformed from K to K according to the following equations:

E = E , E⊥ = γ E − v × B , z z ⊥ 1 B = B , B⊥ = γ B + v × E , (B.17) z z ⊥ c2

where E⊥ and B⊥ are the components of the electric and magnetic ﬁelds perpendic- ular to the velocity v: E⊥ = (Ex , Ey), B⊥ = (Bx , By). The electromagnetic potentials (φ/c, A) are transformed exactly as the 4-vector (ct, r):

= , Ax Ax = , Ay Ay v A = γ A + φ , z z c2 φ = γ(φ + v ). Az (B.18) Appendix B: Lorentz Transformations and the Relativistic Doppler Effect 277

B.5 Lorentz Transformation and Photons

It is often convenient, even in classical electrodynamics, to consider electromagnetic radiation as a collection of photons. How do we transform the parameters of a photon from K to K ? The answer is rather evident: the combination (k, ω) constitutes a 4-vector and is transformed according to Eq. (B.9). This is of course in agreement with the fact that the pair (k, ω) is the momentum-energy 4-vector for the photon. The number of photons in K to K is the same — it is a relativistic invariant. Index

A Dipole magnet, 75 Accelerator tune, 80 Dipole radiation, 202 Action, 4 Dispersion function, 98 Action-angle variables, 33 Dynamic aperture, 117 Adiabatic invariant, 53 Adiabatic process, 52 Anharmonicity, 55 E Electromagnetic ﬁeld pressure, 179 Elliptic polarization, 164 B Energy deviation, 98 Beam emittance, 128 Extended phase space, 125 Beta function, 80 Betatron oscillations, 78 Betatron phase, 79 G Gaussian beams, 165 Generalized coordinates, 4 C Generalized momentum, 9 Canonical transformation, 21, 23 Generating functions, 25, 27 Canonically conjugate variables, 9 Circular polarization, 164 H Classical radius, 204 Hamiltonian, 9 Closed orbit, 63 Hamiltonian ﬂow, 36 Closed orbit distortions, 96 Harmonic oscillator, 3, 47 Coherent radiation, 246 Hill’s equation, 79 Compton scattering, 209 Conﬁguration space, 4 Courant-Snyder invariant, 90 I Curvilinear coordinates, 65 Incoherent radiation, 246 Cut-off frequency, 174 Incompressible ﬂow, 123 Cyclotron frequency, 8 Integral of motion, 12 Cylindrical resonator, 176 Inverse Compton scattering, 209 Inverse FEL acceleration, 255

D Damping partition number, 263 K Diffraction radiation, 218 Kinetic equation, 123

© Springer International Publishing AG, part of Springer Nature 2018 279 G. Stupakov and G. Penn, Classical Mechanics and Electromagnetism in Accelerator Physics, Graduate Texts in Physics, https://doi.org/10.1007/978-3-319-90188-6 280 Index

L Resonance overlapping, 112 Lagrangian, 4 Resonant width, 49 Lawson–Woodward theorem, 251 Retarded potentials, 198 Leontovich boundary condition, 153 Retarded time, 192 Liénard–Wiechert potentials, 195 Robinson’s theorem, 267 Light pressure, 209 Linear polarization, 164 Liouville’s theorem, 38 S Long-thin approximation, 142 Separatrix, 54 Longitudinal formation length, 241 Sextupole magnet, 76 Lorentz detuning, 181 Shielding effect, 244 Lorentz factor, 7 Skew quadrupole magnet, 76 Loss factor, 184 Skin depth, 153 Skin effect, 151 Slater’s formula, 182 M Small amplitude approximation, 68 Mathieu equation, 51 Space charge effect, 144 Mathieu functions, 51 Spectrum of radiation, 216 Spectrum of synchrotron radiation, 226 Spherical electromagnetic wave, 202 N Spherical wave, 169 Non-conservative forces, 39 Standard map, 113 Nonlinear oscillator, 53 Surface impedance, 153 Nonlinear resonance, 53 Symplectic map, 36, 37 Synchrotron radiation, 221 P Parametric resonance, 51 T Paraxial approximation, 166 Third-order resonance, 107 Pendulum equation, 4, 53 Thomson cross section, 206 Perfect conductivity, 154 Transition radiation, 213 Phase mixing, 129 Transverse electric (TE) modes, 175 Plane electromagnetic waves, 163 Transverse formation length, 244 Poisson bracket, 12, 23 Transverse magnetic (TM) modes, 173

Q Quadrupole errors, 99 U Quadrupole magnet, 76 Undulator, 231 Quality factor, 48, 178 Undulator parameter, 232

R V Radiation damping time, 263 Vlasov equation, 124 Radiation ﬁeld, 191 Vlasov-Fokker-Planck equation, 131 Radiation reaction force, 204 Random force, 49 Rayleigh dissipation function, 39 W Rayleigh length, 167 Weak focusing, 78 Resonance, 49 Wiggler, 232