1 Vectors and Vector Analysis 2 Electrostatics

1 Vectors and Vector Analysis

• Scalar and Vector Products: ¡ ¢ a · b = axbx + ayby + azbz , a × b = ay bz − az by , az bx − axbz, axby − aybx .

a · b is a scalar, a × b is a pseudovector. • Multiple Products:

a · (b × c) = (a × b) · c , a × (b × c) = b(a · c) − c(a · b) .

• Scalar and vector operators:

¡ ∂ ∂ ∂ ¢ ∂2 ∂2 ∂2 r = ,, , Δ = r · r = + + . ∂x ∂y ∂z ∂x2 ∂y2 ∂z2

• Gradient, divergence, and curl of ﬁelds: ¡ ∂f ∂f ∂f ¢ grad f(r) = rf(r) = ,, , ∂x ∂y ∂z ∂a ∂a ∂a div a(r) = r · a(r) = x +y + z , ∂x ∂y ∂z ¡ ∂a ∂a ∂a ∂a ∂a ∂a ¢ rot a(r) = r × a(r) = z − y , x − z , y − x . ∂y ∂z ∂z ∂x ∂x ∂y

• Second derivatives: ¡ ¢ rot grad f(r) = r × rf(r) = o , ¡ ¢ div rot a(r) = r · r × a(r) = 0 , ¡ ¢ div grad f(r) = r · rf(r) = Δf(r) , ¡ ¢ ¡ ¢ ¡ ¢ rot rot a(r) = r × r × a(r) = r r · a(r) − r × r a(r) = grad div a(r) − Δa(r) .

• Integral theorems of Gauss and Stokes: If Ω is a ﬁnite volume, and ∂Ω its closed surface, I I ZZZ £ ¤ b(r) · da(r) = b(r) · nˆ(r) da(r) = div b(r) d3 r . ∂Ω ∂Ω Ω If A is a ﬁnite surface, and ∂A is its closed boundary, I ZZ £ ¤ b(r) · dr = rot b(r) · nˆ(r) da(r) . ∂A A 2 Electrostatics

• Elementary charge e and dielectric susceptibility ² 0:

e = 1.60217733 · 10−19 As , As ² = 8.854187818 · 10−12 . 0 Vm • Coulomb’s law (point particle, charge q, at position r0):

q r − r0 E(r) = 0 3 . 4π²0 |r − r | 1 • Coulomb’s law (charge distribution ρ(r)):

ZZZ 0 0 1 ρ(r )(r − r ) 3 0 E(r) = 0 3 d r . 4π²0 |r − r |

• Coulomb’s law in restricted geometries (surface charge density σ(r), line charge density λ(r)):

ZZ 0 0 Z 0 0 1 σ(r )(r − r ) 0 1 λ(r )(r − r ) 0 E(r) = 0 3 da(r ) , E(r) = 0 3 ds(r ) . 4π²0 surface |r − r | 4π²0 line |r − r |

• Electric ﬂux φΩ through closed surface ∂Ω of ﬁnite volume Ω: I I £ ¤ φΩ(t) = E(r, t) · da(r) = E(r, t) · nˆ(r) da(r) . ∂Ω ∂Ω

• Gauss’ law (integral and diﬀerential form): I ZZZ 1 3 Q(t) ρ(r, t) φΩ(t) = E(r, t) · da(r) = ρ(r, t)d r = , div E(r, t) = . ∂Ω ²0 Ω ²0 ²0

• Electrostatic potential Φ(r) (point particle, charge q, at position r0): q Φ(r) = 0 . 4π²0|r − r |

• Electrostatic potential Φ(r) (charge distribution ρ(r)):

ZZZ 0 1 ρ(r ) 3 0 Φ(r) = 0 d r . 4π²0 |r − r |

• Electrostatic potential in restricted geometries (surface charge density σ(r), line charge density λ(r)):

ZZ 0 Z 0 1 σ(r ) 0 1 λ(r ) 0 Φ(r) = 0 da(r ) , Φ(r) = 0 ds(r ) . 4π²0 surface |r − r | 4π²0 line |r − r |

• Relation to electric ﬁeld E(r), potential equation: ρ(r) E(r) = − grad Φ(r) , ΔΦ(r) = − . ²0 3 Magnetism

• Magnetic permeability µ0: Vs Vs µ = 4π · 10−7 = 1.256637061 · 10−6 . 0 Am Am

• Current IA through surface A, current density j(r): ZZ

IA(t) = j(r, t) · da(r) , j(r, t) = ρ(r, t)v(r, t) . A

• Continuity equation (conservation of charge, integral and diﬀerential form): If Ω is a ﬁnite volume and ∂Ω its closed surface, then I ZZZ d 3 dQΩ(t) ∂ρ(r, t) I∂Ω(t) = j(r, t) · da(r) = − ρ(r, t)d r = − , + div j(r, t) = 0 . ∂Ω dt Ω dt ∂t 2 • Electromagnetic force (point particle, charge q, velocity v):

F(t) = qE(r, t) + qv(r, t) × B(r, t) .

• Electromagnetic force density (charge and current distribution):

f(r, t) = ρ(r, t)E(r, t) + j(r, t) × B(r, t) .

• Motion in uniform ﬁelds (cyclotron frequency ωC , drift velocity vD): qB 1 ω = , v = (E × B) . C m D B2

• Law of Biot-Savart (static current distribution j(r)): ZZZ µ j(r0) × (r − r0) B(r) = 0 d3 r0 . 4π |r − r0|3

• Law of Biot-Savart (static current I in a wire loop, local direction of current ˆt(r)): I µ I ˆt(r0) × (r − r0) B(r) = 0 ds(r0 0 3 ) . 4π loop |r − r |

• Vector potential (static currrent density distribution): ZZZ µ j(r0) A(r) = 0 d3 r0 . 4π |r − r0|

• Relation to magnetic ﬁeld B(r), potential equation (magnetostatics):

B(r) = rot A(r) , ΔA(r) = −µ0j(r) .

• Magnetic ﬂux through closed surface ∂Ω: I I £ ¤ B(r, t) · da(r) = B(r, t) · nˆ(r) da(r) = 0 , div B(r, t) = 0 . ∂Ω ∂Ω

• Amp`ere’s law (integral and diﬀerential forms): For a closed loop ∂A enclosing a surface A, I ZZ ZZ d B(r, t) · dr = µ0 j(r, t) · da(r) + ² 0µ0 E(r, t) · da(r) , ∂A A dt A ∂E(r, t) rot B(r, t) = µ j(r, t) + ² µ . 0 0 0 ∂t 4 Electromagnetic Induction

• Faraday’s law (integral and diﬀerential forms): For a closed loop ∂A enclosing a surface A, I ZZ d ∂B(r, t) E(r, t) · dr = − B(r, t) · da(r) , rot E(r, t) = − . ∂A dt A ∂t

3 5 Maxwell’s Equations, Electromagnetic Waves

• Maxwell’s laws (diﬀerential form):

ρ(r, t) ∂B(r, t) div E(r, t) = , rot E(r, t) = − , ²0 ∂t ∂E(r, t) div B(r, t) = 0 , rot B(r, t) = µ j(r, t) + ² µ . 0 0 0 ∂t • Plane waves in a vector ﬁeld: (wave vector k, angular frequency ω, velocity c, amplitude u):

u(r, t) = u0 cos(k · r − ωt + φ) , 2π ω λ = , f = , ω = c|k| . |k| 2π

• Speed of light in vacuum: 1 8 m c0 = √ = 2.99792458 · 10 . ²0 µ0 s

• Electromagnetic waves (linearly polarized, in vacuum):

E(r, t) = E0 cos(k · r − ωt + φ) , B(r, t) = B0 cos(k · r − ωt + φ) ,

1 |E(r, t)| E0 ⊥ k , B0 = (k × E0) , |B(r, t)| = . ω c0 • Energy density in the electromagnetic ﬁeld: ² 1 u(r, t) = 0 E(r, t)2 + B(r, t)2 . 2 2µ0

• Energy ﬂux density in the electromagnetic ﬁeld (Poynting vector): 1 £ ¤ P(r, t) = E(r, t) × B(r, t) . µ0

• Energy conservation in the electromagnetic ﬁeld (continuity equation):

∂u(r, t) + div P(r, t) = −j(r, t) · E(r, t) . ∂t 6 Potentials

• Representation of ﬁelds through potentials:

∂A(r, t) E(r, t) = − grad Φ(r, t) − , B(r, t) = rot A(r, t) . ∂t

• Gauge transformations: ¡ ¢ For an arbitrary gauge ﬁeld χ(r, t), the potentials Φ(r, t), A(r, t) and:

∂χ(r, t) Φ0(r, t) = Φ(r, t) − , A0(r, t) = A(r, t) + grad χ(r, t) , ∂t represent the same ﬁelds E(r, t), B(r, t).

4 • Field equations for the potentials: µ ¶ µ ¶ 1 ∂2Φ(r, t) ∂ 1 ∂Φ(r, t) ρ(r, t) 2 2 − ΔΦ(r, t) − 2 + div A(r, t) = , c0 ∂t ∂t c0 ∂t ² 0 µ ¶ µ ¶ 1 ∂2A(r, t) 1 ∂Φ(r, t) 2 2 − ΔA(r, t) + grad 2 + div A(r, t) = µ0j(r, t) . c0 ∂t c0 ∂t • Four-vector notation: µ Covariant vectors A and contravariant vectors Aµ (µ = 0, 1, 2, 3):

0 1 2 3 0 0 (A ,A ,A ,A ) = (A , A) = (A ,Ax,Ay ,Az) ,

0 0 (A0,A1,A2,A3) = (A , −A) = (A , −Ax, −Ay, −Az) . Contraction (inner product) of two four-vectors Aµ, Bµ:

µ 0 1 2 3 0 0 1 1 2 2 3 3 0 0 Aµ B = A0B + A1B + A2B + A3B = A B − A B − A B − A B = A B − A · B .

Invariant length s(A) of a four-vector Aµ:

µ 0 2 1 2 2 2 3 2 0 2 2 s(A) = AµA = (A ) − (A ) − (A ) − (A ) = (A ) − A .

µ Four-gradients ∂ , ∂µ: µ ¶ µ ¶ ¡ ¢ 1 ∂ 1 ∂ ∂ ∂ ∂ ∂0, ∂1, ∂2, ∂3 = , −r = , − , − , − , c0 ∂t c0 ∂t ∂x ∂y ∂z µ ¶ µ ¶ ¡ ¢ 1 ∂ 1 ∂ ∂ ∂ ∂ ∂0, ∂1, ∂2, ∂3 = , r = , , , , c0 ∂t c0 ∂t ∂x ∂y ∂z Four-divergence: 0 1 2 3 0 µ 1 ∂A ∂A ∂A ∂A 1 ∂A ∂µ A = + + + = + r · A . c0 ∂t ∂x ∂y ∂z c0 ∂t D’Alembert operator ¤ and wave equation:

2 µ 1 ∂ ¤ = ∂µ∂ = 2 2 − Δ , c0 ∂t

µ 2 ¶ µ 1 ∂ ¤ψ(r, t) = ∂µ∂ ψ(r, t) = 2 2 − Δ ψ(r, t) = 0 . c0 ∂t • Potential and current four-vectors:

µ ¡ Φ(r, t) ¢ µ ¡ ¢ A (r, t) = , A(r, t) , j (r, t) = c0ρ(r, t), j(r, t) . c0

• Continuity equation for charge (four-vector form):

µ ∂µj (r, t) = 0 .

• Gauge transformation of potential (four-vector form):

(A0)µ(r, t) = Aµ(r, t) − ∂µχ(r, t) .

• Field equations (covariant form): For each ν = 0, 1, 2, 3,

¡ µ¢ ν ν ¡ µ ¢ ν ∂µ∂ A (r, t) − ∂ ∂µA (r, t) = µ0j (r, t) . 5 • Field tensor:

µν νµ µ ν ν µ µν ν F (r, t) = −F (r, t) = ∂ A (r, t) − ∂ A (r, t) , ∂µF (r, t) = µ0j (r, t) .

• Lorentz gauge and wave equation for potentials:

µ µ ν ν ∂µA (r, t) = 0 , ∂µ∂ A (r, t) = µ0j (r, t) .

• Retarded (−) and advanced (+) solutions of wave equations:

ZZZ 3 0 µ 0 ¶ µ µ0 d r µ 0 |r − r | A (r, t) = 0 j r , t ± . 4π |r − r | c0

• Far ﬁelds of moving particle 0 (charge q, distance R = |R| = |r − r |, acceleration at retarded time a = a(t − R/c0)): µ q eˆ × (eˆ × a) µ q eˆ × a E(r, t) ∼ 0 R R , B(r, t) ∼ − 0 R . 4π R 4πc0 R

• Energy ﬂux distribution in the far ﬁeld limit (angle θ between R and a):

µ q2 a2 sin2 θ 0 ˆ P(r, t) ∼ 2 2 eR . 16π R c0

• Total radiated power (Larmor formula):

µ q2 a 2 P (t) = 0 . 6πc0 7 Special Relativity

• Space-time four-vector xµ:

µ ¡ 0 1 2 3¢ ¡ ¢ ¡ ¢ x = x , x , x , x = c0t, x, y, z = c0t, r .

• Invariant distance s between events:

µ 2 2 2 s = (Δx)µ(Δx) = c 0 (Δt) − (Δr) .

• The abbreviations β and γ: v 1 1 β = , γ =p = p . c 2 2 2 0 1 − β 1 − v /c0

• Time dilation: If Δτ is the “proper” time interval in the stationary frame (Δr = o), then the time interval Δt0 in a moving frame is: Δτ Δt 0 = γΔτ = p . 2 2 1 − v /c0

• Length contraction: If Δx 0 is the “proper” length of a moving yardstick (with ends marked synchro- nously, Δt0 = 0), then the length Δx of the yardstick in its rest frame is given by:

Δx0 Δx = γΔx 0 = p . 2 2 1 − v /c0 6 • Lorentz¡ transformation¢ (for space-time events): If c0Δt, Δr is the separation of two space-time events for one observer, then another observer¡ moving¢ 0 0 with uniform relative velocity v in x direction (Lorentz boost) will ﬁnd the separation c0Δt , Δr :

0 0 0 0 c0Δt = γ (c0Δt + βΔx) , Δx = γ (Δx + vΔt) , Δy = Δy , Δz = Δz .

• Lorentz transformation¡ ¢ (for general four-vectors): µ 0 If A = A , A is a four-vector, it will transform under a change into a ¡frame moving¢ at uniform relative velocity v in x direction (Lorentz boost) to the four-vector (A0)µ = (A0)0 , A0 : ¡ ¢ ¡ ¢ (A0)0 = γ A0 + βA1 , (A1)0 = γ A1 + βA0 , (A2)0 = A2 , (A3)0 = A3 .

• Relativistic velocity addition: 0 If u = (ux, uy, uz ) is the velocity of an object in the “unprimed” frame, then the velocity u of the same object observed in a frame moving at uniform relative velocity v in x direction (Lorentz boost) is: 0 ux + v 0 uy 0 uz ux = 2 , u y = 2 , u z = 2 . 1 + uxv/c0 γ(1 + uxv/c0 ) γ(1 + uxv/c0 ) • Relativistic transformation of angles: If an object moves with velocity u under an angle θ with respect to the x axis, then the angle θ0 between the direction of motion and the x0 axis in a frame moving at uniform relative velocity v in x direction (Lorentz boost) is: u sin θ tan θ0 = . γ(v + u cos θ)

• Relativistic Doppler eﬀect: If f is the “natural” frequency of a source in its rest frame, then the frequency f 0 seen by an observer receding from the source at a speed v is redshifted: r s c − v 1 − β f 0 = 0 f = f . c0 + v 1 + β

(If the observer approaches the source (“blue shift”), replace v by −v.)

• Wave four-vector kµ, covariant form of plane wave:

µ ¡ ω ¢ µ µ µ k = , k , kµ k = 0 , u(x ) = u0 cos(kµx ) . c0

• Lorentz transformation for frequency and wave vector (boost in x direction): µ ¶ ω0 ω ω = γ + βkx = γ (1 + β cos θ) , c0 c0 c0 µ ¶ 0 ω kx = γ kx + β = γ (1 + β cos θ) kx , c0 0 0 ky = ky , kz = kz .

• The velocity four-vector uµ:

µ ¡ 0 ¢ 0 1 v/c0 µ u = u , u , u = γ = p , u = βγ = p , uµu = 1 . 2 2 2 2 1 − v /c0 1 − v /c0

7 • Relativistic mass increase: The mass m0 of an object as seen from a moving observer is increased with respect to the mass m in its rest frame by: m m 0 = γm = p . 2 2 1 − v /c0 • Relativistic energy and momentum:

mc2 mv E = m0 c 2 = γmc2 = p , p = m 0v = γmv = p . 0 0 2 2 2 2 1 − v /c0 1 − v /c0

• Energy-momentum four-vector pµ:

µ µ ¡ E ¢ 2¡ µ¢ 2 2 2 2 p = mc0 u = , p , c0 pµp = E − (c0p) = (mc0 ) . c0

• Lorentz transformation for energy and momentum (Lorentz boost in x direction):

0 ¡ ¢ 0 ¡ E ¢ 0 0 E = γ E + βc0px , p x = γ px + β , py = py , pz = pz . c0 8 Elementary Particle Physics