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1 Vectors and Vector Analysis 2 Electrostatics 1 Vectors and Vector Analysis ² 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. ² Multiple Products: a ¢ (b £ c) = (a £ b) ¢ c ; a £ (b £ c) = b(a ¢ c) ¡ c(a ¢ b) : ² Scalar and vector operators: 2 2 2 ¡ @@@ ¢ @ @ @ r = ;; ; ¢ = r ¢ r = 2 +2 + 2 : @x @y @z @x @y @z ² Gradient, divergence, and curl of ¯elds: ¡ @f @f @f ¢ grad f(r) = rf(r) = ;; ; @x @y @z @ax @ay @az div a(r) = r ¢ a(r) = + + ; @x @y @z ¡ @az @ay @ax @az @ay @ax ¢ rot a(r) = r £ a(r) = ¡ ; ¡ ; ¡ : @y @z @z @x @x @y ² 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) = ¢f(r) ; ¡ ¢ ¡ ¢ ¡ ¢ rot rot a(r) = r £ r £ a(r) = r r ¢ a(r) ¡ r £ r a(r) = grad div a(r) ¡ ¢a(r) : ² Integral theorems of Gauss and Stokes: If ­ is a ¯nite volume, and @­ its closed surface, I I ZZZ £ ¤ 3 b(r) ¢ da(r) = b(r) ¢ n^(r) da(r) = div b(r) d r : @­ @­ ­ If A is a ¯nite surface, and @A is its closed boundary, I ZZ £ ¤ b(r) ¢ dr = rot b(r) ¢ n^(r) da(r) : @A A 2 Electrostatics ² Elementary charge e and dielectric susceptibility ²0 : ¡19 e = 1:60217733 ¢ 10 As ; ¡12 As ²0 = 8:854187818 ¢ 10 : Vm 0 ² Coulomb's law (point particle, charge q, at position r ): 0 q r ¡ r E(r) = 0 3 : 4¼²0 jr ¡ r j 1 ² Coulomb's law (charge distribution ½(r)): ZZZ 0 0 1 ½(r )(r ¡ r ) 3 0 E(r) = 0 3 d r : 4¼²0 jr ¡ r j ² Coulomb's law in restricted geometries (surface charge density σ(r), line charge density ¸(r)): ZZ 0 0 Z 0 0 1 σ(r )(r ¡ r ) 0 1 ¸(r )(r ¡ r ) 0 E(r) = 0 3 da(r ) ; E(r) = 0 3 ds(r ) : 4¼²0 surface jr ¡ r j 4¼²0 line jr ¡ r j ² Electric flux Á­ through closed surface @­ of ¯nite volume ­: I I £ ¤ Á­(t) = E(r; t) ¢ da(r) = E(r; t) ¢ n^(r) da(r) : @­ @­ ² Gauss' law (integral and di®erential form): I ZZZ 1 3 Q(t) ½(r; t) Á­(t) = E(r; t) ¢ da(r) = ½(r; t)d r = ; div E(r; t) = : @­ ²0 ­ ²0 ²0 ² Electrostatic potential ©(r) (point particle, charge q, at position r0): q ©(r) = 0 : 4¼²0jr ¡ r j ² Electrostatic potential ©(r) (charge distribution ½(r)): ZZZ 0 1 ½(r ) 3 0 ©(r) = 0 d r : 4¼²0 jr ¡ r j ² Electrostatic potential in restricted geometries (surface charge density σ(r), line charge density ¸(r)): ZZ 0 Z 0 1 σ(r ) 0 1 ¸(r ) 0 ©(r) = 0 da(r ) ; ©(r) = 0 ds(r ) : 4¼²0 surface jr ¡ r j 4¼²0 line jr ¡ r j ² Relation to electric ¯eld E(r), potential equation: ½(r) E(r) = ¡ grad ©(r) ; ¢©(r) = ¡ : ²0 3 Magnetism ² Magnetic permeability ¹0: ¡7 Vs ¡6 Vs ¹0 = 4¼ ¢ 10 = 1:256637061 ¢ 10 : Am Am ² Current IA through surface A, current density j(r): ZZ IA(t) = j(r; t) ¢ da(r) ; j(r; t) = ½(r; t)v(r; t) : A ² Continuity equation (conservation of charge, integral and di®erential form): If ­ is a ¯nite volume and @­ its closed surface, then I ZZZ d 3 dQ­(t) @½(r; t) I@­(t) = j(r; t) ¢ da(r) = ¡ ½(r; t)d r = ¡ ; + div j(r; t) = 0 : @­ dt ­ dt @t 2 ² Electromagnetic force (point particle, charge q, velocity v): F(t) = qE(r; t) + qv(r; t) £ B(r; t) : ² Electromagnetic force density (charge and current distribution): f(r; t) = ½(r; t)E(r; t) + j(r; t) £ B(r; t) : ² Motion in uniform ¯elds (cyclotron frequency !C , drift velocity vD): qB 1 ! = ; v = (E £ B) : C m D B2 ² Law of Biot-Savart (static current distribution j(r)): ZZZ ¹ j(r0) £ (r ¡ r0) B(r) = 0 d3 r0 : 4¼ jr ¡ r0j3 ² Law of Biot-Savart (static current I in a wire loop, local direction of current ^t(r)): I ¹ I ^t(r0) £ (r ¡ r0) B(r) = 0 ds(r0 0 3 ) : 4¼ loop jr ¡ r j ² Vector potential (static currrent density distribution): ZZZ 0 ¹0 j(r ) 3 0 A(r) = 0 d r : 4¼ jr ¡ r j ² Relation to magnetic ¯eld B(r), potential equation (magnetostatics): B(r) = rot A(r) ; ¢A(r) = ¡¹0j(r) : ² Magnetic flux through closed surface @­: I I £ ¤ B(r; t) ¢ da(r) = B(r; t) ¢ n^(r) da(r) = 0 ; div B(r; t) = 0 : @­ @­ ² Amp`ere's law (integral and di®erential forms): For a closed loop @A enclosing a surface A, I ZZ ZZ d B(r; t) ¢ dr = ¹0 j(r; t) ¢ da(r) + ² 0¹0 E(r; t) ¢ da(r) ; @A A dt A @E(r; t) rot B(r; t) = ¹0j(r; t) + ² 0¹0 : @t 4 Electromagnetic Induction ² Faraday's law (integral and di®erential 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 ² Maxwell's laws (di®erential form): ½(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) = ¹0j(r; t) + ²0 ¹0 : @t ² Plane waves in a vector ¯eld: (wave vector k, angular frequency !, velocity c, amplitude u): u(r; t) = u0 cos(k ¢ r ¡ !t + Á) ; 2¼ ! ¸ = ; f = ;! = cjkj : jkj 2¼ ² Speed of light in vacuum: 1 8 m c0 = p = 2:99792458 ¢ 10 : ²0 ¹0 s ² Electromagnetic waves (linearly polarized, in vacuum): E(r; t) = E0 cos(k ¢ r ¡ !t + Á) ; B(r; t) = B0 cos(k ¢ r ¡ !t + Á) ; 1 jE(r; t)j E0 ? k ; B0 = (k £ E0) ; jB(r; t)j = : ! c0 ² Energy density in the electromagnetic ¯eld: ² 0 2 1 2 u(r; t) = E(r; t) + B(r; t) : 2 2¹0 ² Energy flux density in the electromagnetic ¯eld (Poynting vector): 1 £ ¤ P(r; t) = E(r; t) £ B(r; t) : ¹0 ² Energy conservation in the electromagnetic ¯eld (continuity equation): @u(r; t) + div P(r; t) = ¡j(r; t) ¢ E(r; t) : @t 6 Potentials ² Representation of ¯elds through potentials: @A(r; t) E(r; t) = ¡ grad ©(r; t) ¡ ; B(r; t) = rot A(r; t) : @t ² Gauge transformations: ¡ ¢ For an arbitrary gauge ¯eld Â(r; t), the potentials ©(r; t); A(r; t) and: @Â(r; t) ©0(r; t) = ©(r; t) ¡ ; A0(r; t) = A(r; t) + grad Â(r; t) ; @t represent the same ¯elds E(r; t), B(r; t). 4 ² Field equations for the potentials: µ 2 ¶ µ ¶ 1 @ ©(r; t) @ 1 @©(r; t) ½(r; t) 2 2 ¡ ¢©(r; t) ¡ 2 + div A(r; t) = ; c0 @t @t c0 @t ² 0 µ 2 ¶ µ ¶ 1 @ A(r; t) 1 @©(r; t) 2 2 ¡ ¢A(r; t) + grad 2 + div A(r; t) = ¹0j(r; t) : c0 @t c0 @t ² Four-vector notation: ¹ Covariant vectors A and contravariant vectors A¹ (¹ = 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) : ¹ ¹ Contraction (inner product) of two four-vectors A , B : ¹ 0 1 2 3 0 0 1 1 2 2 3 3 0 0 A¹ B = A0B + A1B + A2B + A3B = A B ¡ A B ¡ A B ¡ A B = A B ¡ A ¢ B : ¹ Invariant length s(A) of a four-vector A : ¹ 0 2 1 2 2 2 3 2 0 2 2 s(A) = A¹A = (A ) ¡ (A ) ¡ (A ) ¡ (A ) = (A ) ¡ A : ¹ Four-gradients @ , @¹: µ ¶ µ ¶ ¡ 0 1 2 3¢ 1 @ 1 @ @ @ @ @ ;@ ;@ ;@ = ; ¡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 ¹ 1 @A @A @A @A 1 @A @¹ A = + + + = + r ¢ A : c0 @t @x @y @z c0 @t D'Alembert operator ¤ and wave equation: 2 ¹ 1 @ ¤ = @¹@ = 2 2 ¡ ¢ ; c0 @t µ 2 ¶ ¹ 1 @ ¤Ã(r; t) = @¹@ Ã(r; t) = 2 2 ¡ ¢ Ã(r; t) = 0 : c0 @t ² Potential and current four-vectors: ¹ ¡ ©(r; t) ¢ ¹ ¡ ¢ A (r; t) = ; A(r; t) ; j (r; t) = c0½(r; t); j(r; t) : c0 ² Continuity equation for charge (four-vector form): ¹ @¹j (r; t) = 0 : ² Gauge transformation of potential (four-vector form): 0 ¹ ¹ ¹ (A ) (r; t) = A (r; t) ¡ @ Â(r; t) : ² Field equations (covariant form): For each º = 0; 1; 2; 3, ¡ ¹¢ º º ¡ ¹ ¢ º @¹@ A (r; t) ¡ @ @¹A (r; t) = ¹0j (r; t) : 5 ² Field tensor: ¹º º¹ ¹ º º ¹ ¹º º F (r; t) = ¡F (r; t) = @ A (r; t) ¡ @ A (r; t) ;@¹F (r; t) = ¹0j (r; t) : ² Lorentz gauge and wave equation for potentials: ¹ ¹ º º @¹A (r; t) = 0 ;@¹@ A (r; t) = ¹0j (r; t) : ² Retarded (¡) and advanced (+) solutions of wave equations: ZZZ 3 0 µ 0 ¶ ¹ ¹0 d r ¹ 0 jr ¡ r j A (r; t) = 0 j r ; t § : 4¼ jr ¡ r j c0 ² Far ¯elds of moving particle 0 (charge q, distance R = jRj = jr ¡ r j, acceleration at retarded time a = a(t ¡ R=c0)): ¹0q e^R £ (e^R £ a) ¹0q e^R £ a E(r; t) » ; B(r; t) »¡ : 4¼ R 4¼c0 R ² Energy flux distribution in the far ¯eld limit (angle θ between R and a): ¹ q2 a2 sin2 θ 0 ^ P(r; t) » 2 2 eR : 16¼ R c0 ² Total radiated power (Larmor formula): 2 2 ¹0q a P (t) = : 6¼c0 7 Special Relativity ¹ ² Space-time four-vector x : ¹ ¡ 0 1 2 3¢ ¡ ¢ ¡ ¢ x = x ; x ; x ; x = c0t; x; y; z = c0t; r : ² Invariant distance s between events: ¹ 2 2 2 s = (¢x)¹(¢x) = c 0 (¢t) ¡ (¢r) : ² The abbreviations ¯ and γ: v 1 1 ¯ = ; γ =p = p : c 2 2 2 0 1 ¡ ¯ 1 ¡ v =c0 ² Time dilation: If ¢¿ is the \proper" time interval in the stationary frame (¢r = o), then the time 0 interval ¢t in a moving frame is: 0 ¢¿ ¢t = γ¢¿ = p : 2 2 1 ¡ v =c0 0 ² Length contraction: If ¢x is the \proper" length of a moving yardstick (with ends marked synchro- 0 nously, ¢t = 0), then the length ¢x of the yardstick in its rest frame is given by: 0 0 ¢x ¢x = γ¢x = p : 2 2 1 ¡ v =c0 6 ² Lorentz¡ transformation¢ (for space-time events): If c0¢t; ¢r 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 ¯nd the separation c0¢t ; ¢r : 0 0 0 0 c0¢t = γ (c0¢t + ¯¢x) ; ¢x = γ (¢x + v¢t) ; ¢y = ¢y ; ¢z = ¢z : ² Lorentz transformation (for general four-vectors): ¡ ¢ ¹ 0 If A = A ; A is a four-vector, it will transform under a change into a ¡frame moving¢ at uniform 0 ¹ 0 0 0 relative velocity v in x direction (Lorentz boost) to the four-vector (A ) = (A ) ; A : 0 0 ¡ 0 1¢ 1 0 ¡ 1 0¢ 2 0 2 3 0 3 (A ) = γ A + ¯A ; (A ) = γ A + ¯A ; (A ) = A ; (A ) = A : ² 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 ) ² Relativistic transformation of angles: 0 If an object moves with velocity u under an angle θ with respect to the x axis,
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