8.07 – Fórmulas básicas a · b = ai bi , ijk pqk = δip δjq − δiq δjp , (a × b)i = ijk aj bk det A = i1 i2 ···in A1,i1 A2,i2 · · · An,in F · da , F · dl ∇ · F = ∇ × F = V S Γ S =0 ∇ × (∇φ) = 0, ∇ · (∇ × A) δ(x − xi ) , g(xi ) = 0 δ(g(x)) = (x )| |g i i ∇r = er ∇2 1 = −4πδ(x) r 1 ∂Φ 1 ∂Φ ∂Φ er + eθ + eφ ∂r r ∂θ r sin θ ∂φ ∂2Φ 1 ∂ ∂Φ 1 1 ∂ 2 ∂Φ 2 r + 2 sin θ + 2 2 ∇ Φ= 2 r ∂r ∂r r sin θ ∂θ ∂θ r sin θ ∂φ2 ∇Φ = |E(r)| = 1 q 4π0 r2 V (r) = 1 q 4π0 r ρ · da = Qencl , = −∇V E E ∇2 V = − 0 0 S x)ρ(x ) 1 qi qj 1 0 2 3 3 ρ( W = → → w = |E| d xd x 8π0 |xi − xj | 8π0 |xi − xj | 2 = ρ, ∇·E 0 i=j V (x) = 1 4π0 Q = CV → Qi = Cij Vj 1 ∂V ∂G(x, x ) 3 G(x, x ) − V (x ) da d x G(x, x )ρ(x ) + 4π S ∂n ∂n ∇2x G(x, x ) = −4πδ(x − x ) G(x, x ) = “potencial” en x debido a una unidad de carga en x , cuando ... a Imagen fijada para esfera: (q , y), (q , y ) con y y = a2 , q = − q y cos αx cos βy cosh γz una solución cartesiana de Laplace γ 2 = α2 + β 2 sin αx sin βy sinh γz {rl , r−(l+1) } Pl (cos θ) soluciones acimutales de Laplace ∞ ∞ rl 1 < P (cos γ) , = l+1 l |x − x | r> 1 √ = hl Pl (x) 1 − 2hx + h2 l=0 l=0 1 1 l Pl (1) = 1 Pl (−x) = (−1) Pl (x) Si1 ···in = (det R) Ri1 j1 · · · Rin jn Sji ···jn −1 dx Pl (x)Pl (x) = 2 δll 2l + 1 (pseudo) tensor bajo rotación p · x 1 xi xj 1 Q + Q + · · · + ij 4π0 |x| |x|3 2 |x|5 3 pi = d x ρ(x) xi Qij = d3 x ρ(x)(3xi xj − δij |x|2 ) , V (x) = Q= dip = E d3 x ρ(x) , 1 1 (3( p · r̂)r̂ − p ) 4π0 r3 , da ⇔ I dl J d3 x ⇔ K → dip = E J = ρv , 1 p (2 cos θ r̂ + sin θ θ̂) , 4π0 r3 p = pẑ d p = q v × B dt ∇ · J = 0, × (x − x ) µ0 I dl = µ0 I ∇Ω , B 3 4π |x − x | 4π x ) µ 0 3 J( = 0, B = ∇ × A, A(x) = d x , ∇·A 4π |x − x | · dl = µ0 Ienlazado , B ∇·B =0 ∇ × B = µ0 J , x) = dB( Γ × x = µ0 m , A 4π |x|3 · x = − µ0 ∇ m B , 4π |x|3 1 m = 2 x) , d3 x x × J( dip = µ0 1 (3(m dip = µ0 m (2 cos θ r̂ + sin θ θ̂) , B · r̂)r̂ − m ) → B 3 4π r 4π r3 d · da E = (E + v × B) · dl = − B dt Γ S 1 · J d3 x ·B d3 x = 1 A B W = 2µ0 2 µ0 Mij = 4π Ci Cj dEmec dt i · dl j dl 1 , ij (x) · da = B |xi − xj | Ij Si Mij = Mji = µ0 J + µ0 0 ∂ E ∇×B ∂t ×B , = 1E d3 x J · E S = µ 0 V 2 m = IA m = mẑ dEmec dt d + dt 1 · B) =− d x (E ·D+H 2 + J × B) d3 x (ρE = · da S 3 V dPmec dt S V 1 2 Tαβ = 0 Eα Eβ + c Bα Bβ − (E · E + c B · B)δαβ 2 dP d mec 3 d x 0 (E × B) = + Tij nj da + dt dt V i S 2 =∇×A = −∇Φ − ∂A , B E ∂t | x− x | x , t − |x−x | ρ x , t − J µ 1 0 c c x, t) = , A( d3 x d3 x Φ(x, t) = 4π0 |x − x | 4π |x − x | x, t) = Ee i(k·x−ωt) E( x, t) = Be i(k·x−ωt) B( = √µ n × E × n , B E = Z H = 0, n · E = n · B Z= µ/ ∗ ) ◦ B = 1 Re(E ◦ B E 2 x) x) = 1 ∇ × A( H( µ0 Radiación : e−iωt dP c2 Z0 4 k |(n × p) × n|2 , = dΩ 32π 2 ikr = − iµ0 ω p e , A 4π r radiación dipolar magnética dP dP = ( p → m/c) , dΩ dΩ = 0 E + P , D =ρ ∇·D ρpol = −∇ · P , , = B −M H µ0 = J , ∇×H , Jb = ∇ × M = −∇ΦM H → x) = iZ0 ∇ × H( x) E( k P = c2 Z0 4 2 p| k | 12π P = P ( p → m/c) = E material lineal : D σpol = n · P = µH material lineal : B b = M × n K ∇2 ΦM = −ρM , 3 , ρM = −∇ · M · n σM = M