MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Department Physics 8.07: Electromagnetism II November 5, 207 Prof. Alan Guth FORMULA SHEET FOR QUIZ 2 Exam Date: November 8, 207 A few items below are marked with asterisks,. The asterisks indicate that you won t need this material for the quiz, and need not understand it. It is included, however, for completeness, and because some people might want to make use of it to solve problems by methods other than the intended ones. Index Notation: Unit Vectors: ˆx î ê, ŷ ĵ ê 2, ẑ ˆk ê 3, A A i ê i A B = A i B i, A B i = ɛ ijk A j B k, ɛ ijk ɛ pqk = δ ip δ jq δ iq δ jp Rotation of a Vector: det A = ɛ i i 2 i n A,i A 2,i2 A n,in A i = R ij A j, Orthogonality: R ij R ik = δ jk R T T = I Rotation about z-axis by φ: R z φ ij = Rotation about axis ˆn by φ: j= j=2 j=3 i= cos φ sin φ 0 i=2 sin φ cos φ 0 i=3 0 0 Rˆn, φ ij = δ ij cos φ + ˆn iˆn j cos φ ɛ ijk ˆn k sin φ. Vector Calculus: Gradient: ϕ i = i ϕ = ϕ ϕ ˆx + x y ŷ + ϕ z ẑ, i x i Divergence: Curl: A i A i = A x x + A y y + A z z A i = ɛ ijk j A k Az A = y A y z Ax ˆx + z Laplacian: 2 ϕ = ϕ = 2 ϕ = 2 ϕ x i x i x 2 + 2 ϕ y 2 + 2 ϕ z 2 A z Ay ŷ + x x A x ẑ y
8.07 FORMULA SHEET FOR QUIZ 2, FALL 207 p. 2 Fundamental Theorems of Vector Calculus: Gradient: Divergence: Curl: b ϕ d l = ϕ b ϕ a a A d 3 x = A d a V S where S is the boundary of V A d a = A d l S P where P is the boundary of S Vector Identities: Triple Products: A B C = B C A = C A B A B C = B A C C A B Product Rules: fg = f g + g f A B = A B + B A + A B + B A f A = f A + A f A B = B A A B f A = f A A f A B = B A A B + A B B A Second Derivatives: A = 0 f = 0 A = A 2 A Spherical Coordinates: x = r sin θ cos φ r = x 2 + y 2 + z 2 y = r sin θ sin φ θ = tan x2 + y 2 /z z = r cos θ ˆr = sin θ cos φ ˆx + sin θ sin φ ŷ + cos θ ẑ ˆθ = cos θ cos φ ˆx + cos θ sin φ ŷ sin θ ẑ ˆφ = sin φ ˆx + cos φ ŷ φ = tan y/x ˆx = sin θ cos φ ˆr + cos θ cos φ ˆθ sin φ ˆφ ŷ = sin θ sin φ ˆr + cos θ sin φ ˆθ + cos φ ˆφ ẑ = cos θ ˆr sin θ ˆθ
8.07 FORMULA SHEET FOR QUIZ 2, FALL 207 p. 3 Point separation: Volume element: d l = dr ˆr + r dθ ˆθ + r sin θ dφ ˆφ d 3 x r 2 sin θ dr dθ dφ Gradient: ϕ = ϕ r ˆr + ϕ r θ ˆθ + ϕ r sin θ φ ˆφ Divergence: A = r 2 r 2 A r + r r sin θ θ sin θa θ + A φ r sin θ φ Curl: A = [ r sin θ θ sin θa φ A ] θ ˆr φ + [ A r r sin θ φ ] r ra φ ˆθ + [ r r ra θ A ] r ˆφ θ Laplacian: 2 ϕ = r 2 r 2 ϕ + r r r 2 sin θ ϕ 2 ϕ + sin θ θ θ r 2 sin 2 θ φ 2 Cylindrical Coordinates: x = s cos φ s = x 2 + y 2 y = s sin φ z = z ŝ = cos φ ˆx + sin φ ŷ ˆφ = sin φ ˆx + cos φ ŷ ẑ = ẑ Point separation: Volume element: Gradient: φ = tan y/x z = z ˆx = cos φ ŝ sin φ ˆφ ŷ = sin φ ŝ + cos φ ˆφ ẑ = ẑ d l = ds ŝ + s dφ ˆφ + dz ẑ d 3 x s ds dφ dz ϕ = ϕ s ŝ + ϕ s φ ˆφ + ϕ z ẑ Divergence: A = s s sa s + A φ s φ + A z z [ A z Curl: A = s φ A ] [ φ As ŝ + z z A ] z ˆφ s + [ s s sa φ A ] s ẑ φ Laplacian: 2 ϕ = s ϕ + 2 ϕ s s s s 2 φ 2 + 2 ϕ z 2
8.07 FORMULA SHEET FOR QUIZ 2, FALL 207 p. 4 Delta Functions: ϕxδx x dx = ϕx, ϕ rδ 3 r r d 3 x = ϕ r ϕx d dx δx x dx = dϕ dx x=x δgx = δx x i g, where i is summed over all points for which gx i = 0 x i i 2 r r r r = r r = 4πδ 3 r r 3 i j r Electrostatics: = i ˆrj r 2 i xj = i r 3 = 3ˆr iˆr j δ ij r 3 4π 3 δ ij δ 3 r F = q E, where E r = r r q i r r 3 = r r r r 3 ρ r d 3 x ɛ 0 =permittivity of free space = 8.854 0 2 C 2 /N m 2 = 8.988 0 9 N m 2 /C 2 r V r = V r 0 Energy: r 0 E r d l = E = ρ ɛ 0, E = 0, E = V ρ r r r d3 x 2 V = ρ ɛ 0 Poisson s Eq., ρ = 0 = 2 V = 0 Laplace s Eq. Laplacian Mean Value Theorem no generally accepted name: If 2 V = 0, then the average value of V on a spherical surface equals its value at the center. W = 2 W = 2 Conductors: ij i j q i q j r ij = 2 d 3 xρ rv r = 2 ɛ 0 E 2 d 3 x Just outside, E = σ ɛ 0 ˆn Pressure on surface: 2 σ E outside d 3 x d 3 x ρ rρ r r r
8.07 FORMULA SHEET FOR QUIZ 2, FALL 207 p. 5 Two-conductor system with charges Q and Q: Q = CV, W = 2 CV 2 N isolated conductors: V i = j P ij Q j, P ij = elastance matrix, or reciprocal capacitance matrix Q i = j C ij V j, C ij = capacitance matrix Image charge in sphere of radius a: Image of Q at R is q = a a2 Q, r = R R Separation of Variables for Laplace s Equation in Cartesian Coordinates: V = { cos αx sin αx } { } { } cos βy cosh γz sin βy sinh γz where γ 2 = α 2 + β 2 Separation of Variables for Laplace s Equation in Spherical Coordinates: Traceless Symmetric Tensor expansion: 2 ϕr, θ, φ = r 2 r 2 ϕ + r r r 2 2 ang ϕ = 0, where the angular part is given by 2 ang ϕ sin θ ϕ + 2 ϕ sin θ θ θ sin 2 θ φ 2 2 ang C l i i 2...i l ˆn i ˆn i2... ˆn il where C l i i 2...i l = ll + C l i i 2...i l ˆn i ˆn i2... ˆn il, is a symmetric traceless tensor and ˆn = sin θ cos φ ê + sin θ sin φ ê 2 + cos θ ê 3 unit vector in θ, φ direction General solution to Laplace s equation: V r = C l i i 2...i l r l + C l i i 2...i l ˆn i ˆn i2... ˆn il, r l+ where r = rˆn Azimuthal Symmetry: V r = A l r l + B l r l+ { ẑ i... ẑ il } TS ˆn i... ˆn il where {... } TS denotes the traceless symmetric part of....
8.07 FORMULA SHEET FOR QUIZ 2, FALL 207 p. 6 Special cases: { } TS = { ẑ i } TS = ẑ i { ẑ i ẑ j } TS = ẑ i ẑ j 3 δ ij { ẑ i ẑ j ẑ k } TS = ẑ i ẑ j ẑ k 5 ẑi δ jk + ẑ j δ ik + ẑ k δ ij { ẑ i ẑ j ẑ k ẑ m } TS = ẑ i ẑ j ẑ k ẑ m 7ẑi ẑ j δ km + ẑ i ẑ k δ mj + ẑ i ẑ m δ jk + ẑ j ẑ k δ im + ẑ j ẑ m δ ik + ẑ k ẑ m δ ij + 35 δij δ km + δ ik δ jm + δ im δ jk Legendre Polynomial / Spherical Harmonic expansion: Azimuthal Symmetry: V r = P l cos θ = A l r l + B l r l+ P l cos θ 2l! 2 l l! 2 { ẑ i... ẑ il } TS ˆn i... ˆn il General solution to Laplace s equation: l V r = A lm r l + B lm r l+ Y lm θ, φ Orthonormality: m= l 2π 0 dφ π 0 sin θ dθ Y l m θ, φ Y lmθ, φ = δ l lδ m m Spherical Harmonics in terms of Traceless Symmetric Tensors: where Y l,m θ, φ = C l,m ˆn i... ˆn il, C l,m = { Nl, m{ û + i... û + i m ẑ im+... ẑ il } TS for m 0, Nl, m{ û i... û i m ẑ i m +... ẑ il } TS for m 0, û + 2 êx + iê y, û 2 êx iê y Nl, m = m 2l! 2 m 2l + 2 l l! 4π l + m! l m! for m 0. For m < 0, use Y l, m θ, φ = m Ylm θ, φ, which holds for all m.
8.07 FORMULA SHEET FOR QUIZ 2, FALL 207 p. 7 Electric Multipole Expansion: First several terms: V r = [ Q p ˆr + r r 2 + ˆr ] iˆr j 2 r 3 Q ij +, where Q = d 3 x ρ r, p i = d 3 x ρ r x i Q ij = E dip r = p ˆr r 2 d 3 x ρ r3x i x j δ ij r 2, = 3 p ˆrˆr p r 3 3ɛ 0 p i δ 3 r E dip r = 0, E dip r = ɛ 0 ρ dip r = ɛ 0 p δ 3 r Traceless Symmetric Tensor version: V r = Cl rl+ ˆn i... ˆn il, where C l = 2l!! l! r r = Griffiths version: 2l!! l! ρ r { x i... x il } TS d 3 x r rˆn x i ê i 2l!! 2l 2l 32l 5... = 2l! 2 l l! r l r l+ { ˆn i... ˆn il } TS ˆn i... ˆn i l, for r < r, with!!. Reminder: {... } TS denotes the traceless symmetric part of.... V r = r l+ r l ρ r P l cos θ d 3 x where θ = angle between r and r. r r = r< l r l+ P l cos θ, = λ l P l x > 2λx + λ 2 P l x = l d 2 l x 2 l, Rodrigues formula l! dx P l = P l x = l P l x dx P l xp l x = 2 2l + δ l l
8.07 FORMULA SHEET FOR QUIZ 2, FALL 207 p. 8 Spherical Harmonic version: V r = l m= l where q lm = 4π 2l + Y lmr l ρ r d 3 x q lm r l+ Y lmθ, φ r r = l m= l 4π 2l + r l r l+ Y lmθ, φ Y lm θ, φ, for r < r Properties of Traceless Symmetric Tensors: Trace Decomposition: Any symmetric tensor S i...i l can be written uniquely as S i...i l = S TS [ ] + Sym Mi...i l 2 δ il,i l, where S TS is a traceless symmetric tensor, M i...i l 2 is a symmetric tensor tensor, and Sym [ xxx ] symmetrizes xxx in the indices i... i l. S TS is called the traceless symmetric part of S i...i l. Extraction of the traceless part of an arbitrary symmetric tensor S i...i l : [ { S i...i l } TS = S i...i l + Sym a,l δ i i 2 δ j j 2 S j j 2 i 3...i l ] + a 2,l δ i i 2 δ i3 i 4 δ j j 2 δ j 3j 4 S j j 2 j 3 j 4 i 5...i l +... where a n,l = n l! 2 2l 2n! n!l 2n!l n!2l! The series terminates when the number of i indices on S is zero or one. Integration: dω ˆn i... ˆn i2l where dω 2 l l! = 4π 2l +! π 0 all pairings 2π sin θ dθ dφ 0 δ i,i 2 δ i3,i 4... δ i2l,i 2l The integral vanishes if the number of ˆn factors is odd.
8.07 FORMULA SHEET FOR QUIZ 2, FALL 207 p. 9 [ ] [ ] dω C l { ˆn i... ˆn il } TS C l j...j, { ˆn l j... ˆn jl } TS 4π 2l l! 2 = 2l +! Cl C l if l = l 0 otherwise Other identities: ˆn il { ˆn i... ˆn il } TS = l 2l { ˆn i... ˆn il } TS ẑ il { û + i... û + i m ẑ im+... ẑ il } TS = where û + 2 êx + iê y For any symmetric traceless tensor S i...i l, where δ il+2n,i l+2n [ Sym S i...i l +2n [ = F n, l Sym S i...i l +2n 2 F n, l = Electric Fields in Matter: Electric Dipoles: p = d 3 x ρ r r 2n2l + 2n + l + 2nl + 2n ˆn = any unit vector l + ml m { û + i l2l... û + i m ẑ im+... ẑ il } TS δ il+,i l+2... δ il+2n,i l+2n }{{} n times ] δ il+,i l+2... δ il+2n 3,i l+2n 2 }{{} n times ρ dip r = p r δ 3 r r d, where r d = position of dipole F = p E = p E force on a dipole τ = p E U = p E torque on a dipole potential energy Electrically Polarizable Materials: P r = polarization = electric dipole moment per unit volume ρ bound = P, σ bound = P ˆn ]
8.07 FORMULA SHEET FOR QUIZ 2, FALL 207 p. 0 D ɛ 0 E + P, D = ρ free, E = 0 Boundary conditions: E above E below = σ ɛ 0 E above E below = 0 D above D below = σ free D above D below = P above P below Linear Dielectrics: P = ɛ 0 χ e E, χ e = electric susceptibility ɛ ɛ 0 + χ e = permittivity, D = ɛ E ɛ r = ɛ ɛ 0 = + χ e = relative permittivity, or dielectric constant Clausius-Mossotti equation: χ e = Nα/ɛ 0 Nα, where N = number density of atoms 3ɛ 0 or nonpolar molecules, α = atomic/molecular polarizability P = α E Energy: W = D E d 3 x linear materials only 2 Force on a dielectric: F = W, where W is the potential energy stored in the system. Even if one or more potential differences are held fixed, the force can be found by computing the gradient of W with the total charge on each conductor fixed. Table of Legendre Polynomials P l x: P 0 x = P x = x P 2 x = 2 3x2 P 3 x = 2 5x3 3x P 4 x = 8 35x4 30x 2 + 3
8.07 FORMULA SHEET FOR QUIZ 2, FALL 207 p. Table of Spherical Harmonics Y lm θ, φ: l = 0 Y 00 = 4π 3 Y = sin θeiφ 8π l = 3 Y 0 = 4π cos θ Y 22 = 5 4 2π sin2 θe 2iφ l = 2 5 Y 2 = sin θ cos θeiφ 8π Y 20 = 5 2 4π 3 cos2 θ Y 33 = 35 4 4π sin3 θe 3iφ Y 32 = 05 4 2π sin2 θ cos θe 2iφ l = 3 Y 3 = 2 4 4π sin θ5 cos2 θ e iφ Y 30 = 7 2 4π 5 cos3 θ 3 cos θ Y 44 = 3 35 6 2π sin4 θe 4iφ Y 43 = 3 35 2 2π sin3 θ cos θe 3iφ l = 4 Y 42 = 3 5 8 2π sin2 θ7 cos 2 θ e 2iφ Y 4 = 3 5 2 2π sin θ7 cos3 θ 3 cos θe iφ Y 40 = 3 35 cos 4 θ 30 cos 2 θ + 3 4π For m < 0, use Y l, m θ, φ = m Ylm θ, φ, which is valid for all m.