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Formulas in Electrodynamics Based on course by Yuri Lyubarsky and Edited By Eitan Rothstein Department of Physics, Ben-Gurion University, Beer-Sheva 84105, Israel R 0 n P∞ 1 1 0 0 3 0 Electric multipole expansion: V (r) = 4π n=0 r (n+1) (r ) Pn (cos θ )ρ(r )d r . 0 R 0 0 3 0 P 1 p·r̂ Electric dipole moment potential Vdip (r) = 4π r ρ(r )d r = i qi ri . 2 , with the electric dipole moment p = 0 r 1 1 The electric field of an electric dipole is Edip (r) = 4π [3(p · r̂) · r̂ − p]. 3 0 r µ0 1 The magnetic field of a magnetic dipole is B (r) = dip 4π r 3 [3(m · r̂) · r̂ − m], R with the magnetic moment: m = 21 r × JdV Boundary conditions: 1 E1⊥ = 2 E2⊥ , k k E1 = E2 , B1⊥ = B2⊥ , 1 k 1 k B = B . µ1 1 µ2 2 Poynting vector: S = µ1 E × B. 1 Energy density: u = 2 E 2 + 2µ B2. Radiation angular distribution of a dipole: dI = Liénard-Wiechert potentials: V (r, t) = 1 qc , 4π0 r̃c − r̃ · v A(r, t) = µ0 16π 2 c 2 (p̈ × n̂) dΩ. v V (r, t), c2 where r̃ is the vector from the retarded position to the field point r and v is the velocity of the charge at the retarded time. Lorentz transformation x̄ = γ(x − vt), v t̄ = γ(t − 2 x), c 1 γ = p . 1 − v 2 /c2 eB . Relativistic cyclotron frequency: ωB = mcγ The field tensor: 0 −Ex /c −Ey /c −Ez /c ∂Aν ∂Aµ 0 Bz −By E /c Fµν = − = x µ ν E /c −B 0 Bx ∂x ∂x y z Ez /c By −Bx 0 Transformation of fields Ēk = Ek , B̄k = Bk , Ē⊥ = γ(E⊥ + v × B⊥ ), 1 B̄⊥ = γ(B⊥ − 2 v × E⊥ ). c Energy momentum four vector: P = mcγ(1, v/c).