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1) and with the neglect of spin, the matrix elements, 2p|ˆ ρ|1s : 2q − 3r ρ(r, θ, φ, t) = √ 4 re 2a0 Y00 Y10 e−iωt ba0 We also know the current density. 529 × 10−19 m is the Bohr radius, ω0 = 32πǫ30 ¯ha0 me2 2 c is is the frequency difference of the levels, and v0 = 4πǫe 0 ¯h = αc ≈ 137 the Bohr orbit speed. a. Find the effective transitional “magnetization”, calculate ∇ · M , and evaluate all the non-vanishing radiation multi-poles in the longwavelength limit. The magnetization is 1 M = (r × J) 2 J can be broken up into Jr and Jz components.

22 Use the Kramers-Kr¨ onig relation to calculate the real part of ǫ(ω), given the imaginary part of ǫ(ω) for positive ω as . . The Kramer-Kr¨onig relation states: ℜ ǫ(ω) ǫ0 2 P π =1+ ǫ(ω ′ ) ω′ dω ′ ℑ ω ′2 − ω 2 ǫ0 ∞ 0 a. ℑ ǫ(ω) = λ [Θ(ω − ω1 ) − Θ(ω − ω2 )]. ǫ0 Plug this into the Kramer-Kronig relationship. ℜ ǫ(ω) ǫ0 =1+ 2λ π ω2 ω1 ω′ dω ′ + 0 ω ′2 − ω 2 Notice that the real part of ǫ(ω) depends on an integral over the entire frequency range for the imaginary part! Here, we will use a clever trick.

P = F 1 1 E ×B = Pc = S = A c cµ0 Take the average over time, and factor of one half comes in. We also know that B0 = Ec0 . 106 that the energy density is 12 (E · D + B · H) → 21 ǫ0 (E0 )2 and wait that’s the same as the pressure! 1 P = ǫ0 (E0 )2 = u 2 42 This result generalizes quite easily to the case of a non-monochromatic wave by the superposition principle and Fourier’s theorem. b. 4 kW/m2 . If an interplanetary “sail-plane” had a sail of mass 1 g/m2 per area and negligible other weight, what would be its maximum acceleration in meters per second squared due to the solar radiation pressure?

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