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phys documents (dragged) 18 - Chapter 2 Electricity...

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Chapter 2: Electricity & Magnetism 11 2.5.2 Electromagnetic waves in matter The wave equations in matter, with c mat = ( ε μ ) - 1 / 2 the lightspeed in matter, are: 2 - ε μ 2 t 2 - μ ρ t E = 0 , 2 - ε μ 2 t 2 - μ ρ t B = 0 give, after substitution of monochromatic plane waves: E = E exp( i ( k · r - ω t )) and B = B exp( i ( k · r - ω t )) the dispersion relation: k 2 = ε μ ω 2 + ω ρ The first term arises from the displacement current, the second from the conductance current. If k is written in the form k := k + ik it follows that: k = ω 1 2 ε μ 1 + 1 + 1 ( ρεω ) 2 and k = ω 1 2 ε μ - 1 + 1 + 1 ( ρεω ) 2 This results in a damped wave: E = E exp( - k n · r ) exp( i ( k n · r - ω t )) . If the material is a good conductor, the wave vanishes after approximately one wavelength, k = (1 + i ) μ ω 2 ρ . 2.6 Multipoles Because 1 | r - r | = 1 r 0 r r l P l (cos θ ) the potential can be written as: V = Q 4 πε n k n r n For the lowest-order terms this results in: Monopole: l = 0 , k 0 = ρ dV Dipole: l = 1 , k 1 = r cos( θ ) ρ dV
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