Unformatted text preview: Chapter 2: Electricity & Magnetism 11 2.5.2 Electromagnetic waves in matter
The wave equations in matter, with c mat = (εµ)−1/2 the lightspeed in matter, are:
2 − εµ ∂2 µ∂ − ∂ t2 ρ ∂t E =0, 2 − εµ ∂2 µ∂ − ∂ t2 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: iµω k 2 = εµω 2 + 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 εµ 1+ 1+ 1 and k = ω (ρεω )2 1 2 εµ −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, µω . the wave vanishes after approximately one wavelength, k = (1 + i) 2ρ 2.6 Multipoles
Because 1 1 = r − r  r
∞ 0 r r l Pl (cos θ) the potential can be written as: V = Q 4πε n kn rn For the lowestorder terms this results in: • Monopole: l = 0, k 0 = • Dipole: l = 1, k1 = • Quadrupole: l = 2, k 2 = ρdV r cos(θ)ρdV
1 2 i 2 2 (3zi − ri ) 1. The electric dipole: dipole moment: p = Qle , where e goes from ⊕ to , and F = (p · W = −p · Eout . 3p · r Q Electric ﬁeld: E ≈ − p . The torque is: τ = p × Eout 4πεr3 r2 √ 2. The magnetic dipole: dipole moment: if r A: µ = I × (Ae⊥ ), F = (µ · )Bout 2 mv⊥ , W = −µ × Bout µ = 2B −µ 3µ · r Magnetic ﬁeld: B = − µ . The moment is: τ = µ × Bout 4π r 3 r2 )Eext , and 2.7 Electric currents
The continuity equation for charge is: ∂ρ + ∂t I= · J = 0. The electric current is given by: dQ = dt (J · n )d2 A For most conductors holds: J = E /ρ, where ρ is the resistivity. ...
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This note was uploaded on 01/30/2011 for the course PHYSICS 208 taught by Professor Ye during the Spring '10 term at Blinn College.
 Spring '10
 Ye
 Magnetism, Light

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