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Unformatted text preview: Chapter 2: Electricity & Magnetism 11 2.5.2 Electromagnetic waves in matter The wave equations in matter, with c mat = ()-1/2 the lightspeed in matter, are: 2 - 2 - t2 t E =0, 2 - 2 - t2 t B=0 give, after substitution of monochromatic plane waves: E = E exp(i(k r -t)) and B = B exp(i(k r -t)) the dispersion relation: i k 2 = 2 + 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 = 1 2 1+ 1+ 1 and k = ()2 1 2 -1 + 1+ 1 ()2 This results in a damped wave: E = E exp(-k n r ) exp(i(k n r - t)). If the material is a good conductor, . the wave vanishes after approximately one wavelength, k = (1 + i) 2 2.6 Multipoles Because 1 1 = |r - r | r 0 r r l Pl (cos ) the potential can be written as: V = Q 4 n kn rn For the lowest-order terms this results in: Monopole: l = 0, k 0 = Dipole: l = 1, k1 = Quadrupole: l = 2, k 2 = dV r cos()dV 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: = p Eout 4r3 r2 2. The magnetic dipole: dipole moment: if r A: = I (Ae ), F = ( )Bout 2 mv , W = - Bout || = 2B - 3 r Magnetic field: B = - . The moment is: = Bout 4r3 r2 )Eext , and 2.7 Electric currents The continuity equation for charge is: + t I= J = 0. The electric current is given by: dQ = dt (J n )d2 A For most conductors holds: J = E/, where is the resistivity. ...
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