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Unformatted text preview: ic dipole moment per unit volume) of a material is given by P=χE. The electric susceptibility is in general a complex function of frequency and is zero in €acuum. v
A. ABSORPTION & SCATTERING: LONG
WAVELENGTH LIMIT We first consider the case of wavelengths λ>>a. In this case, an incident electromagnetic wave corresponds to an externally applied electric field that is spatially uniform (over the size of the grain) but time
varying, E ext ∝ e− iωt . The polarization at position x within the grain is simply χE(x), where E(x) is determined by integrating the electric field from a collection of electric dipoles: €
x − x'
3
. E( x ) = E ext − ∇ x ∫
3 ⋅ P( x ' ) d x '
V x − x'
We are fortunate that the electric field introduced by a uniform polarization is: € x − x'
1
1
3
E un ( x ) = −∇ x ∫
d 3 x ' = (P ⋅ ∇ x )∇ x ∫
d 3 x '. 3 ⋅ Pd x ' = ∇ x ∫ (P ⋅ ∇ x )
x − x'
x − x'
V x − x'
V
V € The last integral is the potential from a uniformly charged sphere with charge density ρ=1. Its gradient is (negative) the electric field of such a sphere, which by Gauss’s law is −4πx/3 inside the sphere. Thus we see that the electric field from a sphere of uniform polarization is 4
E un ( x ) = − πP. 3
We see that the dielectric sphere in a uniform external field has a solution for uniform polarization P where: €
P = χE = χ (E ext − 4 πP), 3
or χ
P=
E ext . 1 + 4 πχ
3
€
€ 2 The overall dipole moment of the sphere is (4/3)πa3P, so the emitted (i.e. scattered) power is given by the dipole formula, 2 2 ω4 4
χ
ω4
ε −1
Psc = 3 πa 3
E ext = 3 a 3
E ext . 4
3c 3
1 + 3 πχ
3c
ε+2
If we recall that the incident flux F is cEext2/8π, we conclude that...
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This document was uploaded on 03/08/2014 for the course AY 102 at Caltech.
 Winter '08
 Sargent,A

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