4.8. Slowing of a charged particle in a dielectric medium
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We consider a charged particle with velocity v passing through an
infinite medium with complex dielectric coefficient f(ω). Here ω is
the temporal angular frequency of the electromagnetic field of the
particle. This presumes that the electromagnetic field is amenable
to Fourier analysis. This is shown schematically in Figure 4.8. The
2
Y
T
Figure 4.8: Particle passing through a dielectric medium.
particle has charge q and velocity v. An electromagnetic field is
experienced at the observation point O due to the passing particle.
We adopt the nonrelativistic approximation, in which magnetic effects are negligible. The instantaneous electrostatic potential ϕ(x)
evaluated at any position x obeys Poisson’s equation,
ρ(x)
v
2 ϕ(x) = −
.
(4.180)
f
The charge density ρ is due to the particle, and is given by
ρ(x) = q δ(x − vt).
(4.181)
The potential ϕ(x) can be expressed as a Fourier integral,
ϕ(x) = d
3 k ϕ ˜(k) e
ik·x .
(4.182)