W9_parallel_resonance.eps

Week 10: Maxwell’s Equations and Light 345

A direct consequence of this result is the death of classical physics. Classically, we

expect an electron toorbita proton in a hydrogen atom, much the way the moon

orbits the earth. After all, the forces of attraction between them have a more or

less identical form! But if an actual hydrogen atom were bound in this way, the

electron (like the moon) would be more or less perpetuallyaccelerating. It would

therefore be more or less perpetuallyradiating away energyand dropping into a

lower orbit to provide it. If one considers how long it would take before an electron

in a circular orbit around a proton with an initial radius around 10−^10 meters (one

Angstrom, roughly the size of almost any atom) to spiral in to the proton, it is a

very, very short time (as the further in it gets the more strongly itmus accelerate

and the faster it radiates to a still lower energy orbit with a still smaller radius). In

a tiny fraction of a second, the classical “atom” would collapse!

The fact that this manifestly doesnotoccur, when itmustoccur if both Newton and

Maxwell are correct, is one of several factors that led to the invention of quantum

mechanics and modern physics (including relativity theory). This, then, is the

next course in physics that students beginning a serious study of physics should

undertake, as soon as they complete this one and solidify their understanding of

classical electricity and magnetism and light. Things are getting interesting!

• When one considers a point charge oscillating around is oppositely charged mate
  (a dynamical version of our Lorentz model for an atom that helpedus understand
  dielectric polarization earlier) one can either convert this expression into or derive
  directly from the Poynting vector the following expression for the power cross-
  section:
  dP
  dΩ

= c

2

32 π^2

√

μ 0

ǫ 0

k^4 |pz|^2 sin^2 (θ) (804)

Thek^4 = (2π/λ)^2 is very important, as it is why the sky is blue! Remember it for

later – shorter wavelength/higher frequency light waves have a much larger power

cross-section, all things being equal, than longer ones, because the fields are related

to thetime derivatives of the dipole moments which increase with the frequency.

Again, the actual power radiated away in any direction drops off like 1/r^2.

• Finally, one can (as usual) consider thecollectiveradiation frommany charged
  particles oscillating against a neutral background in, for example, anantenna. An
  antenna is basically a wire that has a current in it such that it forms a macroscopic
  dipole moment (in say thez-direction) that oscillates at some frequencyω. This
  antenna will then radiate away energy in the form of electromagnetic radiation!.
  The power cross section is basically the same as that just given (butfor a much
  larger dipole momentpz), so that theintensityof the radiation field of az-oriented
  dipole antenna located at the origin of a spherical polar coordinate system is usually
  given by:
  I(θ) =

P 0

r^2

sin^2 (θ) (805)

(and is azimuthally symmetric about thez-axis). P 0 has the units of power, and

intensity has units of power per unit area, so this works. It is oftengiven as:

P 0 =Irms^2 Rrad (806)

whereIrmsis the root-mean-square current in the antenna andRradis theradiation

resistanceof the antenna, which can heuristically be thought of as resulting from

thereaction forceexerted on the radiating charges due to their own radiated field!

Deriving these results is beyond the scope of this course, but it is nevertheless

useful to understand and use the terminology when we consider radios (as we saw

last week). Note well that the radiation is most strongly emittedperpendicular to

the dipole moment, and that no energy at all is radiatedalongthe dipole moment.