374 Week 11: Light
11.1: The Speed of Light
We just learned that the speed of light in a vacuum, derived from Maxwell’s Equations,
isc = 1/√ǫ 0 μ 0 = 3× 108 meters/second. However, we havealsolearned that the
permittivity and permeability of bulk polarizable matter are not equalto their vacuum
equivalents. The conclusion is inescapable. The speed of light is notcin a medium.
Weexpectit to bev= 1/√ǫμwhere e.g.ǫ=ǫrǫ 0 (scaled by the dielectric and diamagnetic
constants of the material). It turns out for many reasons that the polarization of the
medium alwaysslows down the wave– in free space it just sweeps along, but in the
medium it has to move all of that bulk charge too, which has mass and cannot respond
as quickly. For most transparent materials,μ≈μ 0 so:v≈√^1
ǫrǫ 0 μ 0=√c
ǫr(917)
To keep life simple, we take all of the contributing properties of the material and roll
them into a single relation:
vmedium=c
n(918)
nis called theindex of refractionof the medium and is roughly equal to√
ǫr(which is
dimensionless, recall).
However, there is a problem with this.ǫris defined in thestatic limitofω= 0. Visible
light has a frequency of 4. 3 × 1014 Hz to 7. 5 × 1014 Hz, and the charges in a dielectric
material simply don’t havetimeto reach their peak polarization before the wave points
the other way! Indeed, it turns out that the index of refraction isafunction of frequency:
n(ω). This means (as we shall see) that different frequencies are bentby different amounts
via Snell’s law at an interface between two dispersive media, splitting white light up
into aspectrumof colors, with the highest frequency (shortest wavelength) lightusually
getting bent themostalthough this is very much dependent on the particular medium
in question.
This is why water droplets break up light into arainbow. Note well that this means that- as far as we can tell examining the world around us or looking back into the remote
past as we look up at the stars – water droplets havealwaysbroken up light into rainbows
when backlit by a local source of light, just as they do if you spray water in a fine mist
away from the sun in your back yard.
This has profound religious and philosophical consequences. At onetime there was a
rather extensive argument concerning the “frangibility of light” where Biblical literalists
argued that this process could not have occurred before the Flood in Genesis, as it clearly
states therein that the rainbow was first createdat a specific antediluvian timeas a sign
that God wouldn’t try to drown the world ever again.
It is worth noting that if light wasn’t “frangible” before this (mythical) Flood, there
would havebeen no lightas the processes that produce it are the same as the processes
that break it up in interaction with matter into colors in rainbows and everywhere else.
Nor would there have been anynormal matter– as we have just learned in considerable
detail, the electromagnetic forces that hold atoms and molecules togetherarethe forces
that are responsible for polarizability, which in turn is responsible fordispersion.
11.2: The Law of Reflection
A perfect conductor in electrostatic equilibrium, we recall, cancels the electric field inside
by arranging charges on its surface to effect the cancellation. Similarly, it creates surface