386 Week 11: Light
coherently into the reflected wave; similarly those polarized molecules create a forward
propagating wave into the medium (although at a different angle according to Snell’s
law). As before, the polarized surface molecules (dipoles)cannot radiate along their own
axisso that light that is reflectedparallelto one of the polarization directions cannot
contain that polarization.
This state of affairs occurs when the reflected ray is perpendicularto the refracted ray,
pictured above. In this case:
n 1 sin(θ) =n 2 sin(φ) (949)
is Snell’s law, but clearly:
φ=π
2−θ (950)so that:
sin(φ) = sin(π/ 2 −θ) = cos(θ) (951)
andBrewster’s formula:
tan(θb) =n 2
n 1(952)
is the condition forθb, the so-calledBrewster angleof incidence (and hence reflection)
where the reflected ray is completely polarized parallel to the surface (and perpendicular
to the plane of reflection, just as was the case with scattered lightabove).
However, the polarization component in the plane of reflection isalwaysreduced at
angles other thanθ= 0 as the component of the polarization gradually lines up with the
reflected ray so reflected light is at leastpartiallypolarized in the plane at all angles other
than 0. Note that the transmitted light ispartiallypolarizedinthe plane of transmission- this is not complete because all of the perpendicularly polarized lightis not reflected
at the surface, some is still transmitted into the medium.
Polaroid Sunglasses
As we have just seen, reflected glare from any smooth surface is likely to be at least
partially polarized parallel to the ground. It is thus blocked by a pair of polaroid sun-
glasses with averticaltransmission axis. Similarly, (scattered) light from the blue sky
viewed near the horizon at midday is predominantly polarized parallel to the ground and
isalsoblocked by a vertical transmission axis, which can make e.g. driving safer and less
stressful on the eye.11.5: Doppler Shift
Since light is a wave, the frequencies picked up by a frequency sensitive receiver (e.g.
the human eye) depend on the original frequency (color) emitted by the source and
Doppler shiftedby the motion of the source and/or the receiver. A complete treatment
of the Doppler shift requires relativity and is beyond the scope of this course, but an
elementary treatment suffices to understand the Doppler shift atvelocities that are
small compared to the speed of light^107.
The idea underlying the Doppler shift is very simple. If the source is moving towards
the receiver, its motion foreshortens the normal wavelength, increasing the frequency
observed by the stationary receiver. If the receiver is moving towards the source, its
motion reduces the time between the wavefronts it receives, increasing the frequency it(^107) At higher speeds, lengths contract and times dilate, so thissimple argument has to be made a bit more complicated.
In this case the correct argument leads to the formula for therelativistic Doppler shiftfor moving source and/or
receiver, but at low speeds the forms for the shifts are approximately (to lowest nontrivial order inv/c) the same