Carl E. Mungan wrote:
None of this is the same as Snell's Law, in
which ALL of the
trajectory change occurs at the interface between two media. For a
physical object traveling in a medium, the trajectory is
continuously altered by the medium, i.e. friction, air resistance,
or water resistance. (Vickie Frohne)
I don't see this either. If we assume drag oppositely directed to
velocity and that the object is cylindrically symmetric, it can
travel straight. Also, isn't your above example of the toy car
analogous to Snell's law? If I think of the axle as a wavefront, it
turns for what seems like the same basic reason as light does, namely
different points along the same line are traveling at different
speeds.
According to the PSSC version of Newton's
corpuscular model, light
particles bend in the same direction as predicted by Snell's law, the
explanation being that attractive forces in the interface between the
less optically dense medium and the more dense medium cause the normal
component of the velocity of the corpuscles to be greater after they
have passed into the more dense medium, while the tangential component
of velocity (and momentum) remains the same.
But does "attractive forces in the interface" mean "surface tension"?
In that case, I still don't see why the object should bend toward the
normal. If the idea is that the object is pulled toward the interface
as it approaches it, shouldn't it also be pulled back as it recedes
from it upon transmission? Kind of like crossing on a skateboard from
a flat, down a half-pipe, and back up to the *same* flat (assuming
the action is purely in the interface)?
A little right triangle
trigonometry
for a velocity diagram shows that sin(theta1)/sin(theta2)=v(2)/v(1),
where 1 refers to the medium of the incident corpuscles and 2 to that
of the refracted corpuscles. If medium 1 is vacuum,
sin(theta1)/sin(theta2)=v(2)/c. Thus the index of refraction is
n=v(2)/c, just the reciprocal of the value
according to the wave theory.
Ah, that's what was opposite in Newton's theory, thanks for the
reminder.
Although Roemer had estimated the speed of
light in vacuum (outer space) in 1676, the direct measurement of the
speed of light in a medium such as water had to wait until the middle
of the nineteenth century. So there was no way to decide on the basis
of
which speed was greater in Newton's day. (I think Carl's assertion is
correct once it is known that light travels slower in the more dense
medium.)
Okay.
PSSC models corpuscles going from a less
dense medium to a more dense
medium with a rolling ball moving on a higher horizontal to a lower
horizontal surface, the surfaces connected by a ramp. (The direction
could be reversed to model a corpuscle going from a more dense to a
less
dense medium.)
This is not the same as the attractive force being only in the
interface. In fact, it relies on a uniform gravitational field in
both media. Shut off gravity and this model no longer works. (The
problem now is how to keep the ball in contact with the surface.
Okay, replace the surfaces with an S-shaped tube in deep space.)
In modern terms, I think the essential point in the rolling ball
analogy to the Descartes-Newton model of refraction is a gravitational
potential difference between the two media represented by the horizontal
planes. The gravitational force only changes the speed of the ball when
it is in the interface (on the ramp). Using a conservative force field
to model the refraction is nice , because it works in both directions
as has been pointed out. The experiment wouldn't work in outer space
even if the ball were constrained to move along an S-shaped
(cross-section?) tube, because there would be no potential energy
difference or force to change the speed of the particles.
Had Descartes and Newton lived in modern times, I think they would have
interpreted the force that
"shoved" (or resisted) the corpuscles across the interface would be the
negative gradient of some kind of conservative potential. I can't
speculate on what kind of field this might have been, since I don't know
the nature of their corpuscles (not photons.) Besides, the corpuscular
hypothesis had to be abandoned, because experiment showed it to be
incorrect.
I am not bothered by the uniform gravitational field throughout the
entire apparatus. About thirty -five years ago, I saw a large -1/r
shaped bowl into which a ball bearing was thrown to simulate the orbit
of a satellite (or planet). I believe this was in a space exhibit at
the IBM building in New York. The actual gravitational field was
essentially uniform over the dimensions of the apparatus. However, they
were modeling motion in a central inverse-square force field --
gravity in this case.
In principle, it would be possible to model refraction of particles
with a beam of electrons, causing them to enter an "interface" in which
there is a uniform electric field. It might be difficult to model the
media on either side of the interface.
There are photographs of this in Chapter 14
of the 2nd ed. of _PSSC
Physics_ and Chapter 5 of _College Physics, Physical Science Study
Committee_. This is actually a basis for a PSSC experiment, "The
'Refraction" of Particles" to see if they change direction according to
Snell's law. It is in the PSSC _Laboratory Guide_ at least through the
4th ed. I could not find the corpuscular model of refraction in either
the text or lab guide for the latest (7th) edition from around 1991.
I would like to see such photos. Maybe I can track down one of these
references. Did they smooth off the sharp corners getting onto and
off of the ramp?
>From what I can see of the pictures, the corners do not appear to
be
rounded off. The ramp seems to be rather steep so that the interface is
narrow. Ball bearings were used for the "corpuscles." The apparatus was
actually manufactured for student use in PSSC experiments. I recall
doing this experiment as a TA at FSU and later while teaching PSSC
Physics. One of the FSU professors designed
apparatus for PSSC labs such as the angular momentum experiment for the
Advanced Topics. The PSSC refraction of particles apparatus is still
listed at science education supply houses such as Science Kit and
Boreal Laboratories (equipment for the 6th ed. of PSSC, so refraction
of particles must have survived through that edition.) There may be
some of this type of apparatus lying around at schools that once used
PSSC.
I've always associated the question of
changes in direction in terms
of a boundary, not in terms of the actualy media, at least for
"corpuscles".
For example, consider a golf ball rolling (without friction) at a
steady speed on a level surface. Now suppose the ball obliquely
hits a ramp to a higher, but also level surface. Once it reaches
the top surface, it will be a) travelling at a slower, steady speed,
and b) heading farther away from the normal. The higher the ramp
and the slower the ball is going at the end, the more it bends. If
the ramp is high enough, you observe "total external reflection".
This is the opposite of how waves behave. (Tim F)
As Hugh points out, this depends on which side of the ramp you
associate with which medium.
According to Eugene Hecht's _Optics_, 4th
ed., p. 141, the derivation of
Snell's law on the basis of Newton's corpuscular law [ I meant
"model."] was actually first
published by Rene' Descartes in 1637. Hecht gives a modern, quantum
mechanics version of this for photons which he regards as being "a bit
simplistic," but of some pedagogical value. Equating the tangential
components of the momenta of the incident and refracted photons as
Descartes did for Newtonian corpuscles,
p(1)*sin(theta1)=p(2)*sin(theta2). Recognizing that the momentum of a
photon is p=h/lambda, this becomes
[h/lambda1]*sin(theta1)=[h/lambda2]*sin(theta2) .
Multiplying through by c/f, where f is the frequency, and using
f*lambda=v, one gets
[h/v(1)]*sin(theta1)=[h/v(2)]*sin(theta2)
or sin(theta1)/sin(theta2)=v(1)/v(2), the correct result (upside down
from the Newton-Decartes result.) Thus
sin(theta1)/sin(theta2)=n(2)/n(1) where the index of refraction is
defined as in the wave theory (n=c/v).
According to Hecht, the fact that light travels slower in the denser
medium was probably first inferred from experiment by Thomas Young in
1802, from the fact that the measured wavelength was shorter in the
more
optically dense medium -- before the definitive experiments of Foucault
in 1850 (with a rotating mirror and a long column of water.) (Hugh
Logan)
Hugh Logan
Flagler Beach, FL
(where one has to do physics near the eye of a hurricane and wait
twenty hours for
power to return.)
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