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*From*: Hugh Logan <hloganPHY@CFL.RR.COM>*Date*: Thu, 12 Aug 2004 16:06:22 -0400

According to Eugene Hecht's _Optics_, 4th ed., p. 141, the derivation of

Snell's law on the basis of Newton's corpuscular law 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).

This latter "derivation" was first pointed out to me by my major

professor, Dr. Earle K. Plyler of FSU, in connection with some ideas he

had about photons -- in 1965, before I saw it in any text.

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 wrote:

Carl E. Mungan wrote:

I have a dim memory that Newton once argued from his corpuscular view

that light particles should bend *opposite* to the wave prediction of

Snell's law. If someone remembers why that should be so, I'd be

grateful for a primer on the subject.

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. 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. 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.)

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