March 30, 2017

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I don't know the physics behind Snell's Law, but I can do the math

Snell's Law says
sin(angle of incidence)/sin(angle of reflection) = refractive index

since you know the two angles...
sin45/sin27 = refractive index

so for an angle of refection of 25º
sin(incidence angle)/sin25º=1.5575..
sin(incidence angle)=sin25(1.55..)
so angle of incidence = arcsin(.658..)

That's just the application of Snell's, that part I've done. What I need is:

sin(a)/sin(a-25) = RI

I would solve it by iteration. There is no simple solution to it. could graph it on a calculator and see where it crosses the axis. Plot y as a function of x on your graphing calculator.

y= RI - sin x /sin(x-25)

use sin(A-B)=sinAcosB - cosAsinB

sin(a-25)=sina(cos25) - cosa(sin25)
=.9063sina -.4226cosa

so in sina/sin(a-25)=RI

sina = .9063(RI)sina - .4226(RI)cosa
sina - .9063(RI)sina = -.4226cosa
sina(1-.9063(RI)) = -.4226cosa

sina/cosa = -.4226/1-.9063(RI))

Tana = ........

a = arctan(.....)


I'm a hobby gem faceter and am trying to figure out some of the math involved. What I want to know is assuming I have a piece of glass, refractive index of 1.54 (or anything for that matter) at what angle would the light need to enter it to be deviated by a certain amount?

For instance, using Snell's law, I know that if a beam of light goes into glass at a 45 degree angle (from the norm), it will travel through the glass at about 27 degrees. The light has deviated by 18 degrees. What angle would I need it to enter if I wanted it to deviate by 25 degrees?

Thanks for any help.

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