Lectures on Physics has been derived from Benjamin Crowell's Light and Matter series of free introductory textbooks on physics. See the editorial for more information.... 
Home Optics Refraction and Images Snell's law  
See also: Derivation of Snell's law  
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Snell's law
The numerical rule governing refraction was discovered by Snell, who must have collected experimental data something like what is shown on this graph and then attempted by trial and error to find the right equation. The equation he came up with was
The value of the constant would depend on the combination of media used. For instance, any one of the data points in the graph would have sufficed to show that the constant was 1.3 for an airwater interface (taking air to be substance 1 and water to be substance 2). Snell further found that if media A and B gave a constant K_{AB} and media B and C gave a constant K_{BC}, then refraction at an interface between A and C would be described by a constant equal to the product, K_{AC}=K_{AB}K_{BC}. This is exactly what one would expect if the constant depended on the ratio of some number characterizing one medium to the number characteristic of the second medium. This number is called the index of refraction of the medium, written as "n" in equations. Since measuring the angles would only allow him to determine the ratio of the indices of refraction of two media, Snell had to pick some medium and define it as having n=1. He chose to define vacuum as having n=1. (The index of refraction of air at normal atmospheric pressure is 1.0003, so for most purposes it is a good approximation to assume that air has n=1.) He also had to decide which way to define the ratio, and he chose to define it so that media with their rays closer to the normal would have larger indices of refraction. This had the advantage that denser media would typically have higher indices of refraction, and for this reason the index of refraction is also referred to as the optical density. Written in terms of indices of refraction, Snell's equation becomes
but rewriting it in the form n_{1} sin θ_{1} = n_{2} sin θ_{2} [relationship between angles of rays at the interface between media with indices of refraction n_{1} and n_{2}; angles are defined with respect to the normal] makes us less likely to get the 1's and 2's mixed up, so this the way most people remember Snell's law. A few indices of refraction are given in the back of the book.


Home Optics Refraction and Images Snell's law 