PhysicsLAB Resource Lesson
Snell's Law: Derivation

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Consider three incident rays of light encountering an interface between two media. In this example, the second medium is the slower medium and the rays are refracted towards the normal - note that angle A is greater than angle B in the diagram.
 
Since all rays are perpendicular to their respective wavefronts,

 
 
 
Since all normals are perpendicular to their respective interfaces,

 
 
 
Therefore, and so we can now examine the following new relationships:


where L is the distance along the interface between points P1 and P2 as shown in the diagram below.

 
Solving each equation for L yields:


Therefore

 
If d1 and d2 represent the distances traveled in the respective mediums during the same amount of time, then we can replace them with the expressions


But v1 and v2 represent the speed of the waves in each medium and can be replaced with the expressions
 
 
where n1 and n2 are the respective indices of refraction and c is the speed of light.
 
At this junction, we can now write
 
 
Canceling the common terms (c and t) yields
 
 
Or, as Snell's Law is more commonly expressed:
 
 
Notice that Snell's Law shows that the index of refraction and the sine of the angle of refraction are inversely proportional - that is, as the refractive index gets larger [n2 > n1] the sine of the refracted angle gets smaller [sinθ2 < sinθ1], since the product of the two terms must remain a constant.



 
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