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 P
_{1}
and P
_{2}
as shown in the diagram below.
Solving each equation for L yields:
Therefore
If d
_{1}
and d
_{2}
represent the distances traveled in the respective mediums during the same amount of time, then we can replace them with the expressions
But v
_{1}
and v
_{2}
represent the speed of the waves in each medium and can be replaced with the expressions
where n
_{1}
and n
_{2}
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 [n
_{2}
> n
_{1}
] the sine of the refracted angle gets smaller [sinθ
_{2}
< sinθ
_{1}
], since the product of the two terms must remain a constant.
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