Angle of refraction
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Angle of refraction refers to the angle a wave makes to the line of normal incidence when a wave passes from one medium to another. The line of normal incidence is a line perpendicular to the medium the light is passing into. Or in common terms, the wave is heading directly towards the material. While angle of refraction normally refers to light, it can be applied to any wave that propagates through a medium like a sound wave. There is a simple formula for determining the refraction angle, this formula is referred to as Snell's Law. Snell's law states: n1sinθ1 = n2sinθ2 where n1 and n2 are the refractive indices of the original medium and the medium the wave is passing into respectively while θ1 and θ2 are the angles the wave makes with respect to the line of normal incidence in the first and second medium respectively. So normally the angle of refraction would be referred to as θ2 in the above equation and can be easily solved if the other three components are known.
[edit] Derivation and Meaning
There are several ways to derive Snell's Law, and therefore the Angle of Refraction. The first way it was discovered was by an application of Fermat's principle which states that a light wave must take a path that is an extremum in time subject to the constraints present. Normally this is translated into "Light will always take the quickest path it can." From this principle, and using a bit of differential calculus, Snell’s Law can be derived thus leading to the Angle of Refraction. If one looks into the meaning of Fermat’s principle, other uses can be seen for the formula as well. For instance, if a truck needs to drive across a parking lot and a grassy field and traverse in a direction parallel to the place the asphalt and the grass meets, Snell’s law can be used to find at what angle you should drive on both the asphalt and the grass exchanging the refractive indices with the speeds the truck can drive on the two surfaces. Using Snell’s Law will ensure the quickest path possible.
There are several other ways to derive Snell’s Law. One of the easiest involves an application of the general boundary conditions of Maxwell equations for electromagnetic radiation. It can also be done geometrically by applying Fermat’s principle through a prism.
The result of this is that the angle a wave is heading will decrease according to the normal if it passes from into a denser medium from a less dense, and will increase if it passes into a less dense medium from a denser one. If a wave passes into and then out of a material, it will leave with the same angle it entered with.
[edit] Implications
Since the refractive index of a material changes with wavelength, the angle of refraction will also change with different wavelengths. This is why light shone through a prism will “spread out” showing an entire rainbow of colors. The amount the angle of refraction changes with wavelength also leads to the resolving power of a telescope.
Looking at Snell’s Law shows that there are situations where the angle of refraction is equal to or greater than 90 degrees. When this situation occurs, there is said to be total internal reflection (or TIR) because none of the light can escape out of the lens (Except an imaginary component that quickly decays). The minimum angle at which TIR occurs is called the critical angle.
