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Law of refraction

[link] shows how a ray of light changes direction when it passes from one medium to another. As before, the angles are measured relative to a perpendicular to the surface at the point where the light ray crosses it. (Some of the incident light will be reflected from the surface, but for now we will concentrate on the light that is transmitted.) The change in direction of the light ray depends on how the speed of light changes. The change in the speed of light is related to the indices of refraction of the media involved. In the situations shown in [link] , medium 2 has a greater index of refraction than medium 1. This means that the speed of light is less in medium 2 than in medium 1. Note that as shown in [link] (a), the direction of the ray moves closer to the perpendicular when it slows down. Conversely, as shown in [link] (b), the direction of the ray moves away from the perpendicular when it speeds up. The path is exactly reversible. In both cases, you can imagine what happens by thinking about pushing a lawn mower from a footpath onto grass, and vice versa. Going from the footpath to grass, the front wheels are slowed and pulled to the side as shown. This is the same change in direction as for light when it goes from a fast medium to a slow one. When going from the grass to the footpath, the front wheels can move faster and the mower changes direction as shown. This, too, is the same change in direction as for light going from slow to fast.

The figures compare the working of a lawn mower to that of the refraction phenomenon. In figure (a) the lawn mower goes from a sidewalk to grass, it slows down and bends towards a perpendicular drawn at the point of contact of the mower with the surface of separation. An imaginary line along the mower when it is on sidewalk is taken to be the incident ray and the angle which the mower makes with the perpendicular is taken to be theta one. As it goes into the grass, the mower turns and the imaginary line moves towards the perpendicular line drawn and makes an angle theta two with it. The imaginary line drawn along the mower when the mower is in the grass is taken to be the refracted ray. Sidewalk is taken to be a medium of refractive index n one and that of grass to be taken as n two. In figure (b), the situation is the reverse of what has happened in figure (a). The mower moves from grass to sidewalk and the ray of light moves away from the perpendicular when it speeds up.
The change in direction of a light ray depends on how the speed of light changes when it crosses from one medium to another. The speed of light is greater in medium 1 than in medium 2 in the situations shown here. (a) A ray of light moves closer to the perpendicular when it slows down. This is analogous to what happens when a lawn mower goes from a footpath to grass. (b) A ray of light moves away from the perpendicular when it speeds up. This is analogous to what happens when a lawn mower goes from grass to footpath. The paths are exactly reversible.

The amount that a light ray changes its direction depends both on the incident angle and the amount that the speed changes. For a ray at a given incident angle, a large change in speed causes a large change in direction, and thus a large change in angle. The exact mathematical relationship is the law of refraction , or “Snell’s Law,” which is stated in equation form as

n 1 sin θ 1 = n 2 sin θ 2 . size 12{n rSub { size 8{1} } "sin"θ rSub { size 8{1} } =n rSub { size 8{2} } "sin"θ rSub { size 8{2} } } {}

Here n 1 size 12{n rSub { size 8{1} } } {} and n 2 size 12{n rSub { size 8{2} } } {} are the indices of refraction for medium 1 and 2, and θ 1 size 12{q rSub { size 8{1} } } {} and θ 2 size 12{q rSub { size 8{2} } } {} are the angles between the rays and the perpendicular in medium 1 and 2, as shown in [link] . The incoming ray is called the incident ray and the outgoing ray the refracted ray, and the associated angles the incident angle and the refracted angle. The law of refraction is also called Snell’s law after the Dutch mathematician Willebrord Snell (1591–1626), who discovered it in 1621. Snell’s experiments showed that the law of refraction was obeyed and that a characteristic index of refraction n size 12{n} {} could be assigned to a given medium. Snell was not aware that the speed of light varied in different media, but through experiments he was able to determine indices of refraction from the way light rays changed direction.

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The Critical Angle Derivation So the critical angle is defined as the angle of incidence that provides an angle of refraction of 90-degrees. Make particular note that the critical angle is an angle of incidence value. For the water-air boundary, the critical angle is 48.6-degrees.
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Source:  OpenStax, College physics. OpenStax CNX. Jul 27, 2015 Download for free at http://legacy.cnx.org/content/col11406/1.9
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