# Snell's Law, Total Internal Reflection, and Dispersion

by Jared Rovny

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00:01 Now we’re ready to introduce Snell’s law. We said that the incident and reflected angle are the same which leaves us to have to figure out the transmitted angle. We already mentioned that light will bend when it goes into a new medium. So, we’re simply asking ourselves: by how much does the light bend as it goes from one medium with one index of refraction like a vacuum into another medium with a different index of refraction? This is given by Snell’s law. Snell’s law says that the index of refraction times the sine of the incident angle will be equal to the new index of refraction times the sine of that transmitted angle.

00:37 Going into a denser medium, as you can see just by looking at this equation, will give you a smaller theta.

00:44 You can sort of examine this equation. It shouldn’t necessarily be immediately apparent. But just look at this equation for a bit because Snell’s law is very often used. It’s a very important equation.

00:54 Again, it tells us exactly how an incident beam will turn into a transmitted beam with a smaller angle if it’s going into a denser medium. By the opposite token, if it's going into a sparser or lighter or less dense medium, the transmitted angle, the angle at which it leaves that medium will instead be bigger.

01:14 So, we have these two different behaviors. They’re both given by Snell’s law. Now that we know Snell’s law, there’s one other important thing we could derive about Snell’s law. Imagine that we have something like this that we have in the bottom here. A light ray is going from some dense medium, maybe it’s water into air or maybe it’s air going into a vacuum, going into a lighter medium. As we increase the angle of incidence, the angle at which the light hits the less dense medium, the transmitted beam is going to change how much it’s transmitted by in the reflected angle, the transmitted angle that is.

01:51 Looking at the Snell’s law for this particular scenario, we can see that we increase the incident angle.

01:57 We could ask ourselves, "At what incident angle will the beam now have a 90-degree of reflecting or transmitting angle?" So, looking at the middle case here, what angle, what incident angle would we need in order for the transmitted beam to not be transmitted out at an angle anymore but be transmitted horizontal to or at a perpendicular angle to the new medium? All we have to do is actually plug this in.

02:22 So, we take this middle case. We take θ2, the new angle to be 90 degrees. Then we remember that the sine of 90 degrees is just one. So, we have the n1 times the sine of the incident angle will be equal to n2, the new medium since the sine of theta transmitted which is 90 degrees is just one.

02:41 We can rearrange this equation and try to solve for the incident angle that we need for this effect to occur.

02:47 We get something like this. That the sine of the incident angle, the sine of θ1 has to be equal to the ratio of the indices of refraction of the two media. A few important things about this: First of all, look at the equation for the sine of this angle. We call this angle the critical angle, the angle at which the light, as it’s trying to leave this medium, will bend and be horizontal to or perpendicular as you can see here to the new surface. This is the critical angle. But notice that the sine of the critical angle is equal to the ratio of these two indices of refraction. But we already know about the sine of theta, that the sine of an angle can never be greater than one. For this reason, we need for n1, the index of refraction of the initial medium that it’s starting from to be greater than n2, the index of refraction of the medium it’s going into because otherwise, we’ll have sine of θ equals some number greater than one. That can never be the case. For this reason, the initial medium, the medium that the light is starting in has got to be more dense than the medium it is going into. This is the only way you can ever get what we call total internal reflection. We call it total internal reflection because this beam, as it’s trying to leave this dense medium can never get out so long as the angle, this incident angle is always greater than this number that we found here for the critical angle. For example, this could be used in fiber optics. If you have a fiber optic cable made of some sort of glass material and light is traveling through that cable, if it’s trying to get out of that cable and go to a less dense medium than the glass, it will not be able to as long as we keep the angle shallow enough, as long as we keep that incident angle beyond this critical angle requirement. Then we can maintain our total internal reflection. Finally, we have one last topic with this introduction to optics which is to point out that in fact for different wavelengths of light, there can actually be different indices of refraction for each wavelength. For example, the index of refraction for red light might be less than the index of refraction for blue light. By Snell’s law, this means that different colors would bend by different amounts when they enter a new medium. We can use this effect which we call dispersion by sending light into some denser medium expecting that light, if it has different components to its color, maybe white light that has many different colors in its spectrum as you can see here and use this effect, this dispersion effect of each color having a different index of refraction and a tool called a prism. You might be familiar with the idea of a prism. You send the light in one side which has many components to its color. Then because of this dispersion effect, we have all the different wavelengths separated on the other end, each one at its own frequency. This, in fact, can be used to analyze light coming in from some source to see what frequency components are present in that source of light. This summarizes or wraps up our initial intro to optics. We’re going to be using more ideas of optics when we get to lenses and mirrors especially which is what we’ll do next.

05:58 Thanks for listening.

The lecture Snell's Law, Total Internal Reflection, and Dispersion by Jared Rovny is from the course Geometrical Optics. It contains the following chapters:

• Total Internal Reflection
• Dispersion

### Included Quiz Questions

1. The beam would bend more and more to go directly into the medium
2. The beam would bend more and more to go parallel to the surface of the medium
3. The beam would stop bending and continue on an unchanging path
4. A beam cannot get into a medium with an increasing index of refraction
5. The beam would entirely reflect from the surface
1. From a lower to a higher index of refraction, refracted light bends towards the normal
2. From a lower to a higher index of refraction, refracted light bends away from the normal
3. From a lower to a higher index of refraction, reflected light bends towards the normal
4. From a lower to a higher index of refraction, reflected light bends away from the normal
5. From a lower to a higher index of refraction, light always only is transmitted (no reflection)
1. It can change depending on the color of the light
2. It is independent of the material through which light travels
3. It depends on the speed at which light travels before it enters a material
4. It is a universal constant dependent on the speed of light (c)
5. It changes with the square of the frequency
1. White light enters a prism and splits into many colors
2. A beam of monochromatic light enters a lens and bends
3. Light moving along a path reflects in such a way that none is transmitted past the boundaries
4. Two light rays enter a lens and interfere on the other side
5. A beam of light hits a surface and refracts exactly along the surface

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