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Snell's Law

Teacher/Instructor Jonathan Osbourne
Jonathan Osbourne

PhD., University of Maryland
Published author

Jonathan is a published author and recently completed a book on physics and applied mathematics.

Snell's Law is the quantitative way to analyze refraction. Snell's Law is expressed in the formula angle of refraction 1 x sin(angle 1)= angle of refraction 2 x sin(angle 2), which is the geometrical requirement of parallel light rays.

So let's talk about Snell's law, Snell's law is the quantitative way that we can do refraction basically what we do is we look at a boundary between 2 media we've got 1 index of refraction and 1 another index of refraction and 2 and what we're interested in is the relationship between the angle that the incident ray comes in at off of the normal and the angle that the transmitted or refracted ray comes into the new medium n again off of the normal. I can't emphasis enough how important it is to recognize that these angles are measured off of the normal. So these angles are not the angle off of the surface, they're the angle off of the perpendicular guy to the surface. Alright now what Snell's law says is that if you multiply the index of refraction times the sine of the angle then you get the same value on both sides. So what this allows us to do for example, if we know both of the indices of refraction and we know the angle of incidence then we can use Snell's law to calculate the angle of refraction or the transmitted angle.

Snell's law is actually a Geometric requirement of the fact that parallel light rays stay parallel when they go in, they all turn the same way alright so that if they come in parallel they're also going to be parallel inside the new medium. Alright we can understand this just by looking at for example a flat piece of glass, you've got 2 parallel light rays that come in so they're going to come in at the same index of or at the same or angle of incidence because this is flat. So then they both turn the same amount, the amount that they turn is actually determined by Geometry and the fact that the speed of light is slower in the glass than it is in the air. Remember that we know this qualitative idea because glass is like a crowded hallway so the light is going to bend toward the normal when it goes into the glass. So then it travels through the glass for a little while and then it comes back out these guys are still parallel comes back out and now it's back in the empty hallway so it bends away from the normal again by the same amount. So parallel light rays go through the glass still parallel on the other side and that means I won't see distortion when I look though plain glass alright.

Let's look at some indices of refraction so here's a very simple table gives us the materials and their index of refraction. Remember that the index of refraction is dimensionless, doesn't have any unit. So air 1 is also the same as vacuum there's some other digits over here air is not exactly vacuum but it's close. Water 1.333, benzene, salt are the same at 1.5 and diamond has one largest bulk indices of refraction that's known 2.419. Alright so let's go ahead and just do a problem alright so suppose that we've got light it's coming in from water and it's going out at air. Notice that this light beam is qualitatively doing the right thing. It's going into an empty hallway so it bends away from the normal, now I want to know what's that angle. Well easy enough we'll use Snell's law, so we'll say 1.333 the index of water times the sine of the angle in water has to equal the index in air which is 1 times the sine of the angle in air. So all you're going to do is type in my calculator 1.333 times the sine of 30 degrees you got to make sure you're in degrees when you're going to do this and then I'll do the inverse sine of that.

Alright and what you'll end up getting is 41.8 degrees very simple alright let's do another one. Suppose that I've got light coming in from benzene and going into diamond, alright now again here I've got the lower index of refraction having the bigger angle right and then it comes in bends toward the normal. Alright once again we're just going to use Snell's law so it'll be benzene 1.5 sine of the angle in benzene 60 equals diamond 2.419 times the sine of the angle in diamond alright so again I go into my calculator 1.5 sine 60 and then I got to divide by 2.419 and then I do the inverse sine of that and what we find is that the angle is thirty two and a half degrees alright. Let's do a more qualitative one, suppose that I'm given a diagram like this and I know this material is salt I want to know what could this material be. Well let's look at it, it doesn't look like it bent at all so that means that the angle incidence is the same as the angle of refraction. Snell's law indicates that, that means that the speed got to be the same, so that means the index has to be the same and I look at my table and I see that the guy that has the same index of refraction as salt is benzene.

Not difficult to do but if you don't know what you're doing it can be confusing. Alright let's look at the next one. Suppose that I've got a situation like this where I've got light coming in at some angle from diamond and I don't get a ray coming out in the air. And I want to know what's the minimum angle possible that this could happen at. Well this is total internal reflection the minimum possible angle is going to be the angle at which the refracted ray would go straight along the boundary because then if I try to go further there's no more room left and I'll only reflect total internal reflection. So to get the minimum angle all I need to do is just use Snell's law and I have the refracted angle be 90 degrees. So in air I've got sine 90 times 1 so I'm not going to write it equals in diamond I got 2.419 times sine of the angle. So that means that the angle is the inverse sine of 1 over 2.419 because this is just 1.

Alright and when you go through and do that, you'll find that the angle is 24.42 degrees so that is the maximum angle of incidence that you can leave diamond at if you want to get out. And that's part of the reason that diamonds are so sparkly alright, let's go ahead to the last one suppose that I've got light leaving water at 35 degrees and coming in to some unknown material and I measure the refractive index, alright sorry I've measured the angle of refraction and I see that it's 25 and I want to determine the refractive index. Alright once again Snell's law so we'll say water 1.333 sine of the angle in water right and that's got to equal the index that I'm interested in times the sine of 25 degrees. And all I got to do is divide and I'll end up with an index of I think it's 1.81. So knowing that index I can then go and look on chart like that and determine what could that material be. People who collect gems do this all the time because you can identify a surfire by the index of refraction or a diamond by the index of refraction and so doing it that way it's actually very easy and it cuts way down on mistakes. So that's Snell's law there's lots of stuff that's very qualitative about it. You always want to make sure that you understand when it bends away it's speeding up, when it bend toward it's slowing down. Notice that we ended up with an answer that agrees with that 1.333 to 1.81 you'll always want to make sure that your answer makes sense qualitatively because qualitatively you don't make as many numerical mistakes. Anyway that's Snell's law.