Refraction

We have seen that light can bounce as in reflection but light can also bend in a process called refraction. Refraction occurs because some materials slow light waves down. Not all materials are created equal though. Some materials slow light down a lot while others only a little. The measure of a material’s ability to slow light down is called the index of refraction (n).

The higher the index of refraction the slower the light goes this means that the speed of light can change. We are so used to it being constant at 3 x10⁸ m/s, but this value refers to the speed of light in a vacuum (c). Since c is the fastest that light can ever travel all other mediums will slow light down. As a result, the index of refraction of a vacuum is set at 1. Therefore, all other materials will have an index of refraction of greater than 1.

If we want to calculate the specific value of the index of refraction we can use the following equation where n stands for the index of refraction, c is the lightspeed constant, and v is the actual velocity of light in the material.

[latexpage]
\[
n=\frac{c}{v}
\]

How Do We Bend Then

Great, but how does a speed change correspond to the bending of light. To see how this occurs let’s imagine a photon as a kayaker on a lake. This lake isn’t any old lake though, in fact, it has a very peculiar portion that is filled with hot chocolate fudge and sprinkles. When the kayaker is paddling in the middle of the water portion of the lake both sides of their kayak travel at the same speed and they travel in a straight line.

However, when the kayaker begins to enter the hot fudge portion of the lake with its thick and sticky consistency the kayaker slows down. Since the kayaker entered the hot fudge at a slight angle their top paddle will end up hitting first making the boat slower on the top. However, the bottom paddle stays in the water and continues on at the same speed.

As a result of the top portion of the boat goes slower and the bottom portion of the boat is faster causing the kayak to rotate upwards. Another way of thinking about this is by considering a ball attached to a string. In this example, the string of the ball is anchored where the kayak first hits the fudge lake. Then continues to swing and rotate around that fixed point. In this case upwards indicating the direction that the light will end up bending in.

When the kayaker is fully into the hot fudge lake both sides of their boat will go at the same speed and they will travel straight again.

However when they return back to the regular water again the side that hits the water first is going to speed up before the other causing them to bend.

Quiz with the answer to the above idea.

The exact same thing occurs with the photons in a ray of light, causing the entire ray to bend when entering and exiting materials with different indexes of refraction. When deciding whihc way something will bend always focus on whether or not the light is speeding up or slowing down and which edge of the light beam hits the material interface first.

Snell’s Law

If we want to quantify how much something bends we can do so with Snell’s Law. This equation allows us to quantify the degree to which light bends (θ₂) compared to what is known as the normal. This normal is closely related to the idea of a normal force, which points out perpendicular to whatever surface an object is pushing against. Here the normal is aligned in a similar manner, perpendicular to the interface between the two materials.

[latexpage]
\[
n_1sin(\theta_1)=n_2sin(\theta_2)
\]

The above equation is Snell’s Law which allows us to compare the angle of entry based on the normal (θ₁) to the angle of refraction based on the normal (θ₂) so long as we know the index of refraction of both materials (n₁ and n₂). While it is worth memorizing the equation it is more important to understand how changing the entry angle and the indexes of refraction affect how much our light ray bends.

For example, if n₁ increases then we know that θ₂ must increase. From a conceptual standpoint, this makes sense since the greater the index of refraction the greater the speed difference between the two materials and the more the light will bend.

Additionally, the smaller θ₁ is the smaller θ₂ will be. This also makes sense from a conceptual standpoint. If you hit the hot fudge lake straight on you aren’t going to bend since both paddles hit the slower fudge portion at the same time. However, if you come in nearly parallel one of the paddles is going to spend a very, very long time in the lower portion of the lake before the other one hits resulting in a ton of bending.

Critical Angle

There are a couple of special instances we should be aware of with refraction. Up first is the critical angle or the angle at which the light is refracted to the point that it can’t leave and instead travels along the interface of the two materials. This phenomenon only occurs when light is traveling from a material with a higher index of refraction to a material with a lower index of refraction.

In this scenario, we can calculate the critical angle (θ₁) so long as we remember that θ₂ is always 90°. Then you can plug and chug. I highly doubt that you’ll be required to calculate the value of sines since we don’t get a calculator. Therefore understanding the underlying concept behind the critical angle is far more important.

Total Internal Reflection

Another special instance occurs when we move beyond the critical angle and make θ₁ closer and closer to 90°. Once we move beyond the critical angle we switch from refraction to reflection. In this case, it is called total internal reflection because all of the light is reflected back into the material from which it came.