Lesson Video: The Dispersive Power of a Prism Physics

In this video, we will learn how to calculate the dispersive power of a prism, given the refractive indices of the different colours of light passing through it.

13:45

Video Transcript

In this video, our topic is the dispersive power of a prism. Dispersion is what we see going on in the sketch. We have a beam of light containing both red and blue light, which once it enters this prism separates out into these two different beams. In this lesson, we’ll come up with a quantitative way of describing how prisms do this. And we’ll also see how index of refraction plays a role in the extent to which a prism spreads out light.

We can get started by understanding what this term “dispersion” means. Dispersion is an effect whereby light is spread out by its constituent colors. This happens when light enters a material that treats the light differently by its wavelength. In this instance of white light entering this material here, we can see that the material causes red light to refract differently than yellow light, which refracts differently than green light, which refracts differently than blue light. Because the material affects different wavelengths of light differently, we would say that it is dispersive. This is a characteristic of prisms and a large part of why we’re so interested in them.

But then, why is it that a dispersive material, like a prism, makes light of different wavelengths refract differently? To understand this, we can recall Snell’s law. This law tells us that if we have a ray of light that’s incident on some interface where the index of refraction on either side of the interface is different, then we can expect the angle of incidence, 𝜃 sub 𝑖, of this ray to relate to the angle of refraction, 𝜃 sub 𝑟, like this. 𝑛 one, the index of refraction of the material the light begins in, multiplied by the sine of the angle of incidence is equal to 𝑛 two, the index of refraction of the material the light enters, times the sine of the angle of refraction.

Going back up to our prism, we can say that all this white light incident on one of the prism’s faces is moving through a material, whatever that is, with the same index of refraction. So in the language of Snell’s law, 𝑛 one is the same for all the wavelengths of this white light. In the same way, all of them share the same angle of incidence, 𝜃 sub 𝑖. And that angle, we can identify as this one right here. So for all this light, all these wavelengths, the left side of Snell’s law is the same. But then, as our prism disperses the light according to wavelength, we can see that the angle of refraction varies by color.

This tells us that 𝜃 sub 𝑟 is not the same for all these different wavelengths. And this is because the prism’s index of refraction is different for each wavelength. This fact is the basic insight about dispersive materials, that the index of refraction of a dispersive material varies with wavelength. And that’s why all the different colors of light refract differently because each one is effectively experiencing a different index of refraction. And as we mentioned earlier, there’s actually a way to quantify how much dispersion happens in a given prism.

To see that, we can complete the drawing of this prism. And we’re going to focus just on two particular wavelengths of light, the red and the blue. Recall that the light incident on our prism is white light. That means it consists of all the wavelengths of light our eyes are sensitive to, all of the colors. One way to remember the major colors that are involved in the visible spectrum and their order is to think of this acronym, ROYGBIV. These letters stand for red, orange, yellow, green, blue, indigo, and violet, respectively. The point is, red is at one end of the spectrum, and blue-like colors ⁠— like blue, indigo, and violet ⁠— are at the other.

So when we focus on the red and blue wavelengths of light coming through our prism, we can say we’re looking at the longest and shortest wavelengths of visible light. Red has that longest wavelength, and blue has the shortest. Looking at these red and blue rays, we can see that their paths are clearly deviated by this prism. Originally, they were both moving in this direction here. But then the red ray of light has deviated this much. We can call this deviation angle 𝛼 sub min because it’s the smallest deviation that any visible light will experience, while the blue light, we can see, is deviated by this much larger angle. This is the maximum angular deviation that visible light will undergo.

The reason we bring all this up is because the amount this particular prism disperses visible light can be quantified in terms of these two deviation angles, 𝛼 sub min and 𝛼 sub max. The equation for this looks like this. On the surface of it, it seems like there’s a lot going on here. But what we’re saying is that this term here, which is called the dispersive power of the particular prism we’re considering, is equal to the difference in these two deviation angles, the maximum deviation and the minimum deviation, divided by their average value. That’s what we get when we add them together and divide them by two.

We could think of this equation this way. First we ask, how much does light of the shortest wavelength, we’re considering that to be blue light in our case, deviate? That angular deviation is given by 𝛼 sub max. Then we subtract from that the minimum amount of deviation that any wavelength experiences. And, in our case, we’ve said that that’s the deviation of red light. So 𝛼 sub max minus 𝛼 sub min is the range of possible angular deviations for the light we have coming into our prism. And then, we’re dividing that range by what we’ve said is the average angular deviation.

If we were thinking of this in terms of colors of visible light, this average angular deviation would correspond to either yellow or green light, somewhere in that range. So the maximum span of angular deviation divided by the average angular deviation caused by the prism is equal to the prism’s dispersive power. It’s important to see that even though we’re using this term “power”, which in physics has a very specific meaning of an amount of energy over an amount of time, in this case, we’re not using the word power to mean that. This equation here doesn’t involve energy or time. It’s just a quantitative description of how much a given prism spreads out light.

We can see, thanks to 𝛼 sub min and 𝛼 sub max, how much this prism spreads out visible light that’s incident on it. But let’s imagine this prism had a different effect on incoming visible light. What if, instead, the effect the prism had on red light was to refract it like this, and blue light was refracted like this. If that were to happen, then we can see that, now, the difference between these two angles is much smaller than the original difference between 𝛼 sub min and 𝛼 sub max.

Looking at our equation for dispersive power, this would lead to a smaller numerator, while the average angular deviation of the light passing through this prism might not change all that much. Therefore, a prism that spreads out red and blue light like this would have less dispersive power than one that spreads out red and blue light like this.

But now, let’s return to the way our prism originally refracted light. And let’s recall this idea from earlier, that dispersion in a material means that the index of refraction varies with wavelength. We’re going to see that there’s a way of writing this equation for dispersive power in terms of index of refraction because a certain index of refraction for the material corresponds to this maximum angular deviation, 𝛼 sub max. And then a certain different index of refraction corresponds to this deviation. And as we said, the same material possesses these different indices of refraction. That’s confusing, but that’s what it means for a material to be dispersive.

Now, if we consider the deviation angle of the blue light, the maximum deviation light passing through this prism experiences, then we can say that the reason this light is refracted so much as it passes through the prism is because it experiences the greatest difference in index of refraction as it crosses the faces of the prism. And that means that of all the possible indices of refraction that visible light could experience as it passes through the prism, the light with the shortest wavelength, what we’ve called blue light, experiences the maximum of those possible values. This causes it to refract more, which makes its angle of deviation greatest.

And then, on the other hand, the light that experiences the least angular deviation, the red light, is refracted the least of all visible colors, which means it experiences a smaller index of refraction than any other of the colors of light. So in writing an equation for dispersive power in terms of index of refraction, we can almost use this equation and simply replace 𝛼 sub max with 𝑛 sub max and 𝛼 sub min with 𝑛 sub min ⁠— almost, but not quite

In our numerator, we again find the difference between the maximum and minimum value, this time of the index of refraction. And in our denominator, we once again calculate the average value of this index. But here we didn’t subtract one from that average. This is to account for the fact that a prism is typically surrounded by air, which we can approximate as having an index of refraction of one.

So then, whether we know the maximum and minimum deviation angles of a prism or the maximum and minimum indices of refraction, either way, we can calculate the dispersive power of that prism, recalling that this is a measure of how much a given prism spreads out light incident on it. Knowing all this, let’s look now at an example exercise.

Which of the following is the term used to refer to the separation of white light into its component wavelengths due to the variation of the refractive index with the wavelength of an object that the white light passes through? (A) Aberration, (B) diffraction, (C) dispersion, (D) distortion, (E) deviation.

Okay, so in this exercise, we’re looking for a word that defines or describes this process here. We’re told that what’s taking place is white light is incident on some object. And then, because the index of refraction of that object varies with wavelength, it refracts the different colors differently, so they spread out like this. And we want to identify a term from among these five that describes this process.

We can start from the top and work our way down, first aberration. This term describes something about an optical system that causes it to form images in an imperfect way. It could be, for example, that a lens, say, has an aberration. And this makes it refract light improperly, leading to a low quality image or even no image at all. So this term aberration does have to do with light, but not in the way we’re describing it here. So we’ll cross this option off our list.

Next, we get to diffraction. This is a property of light that describes how it spreads out when passing through a narrow opening or around a barrier. For example, if we send coherent light waves through a small opening like this, then diffraction makes those waves spread out through the opening like this. So this term does describe the spreading out of waves of light, but it’s not because of a variation of refractive index in a material. So this term isn’t quite the right one either.

Next, we get to dispersion, and this term agrees completely with what’s written in our question statement. When an object like this prism over here separates out white light according to its component wavelengths, in this case colors, because the refractive index of that prism varies with wavelength, then that is known as dispersion. So option (C) looks like it’ll be our answer. But just to make sure, let’s check options (D) and (E). Option (D) suggests distortion as the word for describing this process. But, in a physics context, this word typically means something different.

Say that we have a wave of light and we want to amplify the wave. If we do that and the amplified wave comes out looking like this, we say there’s been amplitude distortion or simply distortion of this wave. So this term distortion is not a match for describing white light being separated into its component wavelengths. And lastly, let’s think of this term deviation. Now, deviation actually is related to the way a prism interacts with light. It describes how light is refracted at this face here and then at this face here of the prism in such a way that its final trajectory, say in the case of our beam of red light, is deviated by some amount from the ray’s original direction.

However, this term does not describe how the refractive index of an object can vary with wavelength. And, therefore, though it describes how a particular wavelength of light is deviated as it passes through some object, it doesn’t explain the separation of white light into its component wavelengths. So we won’t choose option (E) either. And this confirms to us that option (C), dispersion, is the correct choice. This term refers to the separation of white light into its component wavelengths due to the variation of the refractive index with the wavelength of an object the white light passes through.

Let’s now summarize what we’ve learned about the dispersive power of a prism. In this lesson, we learn that prisms disperse white light, which means to spread it out by its wavelength. And dispersion is caused by a prism’s refractive index varying with wavelength. This means that light of one wavelength doesn’t experience the same refractive index as light of another. Therefore, it refracts differently, and that causes dispersion. And lastly, we saw that dispersion can be quantified by what’s called the dispersive power of the prism.

We can express this power in terms of the maximum and minimum angles of deviation experienced by light passing through the prism. Or, equivalently, we can write it in terms of the maximum and minimum indices of refraction that that light encounters as it moves through the prism. This is a summary of the dispersive power of a prism.

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