Video Transcript
The diagram represents red, green, and blue light being deviated through an angle 𝛼 by a prism. Which of the following correctly describes how 𝜆, the wavelength of the light, relates to 𝑛, the refractive index of the prism, for a particular wavelength of light? (A) As 𝜆 increases, 𝑛 increases. (B) As 𝜆 increases, 𝑛 decreases. (C) 𝜆 and 𝑛 are unrelated.
So this question asks us how the refractive index of the prism 𝑛 changes as different wavelengths of light pass through the prism. So let’s start by recalling which colors of light have the longest wavelength, which we represent by the Greek letter 𝜆. Out of all the colors of visible light, we know that red light has the longest wavelength. So we can say that 𝜆 is large for red light. Out of all the colors shown in the diagram, we know that blue light has the shortest wavelength. So we can say that 𝜆 is small for blue light. Finally, green light, which is the final color shown in our diagram, has a wavelength which is somewhere in between the wavelength of red light and blue light.
This means that as the color of light entering the prism changes from blue to green to red, we know that the wavelength 𝜆 is increasing. Now in order to find the relationship between the wavelength 𝜆 and the refractive index 𝑛, we’ll need to see how the refractive index changes for these different colors of light. However, the diagram we’re shown in this question doesn’t directly show us the refractive index for the different colors of light. Instead, the diagram shows us the angle of deviation 𝛼 for the different colors. So to get to the answer of this question, we’re going to need to recall the relationship between the angle of deviation 𝛼 and the refractive index 𝑛.
Let’s recall that the angle of deviation 𝛼 measures the change in direction of the light from when it enters the prism here to when it exits the prism here. If we look carefully at the angles shown here in the diagram, we can see that the angle of deviation for red light is the smallest, the angle of deviation is slightly larger for green light, and the angle of deviation is the largest for blue light.
We can make it even easier to see these angles of deviation if we first extend this dashed line here, which represents the direction the light was traveling when it entered the prism. We can then redraw the angles of deviation for the different colors of light, and it’s easier to see that the angle of deviation for red light, shown here, is smaller than the angle of deviation for blue light, shown here. In fact, since blue light experiences the maximum angle of deviation out of any color of light entering this prism, let’s give a name to this angle of deviation and let’s call it 𝛼 max.
We can recall that the reason these different colors of light are deviated by different amounts is because the index of refraction of the prism is different for each wavelength of light. Let’s think about the angle of deviation of blue light, which we’ve called 𝛼 max, which is the maximum deviation that any light experiences through this prism. 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.
So out of all the possible indices of refraction, this means that blue light experiences the maximum of these possible values. And we can call this maximum index of refraction 𝑛 max.
Let’s now give a name to the angle of deviation of red light. And since this is the minimum angle of deviation for any color, let’s call this 𝛼 min. Just like we said that the maximum angle of deviation 𝛼 max corresponds to the maximum index of refraction 𝑛 max, we can say that the minimum angle of deviation corresponds to the minimum index of refraction, which we can call 𝑛 min.
So we’ve now seen that blue light experiences both the maximum angle of deviation and the maximum index of refraction. And we’ve seen that red light experiences the minimum angle of deviation and the minimum index of refraction. This means that we could say that as we go from blue light to green light to red light, the index of refraction 𝑛 decreases. This gives us the final answer to this question because we can see that if we increase the wavelength 𝜆, then this has the effect of decreasing the refractive index of the prism 𝑛. So we can say that our final answer is (B), as 𝜆 increases, 𝑛 decreases.
