Question Video: Calculating Dispersive Power from Index of Refraction Physics

Video Transcript

White light is dispersed by a prism. For the shortest wavelength light in the white light, the prism has a refractive index of 1.48. And for the longest wavelength light, the prism has a refractive index of 1.44. What is the dispersive power of the prism? Give your answer to three decimal places.

So, for this question, we’re asked to find the dispersive power of a prism given the refractive indices for the longest and shortest wavelengths of light. Let’s start by recalling what the dispersive power of a prism actually is. We usually denote the dispersive power of a prism as an 𝜔 with a subscript 𝛼, like this. So, this is the Greek letter 𝜔, which looks a little bit like a W. And this is the Greek letter 𝛼, and this is the subscript attached to the 𝜔. We treat this as one symbol, and it represents the dispersive power.

This dispersive power is a number that measures the difference in refraction of the light of the highest wavelength and the lowest wavelength that enter a prism. This basically means that the dispersive power 𝜔 𝛼 is a measure of how much a prism spreads out different colors of light. So, when white light enters a prism like this, if the index of refraction for different colors of light is different, as we have in this question, then the prism will refract different colors of light by different amounts. And the spread of these different colors of light is given by the dispersive power of the prism. The larger the value of 𝜔 𝛼, the more these colors spread out.

We can now recall that there is a formula for the dispersive power of a prism written in terms of the minimum and maximum refractive indices of the prism. If 𝑛 subscript max is the maximum refractive index of a prism and 𝑛 subscript min is the minimum refractive index of a prism, then the dispersive power of that prism is given by a fraction, which has 𝑛 max minus 𝑛 min on the top. So, this is the difference between the maximum and minimum refractive indices. And on the bottom of the fraction, we have 𝑛 max plus 𝑛 min divided by two subtract one. We can basically think of this as the average refractive index of the prism, and then we subtract one.

So, to calculate the dispersive power in this question, we first need to identify the maximum refractive index and the minimum refractive index for the prism. Of all the colors that make up white light, we know that the red light has the longest wavelength and will experience the smallest refractive index. We also know that the blue or violet light has the shortest wavelength and will experience the largest refractive index. So, since the question tells us that the shortest wavelength light has a refractive index of 1.48, we know that this will be the maximum refractive index of the prism. And we can set 𝑛 max equals 1.48.

We’re also told that the longest wavelength light has a refractive index of 1.44. So, we know this will be the minimum refractive index of the prism. And we can say that 𝑛 min is equal to 1.44. We now just need to substitute these values into our formula for the dispersive power of the prism to finish this question off. So, the dispersive power is equal to 𝑛 max minus 𝑛 min, which is 1.48 minus 1.44, divided by 𝑛 max plus 𝑛 min over two subtract one, which for us is 1.48 plus 1.44 over two. And we can’t forget to subtract one in the denominator.

We can now start to simplify this expression. And we have 𝜔 𝛼 is equal to 1.48 minus 1.44, which is 0.04 on the top. And this is divided by 1.48 plus 1.44, which is 2.92, which then needs to be divided by two again. And we have the subtract one on the bottom as well. We can then simplify this division here with 2.92 divided by two being equal to 1.46. And if we then do the subtraction on the bottom, we simply have that 1.46 minus one is equal to 0.46.

Finally, to get our final answer, we just need to do the division 0.04 divided by 0.46. We can do this with a calculator and find the answer to be 0.0869 and so on. And this is the value of the dispersive power of the prism in the question. However, the question asked us to give our answer to three decimal places. So, we need to round our answer before we give our final answer. If we do this rounding to three decimal places, we find that the dispersive power of the prism 𝜔 𝛼 is equal to 0.087. And this is our final answer for this question.