Question Video: Determining the Formula for the Dispersive Power of a Prism Using Refractive Indices | Nagwa Question Video: Determining the Formula for the Dispersive Power of a Prism Using Refractive Indices | Nagwa

Question Video: Determining the Formula for the Dispersive Power of a Prism Using Refractive Indices Physics • Second Year of Secondary School

Which of the following formulas correctly relates 𝜔_α, the dispersive power of a prism, to 𝑛_min, the refractive index of the prism for the shortest wavelength of light that passes through it, and 𝑛_max, the refractive index of the prism for the longest wavelength of light that passes through it?

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Video Transcript

Which of the following formulas correctly relates 𝜔 sub α, the dispersive power of a prism, to 𝑛 sub min, the refractive index of the prism for the shortest wavelength of light that passes through it, and 𝑛 sub max, the refractive index of the prism for the longest wavelength of light that passes through it?

Here, we see our answer choices (A), (B), (C), and (D). And if we clear space at the top of the screen, we can see the final answer option (E). All of the answer choices show us formulas involving three variables: the dispersive power of a prism, 𝜔 sub α, the refractive index of the prism for the shortest wavelength of light that passes through it, 𝑛 sub min, and the prism’s index of refraction for the longest wavelength of light that passes through it, 𝑛 sub max.

In this question, we’re being asked to identify the correct equation for the dispersive power of a prism. Recall that when white light is passed through a prism, light rays of different wavelengths are refracted by different amounts. This causes the white light to be separated into its constituent colors. This is called dispersion. The dispersive power of a prism is a measure of how significant a dispersion effect occurs.

Let’s focus on two particular colors of white light: red light, which has the longest wavelength of any visible color, and violet, which has the shortest. The direction of the red light is changed by the smallest angle of any color, α sub min. And the direction of the violet light is changed by the largest angle, α sub max.

The dispersive power of a prism can be defined by these angles using this formula. On the top of the fraction, we have the difference between the maximum angle of refraction and the minimum angle of refraction, which is α sub max minus α sub min. This tells us the range of angles of refraction for the light going into the prism. On the bottom of the fraction, we have the quantity α sub max plus α sub min all divided by two. This is equal to the mean value of α sub max and α sub min. In other words, this is the average angular deviation caused by the prism. But why do different wavelengths have different angular deviations?

Well, in a dispersive material, like a prism, the refractive index of the material varies with wavelength. Air is not a dispersive material. In air, all colors of light experience a refractive index of one. But when the light reaches the prism, the refractive index experienced by the light changes. In fact, this refractive index changes by different amounts for each color. This is what causes the different wavelengths to be refracted through different angles.

For example, the reason that violet light experiences the greatest angle of refraction, α sub max, is because violet light experiences the greatest difference in the refractive index as it enters the prism. This tells us that the index of refraction for violet light is the largest possible value of any color in the prism. So, let’s call the index of refraction for violet light 𝑛 sub max. This means that the largest angle of refraction corresponds to the largest refractive index, α sub max and 𝑛 sub max.

Similarly, red light experiences the smallest angle of refraction, α sub min, because red light experiences the smallest change in the refractive index as it enters the prism. This means that the index of refraction for red light is the smallest value of any color in the prism. We will call the index of refraction for red light 𝑛 sub min. The smallest angle of refraction corresponds to the smallest refractive index, α sub min and 𝑛 sub min.

Let’s go back to the formula for the dispersive power of the prism. Since we know how each angle corresponds to each refractive index, we can swap out all of our α’s for 𝑛’s, like we just described. However, we need to add a minus one into the denominator. This is because we are now thinking about the change in refractive index as the light enters the prism. This one term corresponds to the refractive index of air, which is what prisms are usually surrounded by. So, we end up with this formula. If we compare this to the options given to us in the question, we can see this corresponds to option (D). So, option (D) is the correct answer.

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