### 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.