Video Transcript
White light dispersed by a prism
has wavelengths ranging from 400 nanometers to 700 nanometers. The 400-nanometer-wavelength light
has a minimum angle of deviation of 22.9 degrees. The 700-nanometer-wavelength light
has a minimum angle of deviation of 22.1 degrees. And the 550-nanometer-wavelength
light has a minimum angle of deviation of 22.5 degrees. What is the dispersive power of the
prism? Give your answer to three decimal
places.
So in this question, we have a
prism, which we’ll represent by this triangle. White light enters this prism and
is dispersed because the different wavelengths of light that make up the white light
are deviated by different amounts when they pass through the prism. For example, we see that red light,
which is the longest wavelength light, is deviated much less than blue light, which
has a much shorter wavelength.
The dispersive power of a prism is
a number which tells us how much the prism spreads out white light, and this is
exactly what we’re asked to calculate in this question. The larger the dispersive power of
the prism, the more the prism spreads out or disperses white light. We represent the dispersive power
of a prism with the Greek letter 𝜔, along with a subscript 𝛼. We put an 𝛼 in the subscript
because 𝛼 is the symbol we usually use to denote the angle of deviation for a
particular wavelength of light.
If we draw a dashed line to show
the direction the white light was traveling when it entered the prism, then the
angle of deviation for a given wavelength of light is the angle between this dashed
line and the direction that wavelength of light is traveling when it leaves the
prism. For example, the angle of deviation
for red light is shown here. And we can label this 𝛼 subscript
red. We can also see that the angle of
deviation for blue light is much larger. And we can label this angle 𝛼
subscript blue.
Let’s now recall the formula that
lets us calculate the dispersive power of a prism using the angles of deviation for
the different wavelengths of light passing through the prism. This formula is written like this
and is often given in terms of the symbols 𝛼 max and 𝛼 min, where 𝛼 max is the
largest angle of deviation for the prism and 𝛼 min is the smallest angle of
deviation for the prism. In terms of these two quantities,
the formula for the dispersive power of a prism is written as a fraction. And on the top of that fraction, we
have 𝛼 max minus 𝛼 min. So this is the difference between
the largest angle of deviation and the smallest angle of deviation. And on the bottom of the fraction,
we have 𝛼 max plus 𝛼 min all divided by two, which actually gives us the average
angle of deviation for the prism.
So to calculate the dispersive
power for our prism, we just need to identify 𝛼 max and 𝛼 min for our prism and
then substitute these values into the formula for 𝜔 𝛼. The question tells us that the
longest wavelength of light passing through the prism is 700 nanometers. And we’re also told that this
wavelength of light will experience an angle of deviation of at least 22.1
degrees. 700-nanometer light actually
corresponds to red light. And since we know that this is the
longest wavelength of light passing through the prism, we know it will be deviated
by the smallest or minimum amount.
This is exactly what we saw in our
original sketch of a prism, and it means that we can identify the minimum angle of
deviation with the angle of deviation for red light. Since we’re told this value in the
question, we can then simply write that 𝛼 min is equal to 22.1 degrees. We’re also told that the shortest
wavelength light passing through the prism has a wavelength of 400 nanometers and
that this light will experience an angle of deviation of at least 22.9 degrees. This wavelength of light
corresponds to blue light. And just as we saw earlier, blue
light will experience the maximum deviation out of all the colors that make up the
white light.
This means we can say that the
maximum angle of deviation is the same as the angle of deviation for blue light. And again, since we’re told this
value in the question, we can say that 𝛼 max is equal to 22.9 degrees. At this point, all we need to do is
substitute the values we’ve just found for 𝛼 min and 𝛼 max into the formula we
have for the dispersive power 𝜔 𝛼. Doing this gives us this formula
here. In the numerator, we had 𝛼 max
minus 𝛼 min, and this reads 22.9 degrees minus 22.1 degrees. And in the denominator, we have 𝛼
max plus 𝛼 min divided by two, which now reads 22.9 degrees plus 22.1 degrees all
divided by two.
If we now calculate the top and
bottom of this fraction separately, we find that 𝜔 𝛼 is equal to 0.8 degrees
divided by 22.5 degrees. We can note that the denominator of
this fraction 22.5 degrees, which we found by calculating the average value of the
angles of deviation of the prism, is also the angle of deviation for light with a
wavelength of 550 nanometers. And we’re told this in the
question. This is because 550 is the average
of 400 and 700 nanometers. And so we see that the average of
the angles of deviation of the prism is the same as the angle of deviation for the
average wavelengths of light passing through the prism. And since we were told the angle of
deviation for the average wavelengths of light passing through the prism, we
could’ve used this as a shortcut to know the denominator of this formula.
Either way, all that’s left to do
now is calculate the fraction 0.8 degrees divided by 22.5 degrees. If we do this calculation, we find
that 𝜔 𝛼 is equal to 0.0355 and so on. Notice that this number doesn’t
have any units attached to it because the degrees on the top and bottom of the
fraction have canceled out. Finally, we just need to round our
answer to three decimal places as we’re told in the question. Doing this gives us our final
answer to the question. We found that the dispersive power
of the prism, 𝜔 α, is equal to 0.036.
