Video Transcript
A prism has a dispersive power of 0.044. White light is dispersed by the prism. The longest wavelength light in the white light has a minimum angle of deviation of 25.8 degrees. What is the minimum angle of deviation of the shortest wavelength light in the white light? Answer to one decimal place.
Okay, let’s say that this triangle represents the prism we’re asked about in this question and this thick arrow here represents the white light entering the prism. This white light is traveling in some direction when it enters the prism. And we can show this direction by this dashed line here. This represents the direction the light would continue to travel in if it hadn’t interacted with the prism. However, since the light hits the prism, we know that the prism will refract the light and change the direction it’s traveling in. We also know the prism will refract the different wavelengths or colors of light by different amounts, resulting in the different colors of light being dispersed or spread out by the prism.
We know that this prism definitely spreads out the different colors of light because we’re told that the dispersive power of this prism is equal to 0.044. And it’s important to note this number is greater than zero. The dispersive power of a prism is a measure of how much the prism spreads out light and the more the prism spreads out the different colors of light, the greater its dispersive power will be. So if a prism has a dispersive power of zero, we know that it doesn’t spread out the colors of light at all. But since our prism has a dispersive power of 0.044, we know that it must spread out the colors of light to some extent.
We usually denote the dispersive power of a prism with the symbol 𝜔 𝛼. And since we’re told the dispersive power of this prism, we can write down that 𝜔 𝛼 equals 0.044 for this prism. The other thing we’re told in this question is the angle of deviation for the longest wavelength of light passing through the prism. The angle of deviation for a certain wavelength of light is given by the angle between the direction the light was traveling in when it entered the prism, which is shown by this dashed line here, and the direction that wavelength of light is traveling in when it exits the prism. So for red light, the angle of deviation is given by this angle here. In fact, since we know that red light has the longest wavelength out of all the colors that make up white light, the angle of deviation we’re told is precisely the angle of deviation for red light.
So we know the angle of deviation for red light is equal to 25.8 degrees. And we denote this angle by the symbol 𝛼 subscript min. And this is because we know that the red light will have the minimum angle of deviation out of all the colors of light passing through the prism. This is because the prism will have the minimum refractive index for the longest wavelength of light, meaning they get deviated the least. On the other hand, the shortest wavelength of light that passes through the prism, which is blue light in this case, will have the largest refractive index and hence will have the largest angle of deviation. Because of this, we denote the angle of deviation for blue light with the symbol 𝛼 max, because it’s the maximum angle of deviation that any color of light will experience from this prism. It’s actually this angle that we’re asked to calculate in this question. So our goal now is to find the value of 𝛼 max. Let’s clear some space on the screen and see how we can do that.
So here we have the two quantities we know. The dispersive power of the prism, 𝜔 𝛼, is equal to 0.044 and the angle of deviation for the longest wavelength of light, 𝛼 min, is equal to 25.8 degrees. And the quantity we want to calculate the angle of deviation for the shortest wavelength of light is 𝛼 max. So let’s begin by recalling the equation that links these three quantities together. This equation reads 𝜔 𝛼 equals 𝛼 max minus 𝛼 min divided by 𝛼 max plus 𝛼 min divided by two. This is great because it gives us the quantity we want, 𝛼 max, in an equation along with all the other things that we do know, such as 𝜔 𝛼 and 𝛼 min. So our goal now will be to rearrange this equation so that 𝛼 max is the subject of the equation. However, since 𝛼 max appears in both the numerator and the denominator of this equation, there’s going to be quite a few steps to do this rearranging.
So let’s first substitute in the two quantities we do know with the values we’re told for them. Since 𝜔 𝛼 equals 0.044, the left-hand side of this equation is just 0.044. And since 𝛼 min equals 25.8 degrees, we’ve replaced 𝛼 min with this value everywhere it appears in the right-hand side of the equation. Next, let’s multiply the right-hand side of the equation by two divided by two. Since two divided by two is just equal to one, this is just like multiplying the right-hand side by one, so it doesn’t change anything about our equation. This might sound like an odd thing to do, but it lets us simplify the denominator of the right-hand side quite nicely. This is because the denominator now reads two times 𝛼 max plus 25.8 degrees divided by two. So the two on the top and bottom cancel out, leaving us with a simplified right-hand side.
We could then simplify the numerator of the right-hand side by expanding out the brackets, which means we multiply two by the two terms inside the brackets. So we have two times 𝛼 max minus two times 25.8 degrees. So the numerator now reads two 𝛼 max minus 51.6 degrees. Next, we want to multiply both sides of the equation by what appears in the denominator of the right-hand side, which is 𝛼 max plus 25.8 degrees. This gives us an equation that looks like this. And this is helpful because the right-hand side now has 𝛼 max plus 25.8 degrees on the top and on the bottom of the fraction. So these two terms cancel out, leaving the right-hand side that just reads two 𝛼 max minus 51.6 degrees. However, on the left-hand side, we now have some brackets that we need to expand. And we do this by multiplying both terms inside the brackets by 0.044. This gives us a left-hand side that reads 0.044 times 𝛼 max plus 1.1352 degrees.
Okay, so our equation is starting to look a little bit simpler. And our next step is to collect both terms involving 𝛼 max on one side of the equation and both terms that are just numbers on the other side of the equation. We can start doing this by adding 51.6 degrees to both sides of the equation. This is helpful because on the right-hand side, we now have minus 51.6 degrees plus 51.6 degrees. So these two terms cancel to zero. And on the left-hand side, we have two terms we can combine together because they’re just numbers of degrees. So we can do 1.1352 degrees plus 51.6 degrees which combine to give us a term that reads plus 52.7352 degrees.
Now let’s subtract 0.044𝛼 max from both sides of this equation. This lets us simplify the left-hand side because we now have 0.044𝛼 max minus 0.044𝛼 max. So these two terms again cancel to zero, leaving a left-hand side that just reads 52.7352 degrees. On the right-hand side, we now have two 𝛼 max minus 0.044𝛼 max. So we’ve achieved our goal of getting both terms involving 𝛼 max on the same side of the equation. And we can go one step further and factorize out 𝛼 max from these two terms, which lets us write the right-hand side as two minus 0.044 times 𝛼 max. And this bracket is now just the difference between two numbers. So we can calculate this to give us the right-hand side, that’s just 1.956, times 𝛼 max.
The final step in all of our rearranging is to divide both sides of our equation by 1.956, because this lets us cancel the number completely out of the right-hand side. And we’re finally left with an equation for which 𝛼 max is the subject. All we need to do now to calculate 𝛼 max is to work out this fraction here, 52.7352 degrees divided by 1.956. And if we do this, we find 𝛼 max is equal to 26.9607 and so on degrees. This is almost our final answer. But recall that the question asked us to give our answer to one decimal place, so we need to round our answer. If we do this rounding, we can give our final answer as 𝛼 max equals 27.0 degrees.
