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? (A) 𝜔 𝛼 equals 𝑛 min plus 𝑛 max divided by 𝑛 min minus 𝑛 max divided by two minus one. (B) 𝜔 𝛼 equals 𝑛 min plus 𝑛 max divided by 𝑛 min minus 𝑛 max divided by two plus one. (C) 𝜔 𝛼 equals 𝑛 max minus 𝑛 min divided by 𝑛 max plus 𝑛 min divided by two minus one. (D) 𝜔 𝛼 equals 𝑛 min minus 𝑛 max divided by 𝑛 min plus 𝑛 max divided by two plus one. (E) 𝜔 𝛼 equals 𝑛 min minus 𝑛 max divided by 𝑛 min minus 𝑛 max divided by two minus one.
So in this question, we want to identify the correct formula for the dispersive power of a prism. The answer options that we’re given are all written in terms of two refractive indices, 𝑛 max and 𝑛 min. We can recall that a prism disperses or spreads out light when it has a different refractive index for different wavelengths of light that passed through the prism. 𝑛 max is the name we’re using for the refractive index of the prism for the longest wavelength of light. And 𝑛 min is the name we’re using for the refractive index for the shortest wavelength of light.
In order to remind ourselves exactly what this looks like, let’s clear some space by removing the question statement and moving the answer options up out of the way. If this triangle represents a prism and this thick arrow represents white light entering the prism, then the prism will disperse or spread out the different colors of the white light by different amounts based on their wavelengths. And the amount that it disperses the light is measured by the dispersive power of the prism. We can use this dashed line which shows the direction the light was originally traveling when it enters the prism to easily see how much the different colors are deviated by the prism.
For example, the light with the shortest wavelength, which here is blue light, will experience the most deviation from the prism. This is because it experiences the greatest difference in index of refraction as it crosses the faces of the prism. And this is precisely why we call the index of refraction for this wavelength of light 𝑛 max, because it’s the maximum value out of any wavelength of light. On the other hand, the color of light with the longest wavelength, which is red light, experiences the least deviation from the prism. This is because the prism has the smallest or minimum value of refractive index for red light. And this is why we call its refractive index 𝑛 min. The dispersive power of a prism measures how much the prism spreads out all of these different colors of light.
So now that we’ve recalled all of this, let’s figure out which of the formulas above is the correct one for the dispersive power. The correct formula is clearly some fraction involving 𝑛 min and 𝑛 max. So let’s start by recalling the correct form for the numerator of that fraction. The correct numerator for the dispersive power is given by the difference between 𝑛 max and 𝑛 min. This makes sense, since the dispersive power is a measure of how much the light is spread out by the prism. So it’s sensible to start by working out the difference between the maximum and minimum refractive index of the prism.
So since the numerator is a difference, we know at the very least it has to include a minus sign. So since the numerator of answer (A) is 𝑛 min plus 𝑛 max, we know this can’t give us a difference, and we can rule out answer (A) as being the correct option. The same is true of option (B). The numerator of this is also not a difference. So we can definitely rule out option (B) as being correct. Usually, when we work out the difference between two numbers, we subtract the smaller number away from the larger number. Because of this, option (C) looks like it might be a good option because it’s the only one whose numerator is 𝑛 max minus 𝑛 min. But let’s not rule out the others just yet. Let’s first think about the correct form for the denominator of the fraction.
The denominator for the formula for the dispersive power of a prism involves the mean of 𝑛 max and 𝑛 min. So recall that the mean of two numbers is found by adding up those two numbers and dividing by two. So the mean of 𝑛 max and 𝑛 min will be 𝑛 max plus 𝑛 min divided by two. This lets us eliminate option (E) because out of our remaining options, it’s the only one whose denominator doesn’t involve the mean of the two refractive indices. We can see that this expression here doesn’t give us the average of 𝑛 min and 𝑛 max. So we can eliminate this option as the correct answer.
Finally, let’s recall the last part of the denominator for the correct formula. By convention, we subtract one from the mean value of 𝑛 min and 𝑛 max. This is to account for the fact that a prism is typically surrounded by air. And we can approximate the refractive index of air to be one. So looking at our remaining answers, we see that we can eliminate option (D) because it has a plus one in the denominator. So we know this can’t be the right answer.
This leaves us with option (C) as our correct answer. And we can see that it satisfies all the conditions we’ve said for the formula for the dispersive power. The numerator is the difference between 𝑛 max and 𝑛 min, while the denominator is the average value of 𝑛 max and 𝑛 min subtract one. And this is the correct answer for the formula of the dispersive power.