### Video Transcript

A very thin triangular prism has an
apex angle of 2.5 degrees. The minimum deviation angle through
the prism is 1.4 degrees. Find the refractive index of the
prism using the small-angle approximation. Answer to one decimal place.

This question is asking us to find
the refractive index of a very thin triangular prism. We are told that the apex angle of
the prism, which we’ll label as 𝐴, is 2.5 degrees. We’re also told that the minimum
deviation angle through the prism, which we’ll label as 𝛼 naught, is 1.4
degrees. To find the refractive index of the
prism, we can use the thin prism approximation. The refractive index 𝑛 is equal to
the sum of 𝛼 naught and 𝐴 divided by 𝐴.

There are two things we have to be
careful about when we use this formula. Firstly, this formula is only valid
for very thin prisms, which have a small apex angle. In this question, we’re explicitly
told that the prism is very thin and that the apex angle is a very small 2.5
degrees. This is what the question is
referring to when it says using the small-angle approximation. To use this formula, we need to be
working with small apex angles, which in this case we know that we are.

The second thing we have to be
careful of when we use this formula is that the apex angle and the minimum deviation
angle must be expressed in radians. In this question, we’ve been given
these values in degrees. So, before we do anything else,
let’s convert them to radians.

To do this, we simply multiply each
angle in degrees by 𝜋 divided by 180 degrees. Now all we need to do is to
substitute these values into the thin prism approximation formula. This tells us that the refractive
index 𝑛 is equal to 1.4 degrees times 𝜋 over 180 degrees plus 2.5 degrees times 𝜋
over 180 degrees all divided by 2.5 degrees times 𝜋 over 180 degrees. Completing this calculation, we
find that the refractive index 𝑛 is equal to 1.56. This question asks us to give our
answer to one decimal place, so we can round this result up, giving a value of
1.6. We should note by the way that the
refractive index is a dimensionless quantity, which means that it doesn’t have any
units.

So our final answer is simply that,
to one decimal place, the refractive index 𝑛 is equal to 1.6.