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Question Video: Using the Thin-Prism Approximation to Find the Refractive Index of a Prism Physics

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

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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 weโ€™ll label as ๐ด, is 2.5 degrees. Weโ€™re also told that the minimum deviation angle through the prism, which weโ€™ll label as ๐›ผ naught, is 1.4 degrees. To find the refractive index of the prism, we can use the thin prism approximation. The refractive index ๐‘› is equal to the sum of ๐›ผ naught and ๐ด divided by ๐ด.

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, weโ€™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, weโ€™ve been given these values in degrees. So, before we do anything else, letโ€™s convert them to radians.

To do this, we simply multiply each angle in degrees by ๐œ‹ 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 ๐‘› is equal to 1.4 degrees times ๐œ‹ over 180 degrees plus 2.5 degrees times ๐œ‹ over 180 degrees all divided by 2.5 degrees times ๐œ‹ over 180 degrees. Completing this calculation, we find that the refractive index ๐‘› 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 doesnโ€™t have any units.

So our final answer is simply that, to one decimal place, the refractive index ๐‘› is equal to 1.6.

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