# Question Video: Determining the Refractive Index from the Critical Angle Physics • 9th Grade

What is the refractive index of a material which has a critical angle of 61ยฐ for a light beam traveling from it to air?

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### Video Transcript

What is the refractive index of a material which has a critical angle of 61 degrees for a light beam traveling from it to air?

To begin, letโs recall the formula for determining the critical angle ๐ ๐. The sin of ๐ ๐ equals ๐ two divided by ๐ one, where ๐ one is the refractive index of the first medium that the incident ray is initially in and ๐ two is the refractive index of the second medium on the other side of the medium boundary. To make these quantities clearer, we can draw a diagram like this. Here, weโre considering a ray of light traveling from an unknown medium to air. So ๐ two is the refractive index of air, and ๐ one is the quantity we want to solve for to answer this question.

Letโs go ahead and rearrange this formula to make ๐ one the subject. To do this, we can multiply both sides by ๐ one over the sin of ๐ ๐. This way, ๐ one cancels out of the right-hand side and the sin of ๐ ๐ cancels out of the left-hand side, leaving ๐ one by itself. The equation now reads ๐ one equals ๐ two divided by the sine of the critical angle.

Weโve been told that the critical angle equals 61 degrees. So this is the value of ๐ ๐. Remember too that ๐ two is the refractive index of air, which is simply equal to one. Since we have values for both of the variables on the right-hand side of this equation, weโre ready to substitute them in to reach the final answer. Doing so, we have that ๐ one equals one over the sin of 61 degrees. Plugging this expression into a calculator gives a result of 1.1434 and so on. Rounding to two decimal places, this becomes 1.14, and so we have our final answer.

Thus, weโve found that the refractive index of this unknown material is 1.14.