# 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 61° 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, let’s recall the formula for determining the critical angle 𝜃 𝑐. The sin of 𝜃 𝑐 equals 𝑛 two divided by 𝑛 one, where 𝑛 one is the refractive index of the first medium that the incident ray is initially in and 𝑛 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, we’re considering a ray of light traveling from an unknown medium to air. So 𝑛 two is the refractive index of air, and 𝑛 one is the quantity we want to solve for to answer this question.

Let’s go ahead and rearrange this formula to make 𝑛 one the subject. To do this, we can multiply both sides by 𝑛 one over the sin of 𝜃 𝑐. This way, 𝑛 one cancels out of the right-hand side and the sin of 𝜃 𝑐 cancels out of the left-hand side, leaving 𝑛 one by itself. The equation now reads 𝑛 one equals 𝑛 two divided by the sine of the critical angle.

We’ve been told that the critical angle equals 61 degrees. So this is the value of 𝜃 𝑐. Remember too that 𝑛 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, we’re ready to substitute them in to reach the final answer. Doing so, we have that 𝑛 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, we’ve found that the refractive index of this unknown material is 1.14.