As an expert in the field of optics, I can guide you through the process of finding the critical angle for a medium with a given refractive index. The critical angle is the angle of incidence at which the angle of refraction is 90 degrees, resulting in total internal reflection. This phenomenon occurs when light travels from a medium with a higher refractive index to one with a lower refractive index.

To find the critical angle, we use Snell's Law, which states that the ratio of the sines of the angles of incidence and refraction is equal to the ratio of the refractive indices of the two media. Mathematically, Snell's Law is represented as:

\[n_1 \sin(\theta_1) = n_2 \sin(\theta_2)\]

where \(n_1\) and \(n_2\) are the refractive indices of the first and second media respectively, and \(\theta_1\) and \(\theta_2\) are the angles of incidence and refraction.

For the critical angle, the angle of refraction \(\theta_2\) is 90 degrees, which means \(\sin(\theta_2) = 1\). The refractive index of the second medium is typically that of air, which is approximately 1. Therefore, the equation simplifies to:

\[n_1 \sin(\theta_c) = 1\]

where \(\theta_c\) is the critical angle. To find \(\theta_c\), we rearrange the equation:

\[\sin(\theta_c) = \frac{1}{n_1}\]

Now, you can calculate the critical angle by taking the inverse sine (arcsin) of the reciprocal of the refractive index.

Let's apply this to the example of water. The refractive index of water is approximately 1.33. Using the formula:

\[\sin(\theta_c) = \frac{1}{1.33}\]

\[\theta_c = \arcsin\left(\frac{1}{1.33}\right)\]

This will give you the critical angle for light passing from water to air.

It's important to note that the refractive index can vary slightly depending on the wavelength of light and the specific properties of the material. However, for most practical purposes, the value of 1.33 for water is sufficiently accurate.

To summarize, finding the critical angle involves understanding Snell's Law and applying it to the specific refractive indices of the media involved. By knowing the refractive index of the first medium and setting the angle of refraction to 90 degrees, you can calculate the critical angle using the inverse sine function.

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The critical angle of a material can be calculated using this equation:refractive index = 1/sin c ... Calculate the refractive index for water, which has a critical angle of 49--.Step 1: Calculate sin c. ... Step 2: Substitute the value of sin c into the equation. ... Step 1: Change the subject of the equation to find c.More items...read more >>