Calculate the critical angle for a sapphire surrounded by air.

To calculate the critical angle, we first need to find the refractive index of the sapphire. A quick search provides us with the value: the refractive index of sapphire is approximately 1.77.

Snell's Law of refraction is given by the formula: \(sin\,θ_1\,=\,n_2\,*\,sin\,θ_2\) where \(θ_1\) is the angle of incidence, \(θ_2\) is the angle of refraction, and \(n_2\) is the ratio of the refractive indices (\(\frac{n_{sapphire}}{n_{air}}\)). For the critical angle, the angle of refraction (\(θ_2\)) is 90 degrees. So, we have: \(sin\,θ_1\,=\,n_2\, * sin\,90^\circ\).

Calculate the ratio of the refractive indices, which is the ratio between the refractive index of sapphire and air. \(n_2\,=\,\frac{n_{sapphire}}{n_{air}}\,=\,\frac{1.77}{1}\,=\,1.77\)

Plug the calculated value of \(n_2\) into our modified Snell's Law formula to find the critical angle: \(sin\,θ_1\,=\,1.77 * sin\,90^\circ\) Since \(sin\,90^\circ = 1\), we have: \(sin\,θ_1\,=\,1.77\) Now, find the inverse sine (arcsin) of 1.77 to get the critical angle. However, it's important to note that the sine of an angle cannot exceed 1. Since the sine of the angle is greater than 1 in our formula, it indicates that light is trapped inside the sapphire and total internal reflection occurs for any angle of incidence, and there is no defined critical angle in this case. In conclusion, since the sine of the angle is greater than 1, there is no defined critical angle for a sapphire surrounded by air. Any incoming light will undergo total internal reflection inside the sapphire.