In a ruby laser, laser light of wavelength \(694.3 \mathrm{nm}\) is emitted. The ruby crystal is \(6.00 \mathrm{cm}\) long, and the index of refraction of ruby is \(1.75 .\) Think of the light in the ruby crystal as a standing wave along the length of the crystal. How many wavelengths fit in the crystal? (Standing waves in the crystal help to reduce the range of wavelengths in the beam.)

To find the wavelength of the light inside the ruby crystal, we need to consider the index of refraction, which is given by: Index of refraction = \(\frac{Speed\:of\:light\:in\:vacuum}{Speed\:of\:light\:in\:medium}\) We can rewrite this equation to find the speed of light in the ruby crystal: Speed of light in ruby = \(\frac{Speed\:of\:light\:in\:vacuum}{Index\:of\:refraction}\) Using the provided index of refraction (1.75), we can find the speed of light in the ruby crystal. The wavelength in the ruby crystal can be found using the following equation: Wavelength in ruby = \(\frac{Wavelength\:in\:vacuum}{Index\:of\:refraction}\) Wavelength in ruby (\(\lambda_{ruby}\)) = \(\frac{694.3 \times 10^{-9}\, m}{1.75}\)

To find the number of wavelengths that fit inside the ruby crystal, we can simply divide the length of the ruby crystal (6.00cm) by the wavelength of the light inside the crystal. Number of wavelengths (n) = \(\frac{Length\:of\:ruby\:crystal}{Wavelength\:in\:ruby}\) Before we substitute the values, we need to convert the length of the ruby crystal to meters: Length of the ruby crystal = \(6.00\, cm = 0.06\, m\) Substituting the values: Number of wavelengths (n) = \(\frac{0.06\, m}{\frac{694.3 \times 10^{-9}\, m}{1.75}}\)

Now, we simply need to compute the value to find the number of wavelengths that fit inside the ruby crystal. Number of wavelengths (n) = \(\frac{0.06\, m \times 1.75}{694.3 \times 10^{-9}\, m}\) Number of wavelengths (n) ≈ \(151.66\) Since wavelengths must be whole numbers, let's round this value to the nearest whole number: Number of wavelengths (n) ≈ \(152\) Hence, approximately 152 wavelengths fit inside the ruby crystal as a standing wave.