12. Calculate the wavelength of {}^ast (a) the sodium line at 589 nm in an aqueous solution with a refractive index of 1.35 . (b)the output of a ruby laser at 694.3 nm when it is passing through a piece of quartz that has a refractive index of 1.55 .

To calculate the wavelength of light in a medium with a refractive index, we can use the formula:<br /><br />\[ \lambda_{\text{medium}} = \frac{\lambda_{\text{vacuum}}}{n} \]<br /><br />where \( \lambda_{\text{medium}} \) is the wavelength in the medium, \( \lambda_{\text{vacuum}} \) is the wavelength in vacuum, and \( n \) is the refractive index of the medium.<br /><br />(a) For the sodium line at 589 nm in an aqueous solution with a refractive index of 1.35:<br /><br />\[ \lambda_{\text{aqueous}} = \frac{589 \, \text{nm}}{1.35} \]<br /><br />\[ \lambda_{\text{aqueous}} \approx 435.19 \, \text{nm} \]<br /><br />(b) For the output of a ruby laser at 694.3 nm when it is passing through a piece of quartz with a refractive index of 1.55:<br /><br />\[ \lambda_{\text{quartz}} = \frac{694.3 \, \text{nm}}{1.55} \]<br /><br />\[ \lambda_{\text{quartz}} \approx 448.71 \, \text{nm} \]<br /><br />Therefore, the wavelength of the sodium line in an aqueous solution is approximately 435.19 nm, and the wavelength of the ruby laser output in quartz is approximately 448.71 nm.