Light Waves and Colors Legacy Problem #17 Guided Solution

In Young's experiment, light is shined through two slits and projected onto a pattern. The laser light creates an interference pattern consisting of alternating nodes and antinodes. When projected onto the screen, what we observe are alternating bright and dark spots resulting from constructive and destructive interference in the laser light. Here Jackson and Melanie are repeating this experiment. They're using a slit that has a slit spacing of 25 micrometers. That would be represented by the variable d in Young's equation. So we can say d equals 25 micrometers. They project the pattern onto a whiteboard located 2.35 meters from the slits, and that's the distance L in Young's equation. So we can say L equals 2.35 meters. We're told that they measure the distance from the third bright band to the third bright band on the opposite side of the pattern. This is a distance of 37 centimeters, and this is represented by y. y equals 37 centimeters. And the m that goes with this value of y is equal to 6 if they measure the distance from the third bright band past the center bright band to the third bright band on the opposite side of the pattern. That would be equivalent to six different spacings between bright bands, and that's why m is equal to 6. Now we wish to calculate the wavelength of the red laser light, so this is a manner of substituting our values of y, d, m, and L into Young's equation. We say lambda equals y times d divided by m divided by L. But we have to make sure that all of our units are consistent with one another, so we need to convert two of these quantities to the same unit as the third one. I'm going to pick the unit meters as my destination conversion unit. So I convert 25 micrometers to meters by dividing by 10 to the 6th, so the 25 micrometers becomes 2.5 times 10 to the negative 5th meters. And I convert the 37 centimeters to meters by moving the decimal place twice, and I get 0.37 meters. Now I'm ready to substitute into Young's equation. So I say wavelength equals y times d divided by m divided by L. So I go y, which is 0.37 meters, times d, which is 2.5 times 10 to the negative 5th meters, divided by m of 6 and divided by L of 2.35 meters. That gives me a value of 6.5603 times 10 to the negative 7th meters, and I'm asked to get this in units of nanometers. So I multiply by 10 to the 9th, which gives me 656.03 meters, and I round this to two significant digits such that it becomes 660 meters.