Light Waves and Colors Legacy Problem #21 Guided Solution

Here we are told monochromatic yellow light with a wavelength of 594 nanometers passes through two slits that are spaced 0.125 millimeters apart and produces an interference pattern with a collection of alternating bright and dark spots. This interference pattern is projected on a screen 14.5 meters away and we are to determine the distance between the fifth bright spots on the opposite side of the pattern. Right in the middle of the pattern is what we call central bright spot. It would be bright yellow and then protruding outwards to the left and the right of this central bright spot would be an alternating dark yellow bright, dark yellow bright, etc. We want to know a distance on the screen measured from the fifth one to the right of that central bright yellow spot to the fifth one to the left of that central bright yellow spot. We want to determine what we call y in the Young's equation. In order to do that we need to take Young's equation lambda equal yd divided by ml and rearrange it so as to solve for y. That would become y equal ml times lambda divided by d. Now what we need to know are the values of ml, lambda, and d in order to solve for y and we need to pay attention to units. So in terms of knowing the values we know that lambda is equal 594 nanometers. We know that the d is equal to 0.125 millimeters and we know that l equal 14.5 meters. Now the m value represents the number of spacings from the fifth bright spot on one side of the pattern to the fifth on the other side. The number of spacings would be 10. That counts 5 to the spacings to the left and 5 spacings between bright spots to the right. So m is equal to 10. Now before I begin to substitute into this rearranged Young's equation in order to solve for y, I should first pay attention to units. To make it simple, the best strategy, at least the simplest strategy, is to convert all units to meters and substitute into the equation in units of meters. Doing so would go as follows. There are 10 to the ninth little nanometers in one meter. So if I take 594 nanometers and I multiply by a converting factor that has one meter on top and 10 to the ninth nanometers on the bottom, I'll get the wavelength in meters. It would be 5.94 times 10 to the negative 7. I can do a similar thing with the d. d is equal to 0.125 millimeters. There's 10 to the third little millimeters in a meter. So I can divide by 10 to the third and that gives me 1.25 times 10 to the negative fourth meters. And finally I can do the same thing with L, but that'll be quite easy because it's already in meters. Now I can take my values of lambda, n of d, n of L, n of n, and substitute into my equation y equal mL times lambda divided by d. When I do so I'll get an answer for y in units of meters. It comes out to be 0.6890 meters. I can also convert that to centimeters if I wish. 68.9 centimeters.