Light Waves and Colors Legacy Problem #20 Guided Solution
Problem*
The Bluebird Library has been celebrating the lives of famous scientists. Each month, a new scientist is selected, and displays are created to feature the discoveries and contribution of the scientist. April's scientist of the month is Thomas Young; the library wishes to develop a Young's experiment display. The library has purchased a blue laser which emits light with a wavelength of 473 nm. They also have purchased a slide with a double slit; the slit spacing is 44 µm. The library's current plans are to project the interference pattern onto a white board which is 3 feet wide and located 28 feet from the slits. What is the maximum number of bright spots which will appear on the board at these distances and what is the spacing distance between each bright spot? Assume that each bright spot is bright enough to see. (GIVEN: 1 m = 3.28 ft, 1 m = 106 mm, 1 m = 109 nm)
Audio Guided Solution
The Blue Bird Library is celebrating the life of Thomas Young. They're setting up a Young's experiment display. They're using blue laser light, and the wavelength, lambda, equals 473 nanometers. They purchase a double slit through which they'll shine the light, and the slit spacing, which is D of Young's equation, is equal to 44 micrometers. The library plans to project the interference pattern onto a whiteboard, which is 3 feet wide and located 28 feet from the slits. So the L equals 28 feet, and the Y value is equal to 3 feet. The question is, within this Y value of 3 feet, how many bright spots can we get to appear within that distance? And so we need to use Young's equation in order to solve for M. Doing so means that we take lambda equal YD divided by ML and rearrange it so that it becomes M equal Y times D divided by lambda divided by L. We need to substitute in values of YD, lambda, and L in order to solve for M, and the big thing is that we have to have them all in a consistent unit so that all of our units cancel. So just for simplicity's sake, I'm going to convert all the quantities to units of meters. So the Y of 3 feet can be divided by the 3.28 feet per meter to get a Y value of 0.9146 meters. And the L of 28 feet can also be divided by 3.28 feet so that we get an L value of 8.5366. The D of 44 micrometers can be divided by 10 to the 6th so that D is 4.4 times 10 to the negative 5th meters. And finally, the lambda of 473 nanometers could be converted to 4.73 times 10 to the negative 7th meters. Now I can take all units and substitute it into my rearranged Young's equation and solve for M. So when I substitute Y times D and divide by lambda times L, I end up getting an M value of 9.9668, just a little short of 10, but so close to 10, we could probably call it 10. Now this M value represents the number of spacings between bright spots. So if asked the question, how many bright spots would appear on the screen, well to get 10 spacings, you need 11 bright spots, and that includes the central one, and 5 to the left, and 5 to the right of that central one. Now if I ask how far apart are they spaced, I would have to take this value of 10 for the value of M and substitute it back into my Young's equation, but use it to solve for Y. And so I can rearrange Young's equation to become Y equal M times L times lambda divided by D. And I would find that they're spaced 0.09177 meters apart, or 0.092 meters. That can be converted to centimeters, and it becomes 9.2 centimeters.
Solution
just barely 11 bright spots (including the central one) they are spaced a little less than 9.2 cm apart
Habbits of an Effective Problem Solver
- Read the problem carefully and develop a mental picture of the physical situation. If necessary, sketch a simple diagram of the physical situation to help you visualize it.
- Identify the known and unknown quantities and record in an organized manner, often times they can be recorded on the diagram itself. Equate given values to the symbols used to represent the corresponding quantity (e.g., \(\descriptive{v}{v,velocity} = \num{3e8}\unit{\meter\per\second}\), \(\descriptive{λ}{λ,wavelength} = 554 \unit{\nano\meter}\), \(\descriptive{f}{f,frequency} = \colorbox{gray}{Unknown}\)).
- Use physics formulas and conceptual reasoning to plot a strategy for solving for the unknown quantity.
- Identify the appropriate formula(s) to use.
- Perform substitutions and algebraic manipulations in order to solve for the unknown quantity.
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