Your artist friend is designing an exhibit inspired by circular-aperture diffraction. A pinhole in a red zone is going to be illuminated with a red laser beam of wavelength 670 nm, while a pinhole in a violet zone is going to be illuminated with a violet laser beam of wavelength 410 nm. She wants all the diffraction patterns seen on a distant screen to have the same size. For this to work, what must be the ratio of the red pinhole’s diameter to that of the violet pinhole?

Video transcript

Hello, fellow physicists today, we're gonna solve the following practice from together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem in an optic slab. Two pinholes are being illuminated by different colored lasers. One is an orange colored laser with a wavelength of 590 nanometers and the other laser is indigo with a wavelength of 445 nanometers. The goal of the experiment is to create identical size diffraction patterns on the distant screen. What should the ratio be between the orange pinhole diameter and the indigos pinhole diameter? So that's our end goal. Our goal, we're trying to figure out what the ratio between the orange pinhole and the indi indigo pinhole's diameter is. So we're trying to figure out what the ratio value is. And that's our final answer that we're ultimately trying to solve for is what is this ratio value? So with that in mind, let's read off our multiple choice answers to see what our final answer might be. A is 0.75 B is 1.0 C is 1.3 and D is none of the above or none of these. So with that in mind, first off, we need to recall and use the equation to help us. So the angular width which states that theta is equal to 1.22 multiplied by pi I mean by lambda. So theta which theta in this case is the angular width. So the angular width is equal to 1.22 multiplied by lambda divided by capital D where LAMBDA is the wavelength and capital D is the diameter. So now we need to set our angular width equal to each other. So we need to say that theta orange, which is the angular width of the orange laser is equal to the angular width of the indigo laser, which we're gonna call capital O for orange and capital I for indigo to denote both of those lasers respectfully. So now we can also take this a step further and we could write that the wavelength of the orange laser divided by the diameter of the orange laser is equal to the wavelength of the indigo ray laser divided by the diameter of the indigo laser, which we can also go ahead and take this a step further and write that the diameter of the orange laser divided by the diameter of the indigo laser is equal to the wavelength of the orange laser divided by the wavelength of the indigo laser. So now we need to note that we can substitute in our known values to solve further ratio. So just focusing on do divided by D I, so we could plug in our known variables for do and D I. So we know that do and D I is equal to. So looking at our for the wavelength of the orange laser, we know that the orange laser is equal to 590 nanometers divided by the wavelength of the indigo laser which is 445 nanometers. So our nanometer units cancel out which is perfect because we want our ratio to be in a unit less value. So when we plug that into our calculator and move round to one decimal place, we will get 1.3 and that is our final answer. Hooray, we did it. So looking at our multiple choice answers, the correct answer has to be the letter C 1.3. Thank you so much for watching. Hopefully that helped and I can't wait to see you in the next video. Bye.