Two slits spaced 0.450 mm apart are placed 75.0 cm from a screen. What is the distance between the second and third dark lines of the interference pattern on the screen when the slits are illuminated with coherent light with a wavelength of 500 nm?

Video transcript

A little fellow physicist today want to solve the following practice problem together. So first off, let's read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. A rectangular board is placed 85 centimeters away from 20.34 millimeter wide slits to observe the interference pattern through the small slits coherent light of wavelength of 610 nanometers pass. And an interference pattern is observed on the screen determine the distance between the interference patterns 1st and 2nd dark lines. OK. So our end goal is to determine the distance between the interference patterns 1st and 2nd dark lines. Awesome. So we're given some multiple choice values here and there are all different units. So let's read them off to see what our final answer might be. So A is delta Y equals 0.55 centimeters. B is delta, Y equals 2.5 millimeters. C is delta Y equals 1.53 millimeters and D is delta Y equals 0.81 centimeters. OK. So first off, let us recall and use the equation to determine the height of the nth dark fringe, let's call it equation one. So Y subscript N plus one divided by two, where N is the order or the nth dark French, which we're trying to find the 1st and 2nd French, which will talk more about that in a second equals N plus one divided by two multiplied by the wavelength multiplied by the distance between the double slit and the screen, which is represented by capital D divided by lower case D where lower case D represents the separation of the two slits. OK. So we also need to recall and use the equation for the separation between the a adjacent dark fringes. So to find the separation between the adjacent dark fringes, we'll call it equation two equation two states that delta Y equals Y subscript M two or I should say N two plus one half minus Y subscript and one plus one half. So let's make a quick note here that N one equals the first dark fringe and N two equals the second dark fringe. OK. And then delta wi the distance between the fringes. OK. So we can now take our known variables and plug them into equation two. So then we can now write delta Y is equal to two plus one divided by two multiplied by the wavelength multiplied by the distance between the double slit and the screen divided by the separation of the two slits subtracted by one plus one divided by two multiplied by the wavelength multiplied by the distance between the double slit and the screen divided by the separation of the two slits. And we can simplify this equation using some algebra to say that delta Y equals the wavelength multiplied by the distance between the double slit and the screen divided by the separation of the two slits. And at this stage, we can plug in all of our known variables to solve for delta Y. So let's do that. OK. So we're given the wavelength in nanometers, but we need to convert nanometers m. So let's do that and it's a quick fix. So 610 multiplied by 10 to the power of negative 9 m. So that's how we convert nanometers to meters multiplied by the distance between the double slit and the screen which was given to us in the problem as centimeters when we need to convert centimeters to meters. So all we have to do is just take 85 multiplied by 10 to the power of negative 2 m. And then all of that divided by the separation between the two slits which was 0.34 millimeters, but we need to convert millimeters to meters. So we take 0.34 multiplied by 10 to the power of negative 3 m. OK? So when we plug that into a calculator, we get 0. m. But we need to convert our answer to millimeters. So when we do that, we should get 1.5 three millimeters. So delta Y equals 1.53 millimeters, which is our final answer array we did it. So that means that our final answer has to be C delta Y equals 1.53 millimeters. Thank you so much for watching. Hopefully that helped and I can't wait to see you in the next video. Bye.