Coherent light with wavelength 450 nm falls on a pair of slits. On a screen 1.80 m away, the distance between dark fringes is 3.90 mm. What is the slit separation?

Video transcript

Fellow physicists today, we're going 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. In young's experiment, the slits are illuminated with coherent light of wavelength 632 nanometers. Careful measurement reveals the distance between the adjacent dark fringes to B 4.3 millimeters. If the screen is 2.2 m from the light source, what should be the slits separation? OK. So our end goal is to determine what the slit separ slit separation should be. OK. So we're given some multiple choice answers. Let's read them off to see what our final answer might be. A is D equals 0.323 millimeters B is D equals 0.58 millimeters C is D equals 0. centimeters and D is D equals 0.45 centimeters. OK. So first off, let us recall and use the equation for the height of the nth dark fringe. So the order of the dark fringe, OK. So let's call that equation one and it states that why subscript N plus one divided by two equals and equals N plus one divided by two, multiplied by the wavelength multiplied by the distance between the double slit and the screen all divided by the separation of the two slits. So the distance between the double slit and the screen is capital D. So it's represented by capital D and lower case D represents the separation of the two slits. So now we need to recall and use the equation for the separation between the adjacent dark fringes. Let's call that equation two. And that states that delta Y equals Y subscript N two plus one divided by two minus Y subscript and one plus one divided by two where delta Y is the distance between the fringes. OK. So now we need to plug in our known variables into equation two. So we get that delta Y is equal to and let's make a quick note here that N two equals two for the second dark fringe order. And that N one represents one for the first order dark fringe. OK. So keeping that in mind plugging in our known variables, N two is 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 minus one plus one is N subscript one, N one equals 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. So we can simplify this using some algebra to state 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. So now we need to take this equation, let's call it equation three and rearrange it to solve for D. So when we do that, we'll get that D equals wavelength multiplied by the distance between the double slit and the screen divided by delta Y where delta lies the distance between the fringes. OK. So at this stage, we can plug in our known variables. And so for D OK. So the wavelength is given to us at 632 nanometers, but we need to convert that to meters which is a quick fix. So we take 632 multiplied by 10 to the power of negative nine to get our nanometer value and to turn it into meters. And we need to multiply that by the distance between the double slit in the screen, which was given to us in the pro as 2.2 m all divided by delta Y which delta Y is given to us in the problem as 4.3 millimeters. But we need to convert millimeters to meters. So all we need to do to convert millimeters to meters is we take four point three multiplied by 10 to the power of negative three M. So that takes our millimeters and converts it to meters. So when we plug that into our calculator, we should get 3.23 multiplied by 10 to the power of negative 4 m. But we need to convert our meter value to millimeters. So when we do that, we'll get that D equals 0. millimeters. Hooray, we did it. We found our final answer. Ok? So that means that our final answer has to be the letter ad equals 0. millimeters. Thank you so much for watching. Hopefully that helped and I can't wait to see you in the next video. Bye.