Two slits spaced 0.260 mm apart are 0.900 m from a screen and illuminated by coherent light of wavelength 660 nm. The intensity at the center of the central maximum 1u = 0°2 is I0. What is the distance on the screen from the center of the central maximum (a) to the first minimum

Video transcript

Hello, fellow physicists today, we're gonna 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 sodium vapor lamp gas discharge lamp produces light with a wavelength of 610 nanometers. By using sodium in a excited state. The light from the source is illuminating the two narrow slits. S one and S two S subscript one and S subscript two that are D equals two D equals 0.24 m millimeters apart. A rectangular screen, 0.89 m from the slits displays the pattern. The intensity of the light at the central maximum beta equals zero degrees is I subscript zero. What should the distance from the center of the central maximum at the first minimum? OK. So our end goal is to determine what the distance from the center of the central maximum at the first minimum is. So what should the distance be? OK? That's what we're trying to find. So we're given some multiple choice answers. Let's read them off to see what our final answer might be A is 1. millimeters. B is 21.3 centimeters. C is 3.12 millimeters and D is 11. centimeters. OK. So first off, let us recall and use the equation to determine the minimum intensity which states that D multiplied by sine theta equals N plus one divided by two multiplied by the, by lambda by the wavelength. OK. So D is the distance theta is the angle N is the order number which is some integer. And of course, lambda is the wavelength. OK. So note that the first dark fringe after the central maxima is N equal zero. So N equals zero for the first dark French. So I'm gonna say first dark fringe. OK. So let's plug that information into the equation for the minimum intensity. So D multiplied by sine theta equals zero plus one divided by two multiplied by Lambda. Note that sign is equal to zero. That sine theta is equivalent. Oh Yeah, I should rewrite that. I apologize. That sign theater is equivalent to beta. OK. So using that tidbit of information, we can write that theta equals 0.5 lambda divided by D. OK. So now we can plug in our known variables to solve your data. So let's do that 0.5 multiplied by the wave length which was 610 nanometers. But we need to convert that to meters. So all we have to do is take 610 multiplied by 10 to the power of negative m divided by D the distance which the distance in this case is 0.24 millimeters, but we need to convert millimeters to meters. So all we need to do is take 0.24 multiplied by 10 to the power of negative three millimeters. OK. So 0.24 multiplied by 10 to the power of negative three. So when we plug that into a calculator, we should get 0. radiance, we can also rewrite it in scientific notation. So it's easier to type into a calculator as 1.3 multiplied by 10 to the power of negative three rad radiant. OK? No, we need to recall and use the equation to determine the distance between the central Maxima and the first dark fringe. And that equation is Y equals capital R multiplied by tangent of theta where capital R is the distance of the rectangular screen from the slits. So let's plug in our known values to solve for A Y. And let's note that the distance of the rectangular screen from the slits was 0.89 m multiplied by tangent of the theta value or tangent of the theta value we just found which was 1. multiplied by 10 to the power of negative three radiance. So when we plug that into a calculator, we should get that Y equals 0. meters. OK? Which we need to convert this to millimeters. And when we do that, we get, when we round up, we get 1.1 six. So 1.16 millimeters. Awesome. So that means that our final answer is 1.16 millimeters. So that leaves us with the correct answer being A A is 1.16 millimeters. Thank you so much for watching. Hopefully that was helpful and I can't wait to see you in the next video. Bye.