Parallel rays of monochromatic light with wavelength 568 nm illuminate two identical slits and produce an interference pattern on a screen that is 75.0 cm from the slits. The centers of the slits are 0.640 mm apart and the width of each slit is 0.434 mm. If the intensity at the center of the central maximum is 5.00x10^-4 W/m2, what is the intensity at a point on the screen that is 0.900 mm from the center of the central maximum?

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. An experiment is designed to provide evidence that light has a wave like character. The experiment is to be based on the phenomena of interference between light waves. The apparatus for the experiment consists of a light source 685 nanometers, two narrow slits acting as coherent sources and an observation screen. The slits are apart by 0.714 millimeters and each slit is 0.423 millimeters wide. When the light source illuminates the slits interference patterns can be seen on the screen that is 70 centimeters for the slits. The central or zeroth fringe is the brightest fringe and has the greatest intensity of 5.4 multiplied by 10 to the power of negative or watts per meter squared, find the intensity of a point on the screen that is 0.800 millimeters from the center of the central maximum. And that is our end goal is to find the intensity of a point on the screen that is 0.800 millimeters from the center of the central maximum. OK. So we're given some multiple choice answers and they're all I equals and they're all in the same units of watts per meter squared. So let's read them off and see what our final answer might be. A is 9.8 multiplied by 10 to the power of negative three. B is 5.3 multiplied by 10 to the power of negative six C is 8.4 multiplied by 10. The power of negative 10 and D is 7.2 multiplied by 10 to the power of negative five. OK. So first off, let us recall the equation for intensity considering that the intensity depends on the single and double slit diffraction. Let's call it equation one. So the intensity I is equal to I subscript zero which is gonna represent the greatest intensity for the brightest fringe multiplied by oh sign squared A F divided by two multiplied by sign beta divided by two. So sin of beta divided by two, all divided by be divided by two all to the power of two, all square where Phi is equal to two pi multiplied by D multiplied by sine beta divided by Lambda. And the phase difference B or I should say phase difference beta is equal to two pi multiplied by a multiplied by sine theta divided by lambda. OK. So now we can plug in Phi and beta into equation one. So let's rewrite intensity. So intensity equals I zero, I subscribe zero, multiplied by pine squared of I multiplied by D multiplied by sine data divided by Lambda multiplied by sign of I multiplied by a multiplied by sine theta divided by Lambda all divided by pi multiplied by a multiplied by sine theta divided by Lambda all square. OK. So we also need to make the following note that tangent of theta is equal to Y divided by capital R where Y is the point of. So it's the OK. So it's the intensity of the point on the screen and capital R is the interference patterns. So it's the difference, it's the distance between the interference pattern seen on the screen from the slits. OK. And we also need to make a falling note that tan tangent of data is also equal to sine of data. So with that in mind, we can write that sine of theta is equal to Y divided by capital R is equal to theta is equal to sign the inverse of sign of. And when we can at this stage, we can plug in our values for A Y and R. So let's do that 0.800. So Y equals 0.800 millimeters, but we need to convert millimeters to meters. So the juices take 0.800 multiplied by 10 to the power of negative three to convert it to meters divided by R which is RS given to us as 70 centimeters. So we need to convert centimeters to meters. So all we have to do is just take 70 multiplied by 10 to the power of negative two to convert it to meters. So when we plug that into a calculator, we get 0.00114 but we can rewrite it in scientific notation to make it quicker to type into a calculator. And that will be 1.1 multiplied by 10 to the power of negative three radians. OK. So using sine theta to find Phi and beta, let's do that. So we can use sine theta to help us find Phi and beta the phase difference. So let's start off by solving for Phi. So Phi is equal and we need to plug in all of our known variables is equal to two pi multiplied by zero point 714 millimeters which is the slit this like the slits are separated by this amount. And we need to convert the millimeters to meters. So all I have to do is take 0.714 and multiply it by 10 to the power of negative three multiplied by the value we just found for sine theta which was 1.1 multiplied by 10 to the power of negative three radiance all divided by the wavelength which lambda represents the wavelength which is given to us as 685 nanometers. But we need to convert nanometers to meters. So like the douche takes 685 and multiply it by 10 to the power of negative nine. OK. So when we plug it into a calculator, we should get 7.20 radiance. So now let's start to solve for the phase difference beta. So beta is equal to two pi multiplied by a which are a value in this case is 0.423 millimeters, but we need to convert millimeters to meters. So all we have to do is just take 0.423 multiply it by 10 to the power of negative three multiplied by sin theta 1.1 multiplied by 10 to the power of negative three radiance all divided by lambda. The wavelength which was 685 multiplied by 10 to the power of negative nine meters. OK? So let me plug that into a calculator. We should get 4.27 radiance. OK? Now we can finally solve for the intensity. So let's plug everything in and sulfur intensity. So intensity is equal to I zero, which is given to us in the problem as 5.4 multiplied by 10 to the power of negative four watts per meters square multiplied by O sine squared up 7.20 radiance divided by two multiplied by sign of 4.27 radiance divided by two all, divided by four point 27 radiance divided by two all squared. OK. So when we plug that into a calculator, we should get the intensity is equal to 7.2 multiplied by 10 to the power of negative five watts per meter squared. And that is our final answer. All right, we did it. OK. So going back to look at our multiple choice answers, the correct answer has to be the letter D I equals 7.2 multiplied by 10 to the power negative five watts per meter square. Thank you so much for watching. Hopefully that helped and I can't wait to see you in the next video. Bye.