A single-slit diffraction pattern is formed by monochromatic electromagnetic radiation from a distant source passing through a slit 0.105 mm wide. At the point in the pattern 3.25° from the center of the central maximum, the total phase difference between wavelets from the top and bottom of the slit is 56.0 rad. (a) What is the wavelength of the radiation?

Video transcript

Hello, fellow physicist 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 monochromatic laser shines through a single slit of width 56.0 micrometers. The resultant diffraction pattern is analyzed at a distance D from the slit using a photocell detector and computer software at a 0.3 0.00 degrees away from the central bright fringe. The total phase difference between the wave received from the top and the wave received from the bottom of the slit is 34.0 radiance determine the laser wavelength. So our end goal is to determine the laser wavelength. OK. So we're given some multiple choice answers here. Let's read them off to see what our final answer might be. And let's also note that all the units are in nanometers. So A is 271 B is 407 C is 542 and D is 813. Awesome. So first off, let's recall the equation for the phase difference and the phase difference states that beta. So phase difference is represented by beta is equal to two pie divided by the wavelength represented by Lambda multiplied by the slit width, which is represented by a multiplied by sine theta where theta represents the degree value. OK. So at this stage, since we're trying to determine the laser wavelength, let's rearrange the phase difference equation to solve for lambda the wavelength. And when we do that using some algebra, we get two pi multiplied by the slit width multiplied by sine theta divided by the phase difference. OK. So at this stage, we can plug in all of our known variables to solve for the wavelength. So let's do that two pi multiplied by the slit width which is given to us in the problem as 56. micrometers. But we need to convert that to meters. So to do that, it's quick and easy, all we do is we multiply by 10 to the power of negative m multiplied by sign. And the degree value given to us in the pro is 3. 3.00 degrees divided by and the phase difference beta, the value that is given to us in the prominence 34 radiant, 34.0 ratings. OK. So when we plug that into a calculator, we should get 5. multiplied by 10 to the power of negative 7 m. But as we should remember and recall all of our multiple choice answers are given to us in units and nanometers. So we need to convert the meters to nanometers. And all we need to do is to quickly multiply our answer that we just got on our calculator by 10 to the power of nine. And when we do that, we should get, when we round up, we should get 542 nanometers. And that is our final answer. Awesome. So that means that our final answer has to be C 542 nanometers. Thank you so much for watching. Hopefully that helped and I can't wait to see you in the next video. Bye.