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. (b) What is the intensity at this point, if the intensity at the center of the central maximum is I0?

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 colum monochromatic beam of light is incident on an aperture of width micrometers. The intensity distribution of light as a function of the diffraction angle, theta relative to the original direction of the beam is measured with a light sensor. The central bright fringe has an intensity of 7.00 multiplied by 10 to the power of negative 66 watts per meter square at an angle of 1.15 degrees. The phase difference between the rays from the top and bottom of the slits is 110 radiant. Calculate the intensity of light diffracted at theta equals 1.15 degrees. OK. So we're given some multiple choice answers and they're all in the same units of watts per meter square. So let's read them off to see what our final answer might be. A is 1.13 multiplied by 10 to the power of negative 12, B is 4.53 multiplied by 10 to the power of negative C is 4.98 multiplied by 10 to the power of negative 10. And D is 2.31 multiplied by 10 to the power of negative nine. OK. So first off, let us recall and use the equation for the intensity in terms of the phase difference beta at any point on the diffraction pattern. And that equation can be written as the intensity I is equal to I subscript zero multiplied by sign of be divided by two all divided by divided by two all squared where I subscript zero is the intensity of the central bright fringe and beta is the phase difference. OK. So we can simplify this equation to say that the intensity I is equal to four multiplied by I subscript zero divided by beta squared multiplied by sign beta divided by two squid. So at this stage, we can plug in all of our known variables to solve for the intensity. So let's do that. So I equals or multiplied by and our I zero is given to us in the problem as 7. multiplied by 10 to the power of negative six watts per meter squared divided by the phase difference, which is given to us in the prom as radiance squared all multiplied by sign of 110 radiant divided by two all squared. Ok. So when we plug that into a calculator, we should get the intensity to equal 2. multiplied by 10 to the power of negative nine watts per meter squared. So the intensity at theta equals 1. degrees is 2.31 multiplied by 10 to the power of negative nine watts per meter squared. Ok. So looking at our multiple choice answers, the correct answer should be D 2.31 multiplied by 10 to the power of negative nine watts per meter squared. Thank you so much for watching. Hopefully that helped and I can't wait to see you in the next video. Bye.