Intensity in YDSE (Visual method-phasors) I =4Io cos^2(phi/2)

in this video let's derive the expression for intensity in the young's double slit experiment so we'll derive expression for intensity of the resulting wave and we'll do that as a function of its phase difference phase difference is usually denoted by phi now just a quick heads up this most of this video is going to be visual so a way for you to logically visualize what's going on all right so let's begin you may be by now familiar with the the setup we have two holes and there's a light source behind it and the light hits the two holes they act like coherent sources of light and then the light meets on the screen at different different sp different different places to produce an interference pattern and what we want to do now is figure out when they meet at every point what will be the intensity or the brightness of the light that's what we need to figure out how do we do that well intensity of any oscillation intensity of any oscillation which is going to write i turns out to be proportional to the amplitude of the oscillation squared to the square of the amplitude of the oscillation and don't worry we'll visualize all these complex terms we'll do that in a second but because i want to calc because intensity is proportional to amplitude square that means to figure out the intensity of the light here i need to figure out what will be the resulting amplitude when the light meets and so let's just assume that the amplitude of the waves produced by these is a so then we'll figure out what the resulting amplitude is going to be when they meet and we'll be able to figure out when you square that we'll get the resulting intensity all right now before we begin a lot of complex terms let's start with something let's start with coherence what do we mean by these sources are coherent here's how i like to visualize it coherent sources means that their phase relationship is locked in time here's what i mean so if you were to look at these oscillations notice they are in sync with each other but what's important is that is locked in time meaning forever they will be in sync with each other that's coherent sources this is another example of coherent sources over here notice when one thing is they're oscillating in the opposite direction why are they coherent because again that relationship stays locked in time they will always oscillate in the opposite direction get that so that's the meaning of coherence phase relationship is locked all right let's consider the simplest example of coherence the one in which they are oscillating in sync with each other and if these are sources of light they will start producing light waves and these light waves are in three dimension but it's easier to visualize in one dimension and now this distance the distance of the peak from the mean position is what we mean by amplitude and if you square that that represents or it's proportional to the intensity of the oscillation or the intensity of the wave okay and if these waves are going to hit a point on the screen imagine this is a point on the screen then the waves are going to make that point oscillate as well and notice that point will oscillate with the same frequency as this one and it will have the same amplitude can you see that same amplitude now what we are interested in is what happens when they meet at a point on the screen then what happens to the resulting amplitude well over here notice their individual contributions their individual oscillations are in sync with each other can you see that because this distance and this distance is exactly the same these two pictures are identical these two oscillations end up becoming in sync and so if these two were to meet at one point let me show you that now then the resulting the resulting oscillation would have twice the amplitude does that make sense look each one has amplitude a and they are in sync it's like two forces which are in the same direction the resulting would be twice that and so the resulting amplitude would just be 2a and the square of that would be the intensity right but here's the thing in this example these two waves travel the same distance to meet but that's not always going to be true and that's what makes things interesting so let me show you what i mean if you consider a point over here where the two light waves are meeting then notice this one has to travel a shorter distance and this one has to travel a longer distance now individually they will not be oscillating this particle in sync with each other again let me show you that again here we go this time i've kept the yellow closer to the particle compared to the blue because that's the case we're dealing with yellow closer to the particle compared to the blue because of this you will see that the wave from the yellow will reach the particle first and as a result before this particle starts oscillating this would have finished some number of oscillations and so we say this has a phase difference compared to this and how do we calculate that well if this has finished one extra oscillation compared to this we say it has two pi extra phase one oscillation corresponds to two pi extra phase if it has half an oscillation extra compared to this one then we would say it has pi extra phase and so on and so forth every extra oscillation corresponds to two pi extra phase and because there is a phase difference notice they are not oscillating in sync with each other so in general when the two waves meet on the screen the two the two particles may not oscillate in sync which is either their contributions may not be in sync that's what i want to mean and because their contributions are not in sync now comes the question what happens when they meet well when they meet because their contributions are not in sync with each other the resulting will not the resulting oscillation will not have an amplitude twice it can have any amplitude between 0 and 2a actually and so our goal now is to figure out what that resulting amplitude is in general because the square of that is going to be the intensity so how do we figure that out there are a couple of ways to do that one is algebraic another one is vectorial vectorial is visual so let's do that there are a couple of ways to do that one is algebraic another one is vectors now i like vectors because again we can visualize it so let's do that one more one last visualization so the idea is you treat oscillations as a rotating vector it's an imaginary vector if you have studied alternating currents same idea we use phasors over there it's the same thing the whole idea is you you draw a vector which has the length equal to the amplitude of the oscillation and it's spinning at the same rate of the vibration so same frequency okay and you set that up in such a way that if you were to shine light from the side horizontally whatever shadow gets casted on a wall imagine there's a vertical wall over here the tip of that shadow represents the position of this particle that's how you represent a rotating vector at any point notice right now if you cast a shadow oh sorry cast shine light you cast a shadow again the tip of that represents the position of this now why this is important because now i don't have to look at the position i don't have to look at the vibration oscillations anymore this vector if i know what the vector is doing i automatically know what the position of my particle is and vectors are easier to add at least at least compared to oscillations when you're adding oscillations so what we're going to do now is we're going to take another vector for this one and when these two waves meet we're just going to keep these two vectors together so we're going to meet now there we go and all we have to do now is figure out when you add these two vectors what happens to the resultant that resultant will represent our resulting amplitude does that make sense that's what we need to do now and what is the angle between them represent the angle between them represents the phase difference if the two particles were oscillating in phase these two vectors would be in line if the two particles were oscillating in the opposite direction 180 then these these two would be opposite to each other giving a phase of 180 degrees so can you see that the angle directly represents the face and so i know how to cal add up vectors using the parallelogram law this now represents the amplitude of the resultant and just to be clear just that we are on the same page what this means now is that if i run this animation then the shadow of this one on the vertical wall that represents the oscillation or the position of the resultant oscillation this is our resultant oscillation so for that all i have to do is calculate what the amplitude what this this length is that represents the amplitude of this and the square of that will give me the intensity so let's set that up here is the first vector that represents the amplitude of the first wave first source this is the second vector that represents the amplitude of the second wave the angle between them is phi that's the phase difference when they reach here and the resultant of that represents the resulting amplitude so we know how to add vectors we know the formula to calculate the resultant of two vectors so why don't you pause the video now and figure out what the resulting amplitude is all right if you're giving this a shot let's see the resulting amplitude is how do you calculate well the formula is a squared plus b squared plus 2 a b cos theta so the square of the amplitude is going to be a squared plus b squared plus 2 a b so a b becomes a square cos phi okay now we just have to simplify this i get a 2 a squared here there's a 2 a squared here i'll take that common and what i end up with inside the bracket is 1 plus cos phi can i simplify this further yes i can using trigonometric relation 1 plus cos phi is 2 cos squared phi over 2. and so if i put them all together if i multiply them i get the resulting amplitude squared equals 4 a squared cos squared phi over 2. all right i have amplitudes but i want intensity notice i already have amplitudes squared so these should be proportional to their intensities so this is proportional to the resulting intensity so i'll write as k times i r ir is the resulting intensity i'm writing k because this is not equal to right there is some proportionary constant similarly this is proportional to the intensity of the incoming wave so it'll be k times i naught times cos squared phi over 2 and k cancels and i get my relationship i now have the intensity resulting intensity becomes 4 times i naught times cos squared phi over 2. and notice i found what i wanted i now have the expression for the intensity as a function of phase difference in another video we will play with this equation more and we'll get a deeper understanding of you know what really happens to the intensity when the waves interfere but for now just reflect on how we derived this we spent most of the time visualizing and the math was just last few minutes so what i like about this method is if you can visualize properly what's going on you can derive most of that stuff with practice in your head which means i don't have to remember any of these equations