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# More on single slit interference

## Video transcript

so I got to be honest with you single slit interference is confusing in fact this argument I gave earlier first time I heard about this I thought this was just mathematical mumbo-jumbo I was like what are you talking about this makes no sense it seems like you could argue anything like this but you can I'm gonna try to show you that in this video specifically I'm going to try to show you that this same argument that we made for the destructive points won't work for the constructive points I think that will help a little bit in other words this half wavelength relationship won't give the constructive points not exactly you'll give it approximately but it won't work exactly why all right well let's look at this so let's get rid of all that so let's say we were going to try to derive the formula for the constructive points the first thing I do is I'd say well all right again each point on here diffracts when it gets to the hole I'll have infinitely many sources but I can't draw infinitely many so let's just consider eight again one two three four eight I'd get my interference pattern on the wall and this is the graphical representation I mean they're really like smudgy lines over here but the graphical representation looks something like this you'd have a big old bright spot in the middle these points on the end and you keep getting them but I can't keep drawing them over here so it keeps on going and I'd pick a constructive point right here this is a bright spot so this if I had to give a guess for what point would be totally constructive for all the ways let's say at that point so I take my topmost wave I'd say it travels a certain distance to this bright spot I take my wave in the middle travels a certain distance to get here I'd imagine my line straight down here to try to figure out the path length difference now remember this here is the path length the difference what should that be in order for these two purple waves to interfere constructively over here it's got to be an integer wavelength so one wavelength two wavelength since this is the first one from the center we'll just say that's one wavelength what's the relationship we know that remember the relationship from the path length difference and the theta the angle that it's at was just D sine theta D is well the whole width of the hole is a W so what width is this this width between this source of light and this source would be W over 2 and the relationship I'd get all right D is W over 2 times sine of theta would equal 4 this first point I'd say it's equal to lambda so let's just assume these two are interfering constructively at this point and that would give me W times sine theta equals 2 lambda gives me a constructive point now I'm already confused what W sine theta equals to lambda constructive we already proved this is a destructive point remember our relationship for the destructive points that we derived was W sine theta equals M lambda as long as M is not zero but one two three four five these are giving destructive for any M equals one two that could even be negative if you want to consider down here any integer it looks like we're just prove these are constructed how are these constructive well they're not really they kind of are but watch what happens just here we go so if I follow this argument through the thing that failed our previous argument is fine the party iment that fails is this current one with constructive because yes these two are constructive there but watch this point here now this number the game we played we said if these two are constructive then the rest of them should all be constructive is that so well let's go down one let's go down one I imagine these two waves getting here so far looking good they're at the same angle they're the same distance between them I mean this length here is still W over two so I'd still get W over to sign of the same angle because it's the same point on the wall so if W over 2 is the same signing sign of the theta is the same then that's going to also be path length difference of lambda which means these two blue waves also interfere constructively so this is looking pretty good which is kind of bad I'll show you why is that these two would also be constructive well is this a constructive point is N equals to a constructive point or a destructive point then and it's a destructive point this arguments failing and it fails because watch this even though these two purple ones interfere constructively over here here's a wave cycle even though the two purple ones meet up constructively let's say the top one was there that means the one in the middle this one here was also at the peak so those two interfere constructively how about the next two well those two are going to interfere now maybe those two are like at this point they're both constructive but they're not necessarily the same as the two purple ones and how about the orange ones orange ones might be constructive because they're both at the same point in the phase but they're not at the same point as all the rest of them you can have more what about these down here oh these might be down here those two together are also constructive but you see the problem even though these two are constructive this one's not constructive with this guy and these all add up in fact for the most part they cancel that's why these are so little these you get these weak you get these really weak fringes on the sides of the single slit cuz you're not going to get points where they all add up really well necessarily you get points where a lot of them sort of cancel out and it doesn't completely cancel those here's where I lied for the diffraction grading number for the diffraction grading let me get rid of this for the diffraction grading we had a single line and we made a ton of holes in it and I said that diffraction gratings are great because if you come over here you make a ton of holes in here instead of getting a smudgy pattern on the wall you you get a big bright spot right in the middle and then a well-defined well defined well defined on each side of them evenly spaced basically just 0 and then extremely sharp and then 0 and then extremely sharp and then 0 and extremely sharp and the whole argument I made for diffraction gratings was that the reason it's 0 in between the reason these are giving 0 everywhere except these constructive points was precisely because if we come back over to here was precisely because of this effect right here this effect where they for the most part cancel now I'm saying and they don't actually completely cancel necessarily so these Wiggles here are actually in a diffraction grading pattern they're just so small and unpronounce compared to these you don't really notice and what I'm saying is if I wanted to draw this more realistically I would definitely have this bright spot right here but I'd have this in between small variations small points where it becomes a little more a little less constructive or destructive what you have for a single slit is this just one Center bright spot it's not going to be as well-defined because it's not a diffraction grating it's a single slit but you still get these you get these weird Wiggles that for the most part you ignore for a diffraction grading but they're there and for a single slit that's kind of all you got so can't really ignore it so much those are going to be there it's because these don't completely cancel our argument does not work it works in the sense that two of these might be constructive that means you can pair these off and constructive but they won't all be at the same point on their phase which would give you a completely constructive point there so that's why we don't it's hard to find an exact formula what's the formula for the constructive points well getting this formula not quite as simple you need to know a little bit more physics to do that and so typically in introductory physics classes you aren't asked to find the exact locations of the most constructive points over here even these most constructive points partially canceled you do know how to find the exact locations of the destructive points though and if you wanted an approximate location of a constructive point well you can find the exact location of two neighboring destructive points which if you really wanted to I mean the constructive zin they're approximately in the middle if you wanted to get a rough a rough idea um I could still see some of you being upset though you might say wait hold on a minute so we're saying this formula is good for the destructive points but is this problem we ran into for constructive points also a problem for destructive points and it's not it doesn't matter if there are different points in their phase for the destructive because each pair cancels in other words when we ran through this argument for the destructive points look at if these two purple ones cancel then they cancel I mean if one was at the peak and then the others at the trough or the valley those add up to zero they're gone any effect they might have had on light hitting this point in the screen is gone completely negated and so what about the next two these two blue ones well those two if these two purple ones cancel remember the argument went that these two blue ones would have to cancel so no matter where they're at there's some different point on the cycle let's say ones here and the others that look like the same so let's say one is here and the other is at this corresponding 180 degree out of phase point well they still cancel that adds up to zero and so it doesn't even matter that there are different points in their phase it doesn't matter no matter where they're at ones 180 degrees out of phase with the other every contribution cancels out you add up a bunch of zeros you get zero so do deductive works fine you don't run into the same problem with constructive it's a problem for the constructive points because these might add up to some big number and then the blue ones add up to a different number and the orange one adds up to a different number and then the red ones might add up to a negative number and you keep getting these different numbers you try to add them all up well what do you get that's why this formula is not so easy to find adding up zeroes that's easy just gives you zero so I hope I showed you that this crazy mumbo-jumbo argument can't say anything whatsoever and hopefully that gives you a little more justification hopefully it makes you believe a little more into this formula that we derived for the destructive points those it does work for and so we can find destructive points just fine one more thing we can find is the width of this Center bright fringe here the center bright spots going to be wide and since this goes to M equals one now that first destructive is over here this is why this is in fact twice as wide as all of these between these destructive points and how wide is this well you can find the angle to this first destructive point up here M equals one you can find it to the M equals negative one you do a little trigonometry you can actually get this length that's another thing you can find exactly is the width of this Center bright spot and the location is right in the center but the location of these constructive points up here the exact location that's a little harder you can find their width again because you can find the locations where they terminate but finding where it actually peeks in here don't have an exact formula we do have an exact formula for the single slit destructive points and that's typically what you're going to have to find in these problems