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# Young's double slit equation

## Video transcript

okay so that's all well and good but we've got a problem I told you these two slits are so close together maybe micrometers or nanometers apart that how are we going to measure how are we ever going to physically measure the difference in path length if I go over to this this barrier these two holes are going to look like they're at the exact same spot that's how close they are so I need some way to determine the path length difference based on something I could measure and that's where we're gonna have to play a trick here we have to figure out a function for this path length difference based on what angle I'm at so the basic idea is this let me get rid of all this and let me put it to you this way so let's say I draw a reference line a reference line that goes straight through the center this center line is my friend this is going to let me measure angles here so I've got this line here let's say I wanted to measure to some point on the wall what angle am i at this is how I'm going to measure the angle from this center line to some point over here let's say all right that so my angle is going to be this so this here would be my angle and my question that I'm asking is based on this angle is there some way to determine the path length difference that's the important thing here how do I determine the path length difference how is the path length difference related to this angle the way we can do it is this if the screen is far away here's what I'm going to do I'm going to draw a line from the center of this bottom slit to that point and I'm going to draw a line from the center of this top slit to that point and if this screen is far away significantly further away then these two holes are spaced which isn't to bunch of a problem because these holes are very close together I can draw a third line and this third line is going to look like this third line is going to go from here down cut through this at a right angle and if I'm far away if my screen is far away what will be true is that if this is a right angle right here then the remainder of these paths will be equal in other words the path from here onward from here forwards will be the same length as the path from here forwards so what would the path length be the path length difference would just be this piece down here whatever is left this would be the path length difference this is Delta X in other words so how do I find this well again if I'm far away here this angle here will equal this angle inside of here so these two angles are the same so now that I know that these two angles are the same it's just basic trigonometry I've got a right triangle in here and I'm going to redraw it over here I'll just draw your right triangle so my right triangle looks like this I've got this distance between the holes which is D I'm going to call that distance D the distance between the two holes Center to Center distance and then I've got this other orange line this represents that line I had to draw to make the right angle and then I've got this path length difference this way so this is my triangle and this is supposed to be a right angle this side is Delta X the path length difference the extra amount that wave from the bottom hole had to travel compared to the way from the top hole well this is trigonometry here's my right angle I can just say if I want a relationship between these I can say that sine of theta because this is theta and that theta is the same as this theta over here sine of theta would be opposite over hypotenuse and the opposite to this theta is Delta X so I have Delta x over the hypotenuse in this case is d this entire distance between the two holes because this sides the right angle the hypotenuse never touches the right angle side the hypotenuse is this other side so that's over D so what's the path length difference the path length difference for a double slit is just D times sine of theta so this is what I wanted now I know Delta X is D sine theta and I can write now I can write the double slit formula let me get rid of this the double slit formula looks like this it says that M times lambda equals D sine theta and Y well remember Delta X for octave points was integers times wavelength so zero one wavelength to wavelength and so on and so in order to get constructive points d sine-theta which is the path length difference has to equal zero lambda two lambda and this is the double slit formula it looks like this what does it give you this m is going to be 0 1 2 and so on the d is the distance between the two slits that would be d theta is the angle from the center line up to the point on the wall where you have a constructive point and lambda is the wavelength lambda is the wavelength the distance between peaks of the wave now I mean theoretically speaking you could plug in 1 halves for M and that give you the angles to the destructive points because we know the Delta X the path length difference should just equal 1/2 lambdas to get to the destructive so this can give you the angles to constructive points and destructive points if you plug in the correct m value the order sometimes is this this is called the order of the constructive point this would be the zeroth order because path length difference is 0 sometimes it's called the first order because it's one wavelength difference the next one might be called the second order because it's two wavelength difference you might object though you might still say wait this was no better because D is really close together this D spacing right here is extremely close we can't measure that well but we can measure theta and we can know that wavelength of the laser we send in and we can count which order we're at so this is a quick way to figure out if you had something with two holes in it you can figure out how close they're separated even if you don't have a ruler that small too quick way send some light in you'll get a diffraction pattern like this an interference pattern you measure the angle now I can figure out how close two holes are to spacings and you have do all kinds of experiments to precisely determine how close two holes are in some sort of crystal lattice or a molecular structure and it's determined by Young's double-slit equation