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- [Voiceover] Okay so that's all well

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and good, but we've got a problem.

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I told you these two slits
are so close together,

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maybe micrometers or nanometers apart,

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that how are we going to measure?

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How are we going to physically measure

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the difference in path length?

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If I go over to this
barrier, these two holes

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are gonna look like they're
at the exact same spot.

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That's how close they are, so
I need some way to determine

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the path length difference based

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on something I could measure.

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And that's where we're gonna
have to play a trick here.

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We're gonna have to figure
out a function for this

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path length difference
based on what angle I am at.

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So the basic idea is this, so
let me get rid of all this.

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And let me put it to you
this way, so let's say

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I draw a reference line that goes

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straight through the center.

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This centerline is my friend.

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This is gonna let me measure angles here.

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So I've got this line here
and let's say I wanted

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to measure to some point on
the wall what angle am I at.

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This is how I'm going to measure the angle

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from the centerline to some
point over here let's say.

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So my angle is going to be this,

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so this here would be my angle.

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And my question that I'm asking is

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based on this angle is there some way to

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determine the path length difference?

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That's the important thing here, how do

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I determine the path length difference.

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How is the path length
difference related to this angle?

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The way we can do it is this.

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If this screen is far away,
here's what I'm gonna do.

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I'm gonna draw a line from
the center of this bottom slit

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to that point and I'm gonna draw

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a line from the center of
this top slit to that point.

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And if this screen is far away,
significantly further away

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than these two holes are
spaced, which isn't too

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much of a problem because these holes are

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very close together, I
can draw a third line

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and this third line's
gonna look like this.

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Third line is gonna go
from here down, cut through

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this at a right angle and
if my screen's far away,

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what'll be true is that
if this is a right angle

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right here, then the remainder
of these paths will be equal.

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In other words, the path from here onward,

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from here forwards,
will be the same length

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as the path from here forwards.

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So what would the path length be?

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The path length difference would

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just be this piece down here.

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Whatever is left this would
be the path length difference.

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This is delta x in other words.

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So how do I find this?

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Well again, if I'm far
away here this angle here

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will equal this angle inside of here.

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So these two angles are the same.

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So now that I know that these two angles

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are the same it's just basic trigonometry.

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I've got a right triangle in here

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and I'm gonna redraw it over here.

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I'll just draw you a right triangle.

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So my right triangle looks like this.

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I've got this distance
between the holes, which is d.

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I'm gonna call that
distance d, the distance

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between the two holes,
center to center distance.

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And then I've got this other orange line.

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This represents that line I had

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to draw to make the right angle.

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And then I've got this path
length difference this way.

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So this is my triangle and this

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is supposed to be a right angle.

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This side is delta x, the
path length difference.

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The extra amount that
wave from the bottom hole

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had to travel compared to
the wave from the top hole.

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Well this is trigonometry,
here's my right angle.

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I can just say if I want a
relationship between these,

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I can say that sine of
theta, because this is theta

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and that theta is the same
as this theta over here.

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Sine of theta would be
opposite over hypotenuse.

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And the opposite to this theta is delta x,

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so I have delta x over the
hypotenuse in this case is d,

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this entire distance between the two holes

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because this side is the right angle.

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The hypotenuse never touches
the right angle side.

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The hypotenuse is this other side.

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So that's over d, so what's
the path length difference?

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The path length difference for a

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double slit is just d times sine of theta.

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So this is what I wanted.

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Now I know delta x is d sine theta.

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Now I can write the double slit formula.

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Let me get rid of this.

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The double slit formula looks like this.

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It says that M times
lambda equals d sine theta.

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And why, well remember delta
x for constructive points

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was integers times wavelengths, so zero,

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one wavelength, two wavelength and so on.

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And so in order to get
constructive points d sine theta,

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which is the path length
difference has to equal

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zero lambda, two lambda and this is

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the double slit formula,
it looks like this.

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What does it give you?

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This M is gonna be zero,
one, two and so on.

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The d is the distance between

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the two slits, that would be d.

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Theta is the angle from
the centerline up to the

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point on the wall where you
have a constructive point.

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And lambda is the wavelength,

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the distance between peaks of the wave.

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Now I mean theoretically
speaking you could plug in

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one halves for M and that
would give you the angles

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to the destructive points
because we know the delta x,

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the path length difference,
should just equal

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half lambdas to get to the destructive.

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So this can give you the
angles to constructive

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points and destructive
points if you plug in the

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correct M value, the
order, sometimes this is

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called the order of
the constructive point.

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This would be the zeroth order because

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the path length difference is zero.

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Sometimes this is called the first order

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because it is one wavelength difference.

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The next one might be
called the second order

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because it's two wavelength difference.

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You might object though,
you might still say

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"Wait this was no better because
d is really close together.

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"This d spacing right
here is extremely close.

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"We can't measure that well."

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But we can measure theta
and we can know that

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wavelength of a laser we send in.

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And we can count which order
we're at, so this is a quick

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way to figure out if you
had something with two

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holes in it you could
figure out how close they're

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separated even if you don't have

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a ruler that small, it's a quick way.

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Send some light in, you'll
get a diffraction pattern

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like this, an interference pattern.

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You measure the angle,
now I can figure out how

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close two holes are, two spacings.

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And you can do all kinds
of experiments to precisely

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determine how close two
holes are in some sort

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of crystal lattice or
a molecular structure.

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And it's determined by
Young's Double Slit Equation.