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- [Child] Look at that!
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So pretty!
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- The other day my son asked me
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why rainbows are curved.
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And I could have given
him a simple explanation
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but instead I made this video.
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With beautiful demonstrations
I've never seen before.
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This is the perspective of a rainbow
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from a single raindrop.
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And the best animations ever created
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on the subject.
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Because I promise you, almost
every explanation out there,
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double rainbow,
(intense music)
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is an oversimplification.
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For example, if raindrops
spread white light
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into colors like a prism,
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then why do you never see a rainbow
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when looking in the direction of the sun?
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Or why is it darker above
a rainbow than under it?
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How can you make a rainbow
disappear with sunglasses?
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What's going on here?
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Why is this rainbow so
much smaller than usual?
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And how did this phenomenon
directly lead to a Nobel Prize?
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Oh yeah, we're going deep on this one
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because the full explanation
is so much more satisfying
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than anything you've seen before.
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(rain splattering)
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To make a rainbow, you need three things,
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raindrops, the sun and you, an observer.
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Okay, this is one of those
experiments that seems so simple,
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but I've never seen anyone do it before.
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I have a glass sphere and
that represents my raindrop
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because as we've learned
in previous videos,
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raindrops, of course, are
essentially spherically shaped.
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(air whooshing)
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Just preparing the particulate
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so that I can see the laser beams.
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Over here I have a laser.
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And that laser beam
represents a ray from the sun.
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Rays of light from the sun
reach a raindrop essentially
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parallel to each other
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because the sun is so far away.
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When light strikes the sphere,
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some of it reflects back
off the front surface
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and some is transmitted into the sphere.
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Then at the back,
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again, some of the light
reflects off the back surface
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and some is transmitted.
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Every time light goes from
one medium into another,
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some of it will be reflected
and some transmitted.
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And exactly how much depends
on the angle of the light,
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its polarization and the
nature of the two media.
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This is actually helpful here
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because I can use the reflections
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to make sure the laser
is lined up properly.
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I think I've got this red laser lined up,
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so it's hitting the middle of the sphere
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and it's reflecting here,
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but some of the light goes through
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and some of the light
reflects at the back surface,
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and then some of the light
goes through for the wall.
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Now I'm going to keep the laser horizontal
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and move it up the sphere.
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So let's call the distance
from the central axis,
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the impact parameter.
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(rhythmic music)
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As I move the laser up,
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the reflection off the
front surface goes up.
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It's just bouncing off that curve surface.
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And I can tell you,
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this is a simple, boring reflection.
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Nothing interesting happens with it.
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It's not involved in
rainbows, so it's there,
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but we're basically going to ignore it
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for the rest of the video.
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What's much more interesting
is the reflection
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off the back surface.
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Now here comes that ray here,
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this spot is on the table.
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As the laser moves up, it goes down.
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In fact, the whole beam
inside the sphere bends down.
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That's because as the
light enters the sphere,
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it slows down, and so it refracts.
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But why does light slow
down when it enters
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a dense medium like glass?
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Well, I think a lot of people can tell you
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that light is an electromagnetic wave
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without really thinking
about what that means.
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You know the electric field
around a charged balloon
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that pulls on your hair or
makes it stick to a wall
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and the magnetic field around a bar magnet
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that makes iron filings line up?
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Well, light is what happens
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if you could rip the electric
field off the charges,
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the magnetic field off the
magnet, smoosh them together,
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and send them out traveling
through space, sort of.
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I mean in practice,
electromagnetic waves are made
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by accelerating charges like
by wiggling them up and down.
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Then the changing electric and
magnetic fields, they create,
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team up as light
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and head off on their own.
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The clearest explanation I know of
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for how light is slowed down
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in a medium comes from
Grant over at 3Blue1Brown.
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I asked him if I could
summarize his explanation
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for this video and he graciously agreed.
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So when those electromagnetic
waves encounter charges
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in a medium like those in the
first layer of our sphere,
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the light pushes them back and forth.
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You can think of each charge
as a little mass on a spring
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and the changing electric
and magnetic fields cause it
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to vibrate at the same
frequency as the light.
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But now you have wiggling
that is accelerating charges.
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So they too must create their
own electromagnetic waves.
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And the net electromagnetic
field is just a sum
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of the incident wave plus this new wave.
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The result is almost exactly
like the original wave,
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except it has shifted back slightly.
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It receives a phase kick
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and each layer of the material
adds another phase kick.
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So the net effect of all
this is the wavelength
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of the radiation decreases
in the new medium.
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And since the frequency stays the same,
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a shorter wavelength
decreases the speed of light
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through the material.
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The speed of light in a vacuum
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divided by the speed of
light in a medium is called
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the refractive index.
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It's around 1.5 for
glass and 1.33 for water.
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So when light enters a
new medium at an angle,
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the part of the wave crest that
enter the new medium first,
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slow down first.
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And this changes the angle
of all the wave crests.
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And since the direction of a
beam of light is perpendicular
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to the wave crests, this means
the light changes direction.
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So to recap, light
causes charges to wiggle,
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so they create their own
electromagnetic wave,
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which gives the light a phase
kick shortening its wavelength
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which slows it down and so it bends.
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There's a simple mathematical expression
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that relates the angles of
incidence and refraction
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to the indexes of
refraction of the two media.
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It's known as Snell's law,
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even though it was
independently discovered
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by a handful of people,
some well before Snell.
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(intense music)
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And this is what we're
seeing in the sphere.
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The higher the laser hits the sphere,
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the larger the angle of incidence
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and hence the more it bends
down due to refraction.
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Most of this light exits
out the back of the sphere
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but some of it is reflected.
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And it's this reflected ray
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that we can see coming out
the front of the sphere
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below the incident beam.
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Now I wanna graph the
angle of this reflected ray
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as I move the beam up the sphere.
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When the laser is dead center,
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the reflected beam comes
straight back at the source.
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So let's call that 0 degrees.
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Then as I move the laser
up, this angle increases.
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So the light's coming back at
5 degrees, then 10 degrees,
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and it keeps increasing the higher I go.
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But now we come to the critical point.
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Watch this spot on the table.
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As I move the incident beam
up, this dot is moving in.
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The reflected ray is coming
closer and closer and closer.
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But there's a certain point right there
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where it stops coming closer.
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Look at that.
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And even as I keep moving
this beam higher and higher,
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it doesn't get any closer.
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(intense music)
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And then it goes back the other way.
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So what we're seeing
there is a maximum angle
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this reflected ray reaches
before it turns around
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and goes back the other way.
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(intense music)
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And this is really important.
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It means that over a range
of impact parameters,
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a range of heights of the laser,
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the reflected beam comes out
at essentially the same angle,
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which means the light
is becoming concentrated
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at that angle.
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And a concentration of light
rays is called a caustic.
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Curved surfaces all
tend to create caustics,
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from coffee mugs to glasses
or even just rippling water.
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Caustics create the light patterns we see
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by concentrating light rays.
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In the case of red light
through a sphere of water,
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the maximum scattering angle,
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and hence this caustic
always occurs at 42 degrees
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below the horizontal.
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Since my sphere is made of
glass rather than water,
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well, the angle is different,
but the principle is the same.
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Now you might ask, why does
this reflected ray reach
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a maximum angle and then turn around?
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Well, the answer is just geometry.
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As I move the laser up the sphere,
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although the ray refracts down,
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the point on the back of the sphere
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where it reflects continues to move up
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until you get to this special point,
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which is about 7/8 the
radius of the sphere.
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And here, the angle of
incidence is so steep
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that the refracted ray stops
hitting the back higher
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and starts hitting it lower.
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That is why the reflection turns around
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and therefore we get a
maximum scattering angle
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and the concentration of
light rays at that angle.
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But the precise maximum
scattering angle depends
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on the color of light.
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To see why, let's go back
to the idea of the charges
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in the sphere as masses on springs.
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They have a natural frequency,
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a frequency at which they would oscillate
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if not driven at any particular frequency.
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And in most materials,
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this natural frequency is pretty high,
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much higher than the
frequencies of visible light.
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Now, when light pushes
a charge back and forth,
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the amplitude of the
resulting vibration depends
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on the difference between
the frequency of light
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and the natural frequency of the charge.
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The closer the two frequencies are,
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the greater the amplitude
of the resulting vibration,
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which makes sense if you've
ever pushed someone on a swing.
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The closer your pushing frequency is
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to the natural frequency of the swing,
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the higher they'll go.
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This means that higher
frequency light like blue light
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will cause the charges to
wiggle with greater amplitude.
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And because of this, the charges produce
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higher amplitude
electromagnetic radiation,
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which creates a bigger phase kick,
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which shortens the wavelength
proportionally more,
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making higher frequency
light travel slower
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and bend more than lower frequency light.
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All right, I'm gonna change lasers.
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So when I repeated the
experiment with green light,
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it refracted more than red light
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and therefore the green
dot turned around sooner
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than the red dot.
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Whoa, oh,
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there's the minimum deflection for green
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and it's significantly
different than for red.
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In other words, its maximum
scattering angle was smaller
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than for red light.
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If the sphere were water,
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it would occur at around 41
degrees below the horizontal.
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Similarly, for blue light,
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the maximum scattering
angle approaches 40 degrees.
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I do have a specialty laser,
00:11:20
which is very bright blue,
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but it's very dangerous
00:11:23
so we're gonna use it very carefully.
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how are we gonna use it
very carefully, Derek?
00:11:28
(tool thuds)
00:11:30
That is a good question.
00:11:31
(tape ripping)
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Maybe we're just gonna tape it on here.
00:11:36
Yeah.
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(playful music)
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So the blue only makes it to
there, green here, red to here.
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So it is a pretty serious spread here.
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I think my experiment may
be a little wonky honestly,
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I think this one wasn't
perfectly horizontal.
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To really see the
importance of the caustic,
00:12:03
imagine we illuminate the
sphere uniformly with red light.
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Well, more light is
going to hit the sphere
00:12:09
at higher impact parameters
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because the further out you go,
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the more area there is.
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So I've adjusted these sections
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so that they all have the same area.
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Then using our graph of
the scattering angle,
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you can see where all of
this red light will end up
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after reflecting off the back surface.
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Most of it ends up at the
maximum scattering angle.
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To make this more obvious,
we can add more light
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and we can do the same
thing for orange and yellow
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and all of the other colors.
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And this is what gives us the rainbow.
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(soft choral music)
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It's not enough to say that a
raindrop spreads white light
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into its component colors
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because all of the light that hits closer
00:12:52
to the middle is spread too.
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But since the reflections
all overlap as they come out,
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the colors mix and produce white again.
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It's only the difference in
maximum scattering angles
00:13:03
and the caustics they produce
that gives us the rainbow.
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(soft music)
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So now we know what happens
along a single radius
00:13:13
of the sphere.
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So what happens if we uniformly illuminate
00:13:18
the whole sphere with white light?
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(bright music)
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Well, I blacked out my window
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and cut a hole just big
enough for afternoon sunlight
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to cover the sphere.
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You can see there's a circle
00:13:32
of white light coming from the reflections
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off the back of the sphere,
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and then around it, there's
a ring of rainbow colors.
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Come on.
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One raindrop creates a cone of light.
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The inside is all white,
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and the ring around
the outside is colored.
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This is the perspective of a
rainbow from a single raindrop.
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All these different light rays coming in
00:14:00
at different places reflect
back off the front surface
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and the back surface,
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that reflection off the
back surface reaches
00:14:07
a maximum angle.
00:14:09
And for blue, green, yellow, and red,
00:14:12
the maximum angle is different.
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So the red maximum angle is the furthest.
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That's why it's on the outside here.
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So good, so good.
00:14:20
(light music)
00:14:22
What I really wanted to see
is if I could observe the cone
00:14:25
of different colored caustics
that produced that ring.
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Ho, ho, ho, ho, ho, ho, ho,
00:14:30
ho, ho, ho, ho.
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You've got to see this color cone.
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You have got to see this.
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My eye is right in the color cone here,
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I can see the color cone.
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That is so cool.
00:14:42
(intense music)
00:14:45
So this is the crazy focus
on the back of the sphere.
00:14:51
If you stick your finger in
there, it gets burnt very quick.
00:14:55
Ow!
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That thing is a focal.
00:14:59
(intense music)
00:15:01
Now this really looks like a rainbow,
00:15:03
but remember, this is
just the light coming away
00:15:05
from a single droplet.
00:15:07
When you see a rainbow, there
are billions of raindrops,
00:15:10
and each one is projecting a rainbow cone
00:15:13
back toward the sun.
00:15:14
So how does all of this create
a single unified rainbow?
00:15:20
Well, for your eye to see a color,
00:15:22
let's pick red in a
certain part of the sky,
00:15:25
then the red caustic from a
raindrop there must go directly
00:15:28
into your eye.
00:15:30
And this only happens when
the angle from the sun
00:15:33
to the raindrop to your eye is 42 degrees.
00:15:37
And this explains why rainbows
take the form of an arch
00:15:40
with a 42 degree angle.
00:15:42
(intense music)
00:15:45
Now, the violet light from these
same raindrops passes above
00:15:48
or beside your eye, so
you can't possibly see it.
00:15:52
But there are raindrops below
00:15:54
and inside the arc of
those red giving raindrops
00:15:58
whose violet caustics
do intersect your eye.
00:16:01
They form a shallower angle of 40 degrees
00:16:04
between the sun and your eye.
00:16:06
And of course, there are raindrops
00:16:08
at all intermediate angles that send you
00:16:10
all the other colors of the rainbow.
00:16:14
So a rainbow really is the
ultimate optical illusion
00:16:17
from billions of droplets each
projecting a rainbow cone.
00:16:22
You see a single static arch of color,
00:16:26
but the droplets sending you
00:16:27
those colors are constantly changing.
00:16:29
A single drop as it falls
might send to your eye
00:16:32
first red, then orange,
yellow, green, blue,
00:16:36
indigo, and violet.
00:16:38
(soft music)
00:16:39
And because a rainbow must form
an angle of 40 to 42 degrees
00:16:43
with your eye,
00:16:44
the center of the arch must be on a line
00:16:47
that passes from the sun
through the back of your head.
00:16:51
So your shadow is the
center of your rainbow.
00:16:56
This means no two people can ever see
00:16:58
the exact same rainbow.
00:17:00
In fact, your left and
right eyes don't even see
00:17:03
the same rainbow.
00:17:04
A rainbow is an optical
illusion made unique
00:17:08
for each perspective.
00:17:10
This also explains why in
most parts of the world,
00:17:12
you can only see a rainbow
in the early morning
00:17:14
or late afternoon, not
in the middle of the day.
00:17:18
The higher the sun is, the
lower the top of the rainbow is.
00:17:22
And when the sun is more than
42 degrees above the horizon,
00:17:25
no rainbow is visible from the ground.
00:17:28
(water splattering)
00:17:31
But even when rainbows are visible,
00:17:33
you can turn them
invisible using sunglasses
00:17:37
that is as long as they are polarized.
00:17:41
Light from the sun is unpolarized,
00:17:43
which means the electric fields
00:17:45
of the light are all randomly oriented,
00:17:47
oscillating back and forth
equally in all directions.
00:17:51
But it just so happens
00:17:53
that when the light in
the rainbow ray reflects
00:17:55
off the back of the droplet,
00:17:57
it does so very close
00:17:58
to a special angle known
as Brewster's angle.
00:18:02
At this angle,
00:18:03
all light with its electric
field oriented parallel
00:18:06
to the plane of reflection is transmitted.
00:18:09
So it passes out the back of the droplet,
00:18:12
and therefore the only light
00:18:14
that is reflected has its
electric field perpendicular
00:18:17
to the plane of reflection.
00:18:19
This is the light that
creates the rainbow.
00:18:23
This means rainbow light is
polarized along the direction
00:18:27
of the rainbow,
00:18:28
so horizontal at the top
00:18:29
and closer to vertical on the sides.
00:18:32
This is why you can
use a polarizing filter
00:18:35
to make a rainbow disappear
00:18:37
or to make it brighter
if you orient the filter
00:18:40
to allow that polarized
light to pass through.
00:18:44
But why is it brighter under
a rainbow than above it?
00:18:48
Well, this is because the raindrops
00:18:50
beneath the rainbow are
reflecting all colors
00:18:52
of light at you off their back surfaces.
00:18:55
This is what created the white disk
00:18:58
in my glass sphere experiment.
00:19:00
In contrast, the raindrops above the top
00:19:02
of the rainbow are not
reflecting any light to you
00:19:05
off their back surfaces.
00:19:07
Your eye is now outside the
maximum deflection angle
00:19:10
of all of the colors.
00:19:13
But if you look up even further,
00:19:15
sometimes you see a second fainter rainbow
00:19:18
with its colors inverted.
00:19:20
Double rainbow.
00:19:22
So where does this come from?
00:19:25
Well, it comes from an
additional reflection
00:19:27
inside the raindrops.
00:19:29
Now, instead of reflecting once
00:19:30
off the inside of the sphere,
00:19:32
light reflects twice.
00:19:33
These reflections also
create colored caustics,
00:19:37
though much fainter because light is lost
00:19:39
with each reflection.
00:19:41
If you look at deflection angles,
00:19:42
this light starts going out
the back of the raindrop.
00:19:45
So at an angle of 180 degrees,
00:19:47
but the further out light
hits from the center,
00:19:50
the smaller the angle
light reflects back at
00:19:53
until it reaches a minimum
of around 50 degrees
00:19:56
for red light.
00:19:58
Then it turns around and
goes back the other way.
00:20:01
So between 42 and 50 degrees,
00:20:04
it is dark because no light
reflected once or twice
00:20:08
inside a raindrop comes out at this angle.
00:20:11
This is known as Alexander's Dark Band.
00:20:15
Now, there is photographic evidence
00:20:17
of third and fourth order rainbows formed
00:20:20
after three or four internal reflections,
00:20:23
but this light comes out
the back of a raindrop.
00:20:26
So they are the only kinds of rainbows
00:20:28
that you could expect to see
00:20:29
when looking in the direction of the sun.
00:20:32
But they're so faint
00:20:33
that conditions would have to be perfect.
00:20:36
Under lab conditions,
00:20:37
up to 200th order rainbows
have been detected,
00:20:41
but that is not what is going on here.
00:20:44
This is known as a supernumerary rainbow.
00:20:48
Multiple rainbow like bands
show up under a primary rainbow,
00:20:53
but this only occurs
00:20:54
when the raindrops are all really small,
00:20:56
just 10ths of a millimeter in diameter.
00:21:00
Now, the light rays that
passed just above and below
00:21:02
the primary rainbow ray end up coming out
00:21:05
at similar angles under 40 degrees,
00:21:08
but they travel slightly
different distances
00:21:10
on the order of a wavelength.
00:21:12
And because of this, those
light rays can interfere
00:21:15
constructively and destructively,
00:21:17
producing a series of light
00:21:19
and dark bands inside the main rainbow.
00:21:22
Different colors in these
supernumerary rainbows overlap
00:21:25
more than in the main rainbow,
00:21:27
so they can produce strange
colors like magenta,
00:21:30
a combination of blue and red.
00:21:33
Supernumeraries also offer a clue
00:21:36
to how these small rainbows work.
00:21:39
Whenever I saw images like this
00:21:41
or even observed this sort
of thing from an airplane,
00:21:44
I wondered how a rainbow
could be so small.
00:21:47
(soft choral music)
00:21:49
These are known as
glories or Brocken bows.
00:21:52
Instead of the usual 42 degrees,
00:21:54
these circles of color are only around
00:21:56
two to four degrees wide.
00:21:59
Well, the key is that just
like in supernumeraries,
00:22:02
glories are due to interference.
00:22:04
So they too require tiny water droplets
00:22:07
just 10ths of a millimeter in diameter.
00:22:10
These are the sorts of droplets
you'd find in fog or clouds.
00:22:15
Light that strikes the edge
00:22:17
of the drop can go around the back
00:22:20
and come straight back at the source.
00:22:23
You can see that with the
laser on the glass sphere,
00:22:26
but effectively in the presence
of parallel light rays,
00:22:30
tiny little droplets become
a ring source of light.
00:22:34
But for these tiny droplets,
00:22:36
the distance from one point to all edges
00:22:39
of the sphere can vary on
the order of a wavelength.
00:22:44
So take for example,
00:22:45
the point right out in front of the drop.
00:22:47
Well, now the distance
to all edges is the same.
00:22:50
So the light interferes
constructively here
00:22:52
and produces a bright spot.
00:22:54
But a little bit off to one side
00:22:57
and now half of the light, on average,
00:22:59
has traveled an extra half a wavelength.
00:23:01
And so we get a dark spot here.
00:23:04
If you go a little further,
00:23:05
well now the light has traveled
an extra whole wavelength.
00:23:08
So now the light is
arriving in phase again,
00:23:10
and we get a bright spot here.
00:23:12
So we can rotate this around 360 degrees
00:23:15
and extend it out in all directions.
00:23:17
And what we get is a
fuzzy bullseye pattern.
00:23:21
And of course, all the different colors
00:23:23
of light have different wavelengths.
00:23:24
And so these
00:23:25
bullseye patterns aren't
completely overlapping.
00:23:27
So when they're superimposed,
00:23:29
what we see is rings of color.
00:23:33
Now, this is just the pattern
from a single droplet,
00:23:36
but just like with a rainbow,
00:23:37
if you have millions or
billions of these droplets,
00:23:40
they all contribute to
produce this same pattern
00:23:43
with your shadow at the center.
00:23:46
(soft music)
00:23:47
And it was just such a
pattern that inspired
00:23:50
a Nobel Prize winning discovery.
00:23:52
In September of 1894,
00:23:54
a scientist named CTR Wilson
was visiting an observatory
00:23:58
in the Scottish Hills.
00:24:00
It was then that he observed
"the coloured rings surrounding
00:24:04
the shadow cast on mist or cloud."
00:24:08
He recalls that these glories,
"greatly excited my interest
00:24:11
and made me wish to imitate
them in the laboratory."
00:24:15
So Wilson invented the cloud chamber
00:24:19
for the explicit purpose
of observing glories.
00:24:23
Of course, once he
discovered the cloud chamber,
00:24:25
made the tracks of
energetic particles visible,
00:24:27
he abandoned his original aim
00:24:30
and was later awarded the Nobel Prize.
00:24:33
But it all started with the
mystery of rings of color
00:24:37
in the fog.
00:24:41
- [Child] Wow.
00:24:42
Wow, looks so nice.
00:24:44
- How did you see that?
00:24:46
So now I hope you know my
son, why rainbows are curved,
00:24:51
and why they're polarized
and why they exist at all.
00:24:54
And even more than that,
00:24:56
I hope you know why I find such enjoyment
00:24:58
in learning about our world,
00:25:00
why it is worth figuring things out.
00:25:03
For millennia, rainbows have been this
00:25:06
blatant challenge held up to us by nature.
00:25:09
But can you figure this out?
00:25:12
And it's satisfying to say we have.
00:25:15
- [Child] I'm looking at a
different rainbow than you.
00:25:18
- It's true.
00:25:19
(celestial music)
00:25:23
When I started researching this video,
00:25:24
I thought I already knew about rainbows.
00:25:27
I mean, I'd learned all of the colors
00:25:29
of the rainbow in school
00:25:30
and that it was caused by light
refracting and reflecting,
00:25:34
but I realized now
00:25:35
that I'd only really just
memorize things about rainbows
00:25:38
and not really understood how they work.
00:25:41
Learning should be about
mastering a subject,
00:25:44
not memorizing a list of facts.
00:25:46
And that's why we asked
Brilliant to sponsor this video.
00:25:49
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