Look at that!
So pretty...
The other day my son asked me why Rainbows are curved
and I could have given him a simple explanation
but instead I made this video
with beautiful demonstrations I've never seen before
this is the perspective of a rainbow from a single raindrop
and the best animations ever created on the subject
because I promise you almost every explanation out there
-double rainbow-
is an over simplification
For example if raindrops spread white light into
colors like a prism then why do you never see a rainbow when looking in the direction of the
Sun or why is it darker above a rainbow than under it how can you make a rainbow disappear
with sunglasses what's going on here why is this rainbow so much smaller than usual and
how did this phenomenon directly lead to a Nobel Prize oh yeah we're going deep on
on this one because the full explanation is so much more satisfying than anything you've seen
before to make a rainbow you need three things raindrops the Sun and you an observer okay this
is one of those experiments that seems so simple but I've never seen anyone do it before I have a
glass sphere and that represents my raindrop because as we've learned in previous videos
raindrops of course are essentially spherically shaped just preparing the particulate so that I
can see the laser beams over here I have a laser and that laser beam represents a ray
from the sun rays of light from the sun reach a raindrop essentially parallel to each other
because the sun is so far away when light strikes the sphere some of it reflects back off the front
surface and some is transmitted into the sphere then at the back again some of the light reflects
off the back surface and some is transmitted every time light goes from one medium into another some
of it will be reflected and some transmitted and exactly how much depends on the angle of
the light its polarization and the nature of the two media this is actually helpful here because I
can use the reflections to make sure the laser is lined up properly I think I've got this red laser
lined up so it's hitting the middle of the sphere and it's reflecting here but some of the light
goes through and some of the light reflects to the back surface and then some of the light goes
through to the wall now I'm going to keep the laser horizontal and move it up the sphere so
let's call the distance from the central axis the impact parameter as I move the laser up the
reflection off the front surface goes up it's just bouncing off that curved surface and I can tell
you this is a simple boring reflection nothing interesting happens with it it's not involved in
rainbows so it's there but we're basically going to ignore it for the rest of the video what's much
more interesting is the reflection off the back surface now here comes that Ray here the spot
is on the table as the laser moves up it goes down in fact the whole beam inside the sphere
bends down that's because as the light enters the sphere it slows down and so it refracts
but why does light slow down when it enters a dense medium like glass well I think a
lot of people can tell you that light is an electromagnetic wave without really thinking
about what that means you know the electric field around a charged balloon that pulls on your hair
or makes it stick to a wall and the magnetic field around a bar magnet that makes iron filings line
up well light is what happens if you could rip the electric field off the charges the magnetic
field off the magnet smush them together and send them out traveling through space sort of I mean
in practice electromagnetic waves are made by accelerating charges like by wiggling them up
and down then the changing electric and magnetic fields they create team up as light and head off
on their own the clearest explanation I know of for how light is slowed down in a medium comes
from Grant over at 3 blue one brown I asked him if I could summarize his explanation for
this video and he graciously agre greed so when those electromagnetic waves encounter charges in a
medium like those in the first layer of our sphere the light pushes them back and forth you can think
of each charge as a little Mass on a spring and the changing electric and magnetic fields cause it
to vibrate at the same frequency as the light but now you have wiggling that is accelerating charges
so they too must create their own electromagnetic waves and the net electromagnetic field is just a
sum of the incident wave plus this new wave the result is almost exactly like the original wave
except it is shifted back slightly it receives a phase kick and each layer of the material adds
another phase kick so the net effect of all this is the wavelength of the radiation decreases in
the new medium and since the frequency stays the same a shorter wavelength decreases the
speed of light through the material the speed of light in a vacuum divided by the speed of light
in a medium is called the refractive index it's around 1.5 for glass and 1.33 for water so when
light enters a new medium at an angle the part of the wave crests that enter the new medium first
slow down first and this changes the angle of all the wave crests and since the direction of
a beam of light is perpendicular to the wave crests this means the light changes Direction
so to recap light causes charges to wiggle so they create their own electromagnetic wave which gives
the light a phase kick shortening its wavelength which slows it down and so it bends there's a
simple mathematical expression that relates the angles of incidence and refraction to the indexes
of refraction of the two media it's known as Snell's law even though it was independently
discovered by a handful of people some well before Snell [Music] and this is what we're
seeing in the sphere the higher the laser hits the sphere the larger the angle of incidence and hence
the more it bends down due to refraction most of this light exits out the back of the sphere but
some of it is reflected and it's this reflected ray that we can see coming out the front of the
sphere below the incident beam now I want to graph the angle of this reflected ray as I move the beam
up the sphere when the laser is dead center the reflected beam comes straight back at the source
so let's call that 0° then as I move the laser up this angle increases so
the light's coming back at 5° then 10° and it keeps increasing the higher I go but now
we come to the critical point watch this spot on the table as I move the incident
beam up this dot is moving in the reflected ray is coming closer and closer and closer
but there's a certain point right there where it stops coming closer look at that and even as I
keep moving this beam higher and higher it doesn't get any closer and then it goes back the other
way so what we're seeing there is a maximum angle this reflected ray
reaches before it turns around and goes back the other way
and this is really important it means that over a range of impact parameters a range of heights
of the laser the reflected beam comes out at essentially the same angle which means the
light is becoming concentrated at that angle and a concentration of light rays is called
acostic curved surfaces all tend to create cosics from coffee mugs to glasses or even just Rippling
water cosics create the light patterns we see by concentrating light rays in the case of red light
through a sphere of water the maximum scattering angle and hence this CTIC always occurs at 42°
below the horizontal since my sphere is made of glass rather than water well the angle is
different but the principle is the same now you might ask why does this reflected ray reach a
maximum angle and then turn around well the answer is just geometry as I move the laser up the sphere
although the ray refracts down the point on the back of the sphere where it reflects continues to
move up until you get to this special point which is about 7/8 the radius of the sphere and here the
angle of incidence is so steep that the refracted Ray stops hitting the back higher and starts
hitting it lower that is why the reflection turns around and therefore we get a maximum scattering
angle and the concentration of light rays at that angle but the precise maximum scattering angle
depends on the color of light to see why let's go back to the idea of the charges in the sphere as
masses on Springs they have a natural frequency a frequency at which they would oscillate if not
driven at any particular frequency and in most materials this natural frequency is pretty high
much higher than the frequencies of visible light now when light pushes a charge back and
forth the amplitude of the resulting vibration depends on the difference between the frequency
of light and the natural frequency of the charge the closer the two frequencies are the greater
the amplitude of the resulting vibration which makes sense if you've ever pushed someone on a
swing the closer your pushing frequency is to the natural frequency of the Swing the higher
they'll go this means that higher frequency light like blue light will cause the charges
to wiggle with greater amplitude and because of this the charges produce higher amplitude
electromagnet itic radiation which creates a bigger phase kick which shortens the wavelength
proportionally more making higher frequency light travel slower and bend more than lower frequency
light all right I'm going to change lasers so when I repeated the experiment with green light
it refracted more than red light and therefore the Green Dot turned around sooner than the Red
Dot there's the minimum deflection for green and it's significantly different than for red in other
words its maximum scattering angle was smaller than for red light if the sphere were water it
would occur at around 41° below the horizontal similarly for blue light the maximum scattering
angle approaches 40° I do have a specialty laser which is very bright blue but it's very dangerous
so we're going to use it very carefully how are we going to use it very carefully Derek
that is a good question maybe we're just going to tape it on here
[Music]
yeah so the blue only makes it to there green to here red to here so it is a pretty serious
spread here I think my experiment may be a little wonky honestly I think this one wasn't perfectly
horizontal to really see the importance of the CTIC imagine we illuminate the sphere uniformly
with red light well more light is going to hit the sphere at higher impact parameters because
the further out you go the more area there is so I've adjusted these sections so that they
all have the same area then using our graph of the scattering angle you can see where all
of this red light will end up after reflecting off the back surface most of it ends up at the
maximum SC scattering angle to make this more obvious we can add more light and we can do
the same thing for orange and yellow and all of the other colors and this is what gives us the
rainbow it's not enough to say that a raindrop spreads white light into its component colors
because all of the light that hits closer to the middle is spread too but since the
reflections all overlap as they come out the colors mix and produce White again it's only
the difference in maximum scattering angles and thetics they produce that gives us the [Music]
rainbow so now we know what happens along a single radius of the sphere
so what happens if we uniformly illuminate the whole sphere with white light well I
blacked out my window and cut a hole just big enough for afternoon sunlight to cover
the sphere [Music] you can see there's a circle of white light coming from the
reflections off the back of the sphere and then around it there's a ring of rainbow
colors come on one raindrop creates a cone of light the inside is all white
and the ring around the outside is colored this is the perspective of a rainbow from a
single raindrop all these different light rays coming in at different places reflect back off
the front surface and the back surface and that reflection off the back surface reaches
a maximum angle and for blue green yellow and red the maximum angle is different so the red
maximum angle is the furthest that's why it's on the outside here so good so good what I really
wanted to see is if I could observe the cone of different colored CICS that produce that ring
H you've got to see this color cone you have got to see this I my eye is right in the color cone
here I can see the color cone that is so [Music] cool so this is the crazy focus on the back of
the sphere if you stick your finger in there it gets burnt very quick ow this thing is a fcal
now this really looks like a rainbow but remember this is just the light coming away from a single
droplet when you see a rainbow there are billions of raindrops and each one is projecting a rainbow
cone back toward the Sun so how does all of this create a single unified rainbow well for your eye
to see a color let's pick red in a certain part of the sky then the red costic from a raindrop
there must go directly into to your eye and this only happens when the angle from the Sun to the
Raindrop to your eye is 42° and this explains why rainbows take the form of an arch with a 42°
Angle now the violet light from these same raindrops passes above or beside your eye so
you can't possibly see it but there are raindrops below and inside the Arc of those red giving
raindrops whose Violet CICS do intersect your eye they form a shallower angle of 40° between the Sun
and your eye and of course there are raindrops at all intermediate angles that send you all the
other colors of the rainbow so a rainbow really is the ultimate optical illusion from billions of
droplets each projecting a rainbow cone you see a single static arch of color but the droplets
sending you those colors are constantly changing a single drop as it falls might send to your eye
first red then orange yellow green blue indigo and violet and because a rainbow must form an
angle of 40 to 42° with your eye the center of the arch must be on a line that passes from the
Sun through the back of your head so your Shadow is the center of your Rainbow this means no two
people can ever see the exact same rainbow in fact your left and right eyes don't even see the same
rainbow a rainbow is an optical illusion made unique for each perspective this also explains
why in most parts of the world you can only see a rainbow in the early morning or late afternoon
not in the middle of the day the higher the sun is the lower the top of the rainbow is
and when the sun is more than 42° above the Horizon no rainbow is visible from the ground
but even when Rainbows are visible you can turn them invisible using sunglasses that is as long
as they are polarized light from the sun is unpolarized which means the electric fields of
the light are all randomly oriented oscillating back and forth equally in all directions but it
just so happens that when the light in the rainbow Ray reflects off the back of the droplet it does
so very close to to a special angle known as Brewster's angle at this angle all light
with its electric field oriented parallel to the plane of reflection is transmitted so it passes
out the back of the droplet and therefore the only light that is reflected has its electric
field perpendicular to the plane of reflection this is the light that creates the rainbow this
means rainbow light is polarized along the direction of the rainbow so horizontal at
the top and closer to Vertical on the sides this is why you can use a polarizing filter to make a
rainbow disappear or to make it brighter if you Orient the filter to allow that polarized light
to pass through but why is it brighter under a rainbow than above it well this is because the
raindrops beneath the rainbow are reflecting all colors of light at you off their back surfaces
this is what created the white disc in my glass sphere experiment in contrast the raindrops above
the top of the rainbow are not reflecting any light to you off their back surfaces
your eye is now outside the maximum deflection angle of all of the colors but if you look up
even further sometimes you see a second fainter rainbow with its colors inverted double rainbow
so where does this come from well it comes from an additional reflection inside the raindrops
now instead of reflecting once off the inside of the sphere light reflects twice these Reflections
also create colored cosics though much fainter because light is lost with each reflection if
you look at deflection angles this light starts going out the back of the Raindrop so at an angle
of 180° but the further out light hits from the center the smaller the angle light reflects back
at until it reaches a minimum of around 50° for red light then it turns around and goes back
the other way so between 42 and 50° it is dark because no light reflected once or twice inside
a raindrop comes out at this angle this is known as Alexander's dark band now there is photographic
evidence of third and fourth order rainbows formed after three or four internal Reflections but this
light comes out the back of a raindrop so they are the only kinds of rainbows that you could expect
to see when looking in the direction of the Sun but they're so faint that conditions would have
to be perfect under lab conditions up to 200th order rainbows have been detected but that is
not what is going on here this is known as a super numerary rainbow multiple rainbow like bands show
up under a primary rainbow but this only occurs when the raindrops are all really small just tths
of a millimeter in diameter now the light rays that pass just above and below the primary rainbow
Ray end up coming out at similar angles under 40° but they travel slightly different distances on
the order of a wavelength and because of this those light rays can interfere constructively
and destructively producing a series of light and dark bands inside the main rainbow different
colors in these super numerary rainbows overlap more than in the main rainbow so they can produ
strange colors like magenta a combination of blue and red supernumeraries also offer a clue
to how these small rainbows work whenever I saw images like this or even observed this sort of
thing from an airplane I wondered how a rainbow could be so small these are known as glories or
brocken bows instead of the usual 42° these Circles of color are only around 2 to 4° wide
well the key is that just like in supernumeraries glories are due to interference so they too
require tiny water droplets just tth of a millimeter in diameter these are the sorts of
droplets you'd find in fog or clouds light that strikes the edge of the drop can go around the
back and come straight back at the source you can see that with the laser on the glass sphere but
effectively in the presence of parallel light rays tiny little droplets become a ring source of light
but for these tiny droplets the distance from one point to all edges of the sphere can vary on
the order of a wavelength so take for example the point right out in front of the drop well now the
distance to all edges is the same so the light interferes constructively here and produces a
bright spot but a little bit off to one side and now half of the light on average has traveled an
extra half a wavelength and so we get a dark spot here if you go a little further well now the light
has traveled an extra whole wavelength so now the light is arriving in Phase again and we get
a bright spot here so we can rotate this around 360° and extend it out in all directions and what
we get is a fuzzy Bullseye pattern and of course all the different colors of light have different
wavelengths and so these Bullseye patterns aren't completely overlapping so when they're super
imposed what we see is rings of color now this is just the pattern from a single droplet but
just like with a rainbow if you have millions or billions of these droplets they all contribute
to produce the same pattern with your Shadow at the center and it was just such a pattern
that inspired a Nobel prizewinning Discovery in September of 1894 a scientist named CTR Wilson
was visiting an observatory in the Scottish Hills it was then that he observed the colored Rings
surrounding the shadow cast on mist or Cloud he recalls that these glories greatly excited my
interest and made me wish to imitate them in the laboratory so Wilson invented the Cloud chamber
for the explicit purpose of observing glories of course once he discovered the cloud chamber made
the tracks of energetic particles visible he aband abandoned his original aim and was later
awarded the Nobel Prize but it all started with the mystery of rings of color in the [Music]
fog that looks so nice how did you see that so now I hope you know my son why Rainbows are curved and
why they're polarized and why they exist at all and even more than that I hope you know
why I find such eny mment in learning about our world why it is worth figuring things out
for Millennia rainbows have been this blatant challenge held up to us by nature but can you
figure this out and it's satisfying to say we have I'm looking at a different rainbow than you it's
true when I started researching this video I thought I already knew about rainbows I
mean I'd learned all of the colors of the rainbow in school and that it was caused by
light refracting and reflecting but I realized now that I'd only really just memorized things
about rainbows and not really understood how they work learning should be about mastering a subject
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