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All of you have looked at rainbows, but very few of you have ever seen one.

Looking at something is very different from seeing it.

And today I will make you see the rainbow in a way that goes way beyond the beauty that we can all experience, a way that you will always remember.

And I would like to start asking you 15 perhaps simple questions about the rainbow.

The first question then is would any one of you remember if you see a bow whether the red color is outside or whether the -- the red color is inside?

And then I wonder about the radius of the bow.

If this is a bow in the sky, something like this, here is the horizon, it's clearly a perfect circle, and so the perfect circle has somewhere a center.

And so that means there must be a radius R.

You can measure that radius in terms of how many degrees and so what is roughly that radius.

You've never measured it but is it 10 degrees, is it 20, 30, 50, 60?

The length of the bow.

Is there a difference, do you sometimes see a very long bow, sometimes a very short one?

What is the width of the bow?

You see colors here.

How wide is that strip of colors in degrees?

Perhaps some of you have noticed that there is a difference in light intensity between inside the bow and outside the bow.

Maybe you've never seen it, and if there is a difference where is it brighter, inside the bow or outside the bow?

What time of the day would you see bows?

Would you see rainbows in the north, east, south or west?

Is there perhaps a second bow in the sky?

And if there is a second one, where should you look for the second bow?

And if there is a second one what is the color sequence of the second bow?

Is the red on the outside or is the red on the inside?

And then you can ask the same question, what would be the radius of the second bow?

And what would be the width of the second bow?

All these first 12 questions in principle you should have been able to answer if you really have seen a rainbow.

The last three is more difficult.

The question is are the bows polarized?

In what direction are they polarized?

And are they weakly polarized or are they strongly polarized?

Who knows the answer to 12 questions, to the first 12 questions?

Who knows the answer to more than 10?

Who knows the answer to nine?

Eight?

Seven?

Six?

Five?

Four?

Do I see a hand at four?

Good for you.

Five, four, three?

Three, good, that's already good.

Two?

One?

And who knows the answer to zero?

Most of you, right?

I haven't seen a lot of hands though.

All right.

So I've made my point.

You've looked at rainbows but you've really never seen them.

And I'm going to make you see them today.

What you see here on the blackboard is one drop of water.

I put the sun for simplicity at the horizon.

Later I will put it a little bit higher in the sky.

Light from the sun hits this raindrop.

I've only drawn one narrow beam which hits the raindrop right there.

And you see here the angle of incidence, which with Snell's law we call theta 1.

I call it I here because it's nicer for me, more descriptive, it means incidence angle.

Right at that point A some of the light will be reflected and some of the light will go into the water.

We call that refraction.

And Snell's law will tell me this angle R.

Whatever goes in there reaches point B where there is a transition back to air and so some of that light will come out here and some of that light will be reflected inside.

And then when it reaches point C again there is a transition from water to air.

Some of that light will be reflected inside the water.

And some of it will come out.

And as far as the geometry is concerned, if this angle is R, then this angle is also R, this is also R, and this is also R.

And the angle here is I.

That follows from Snell's law, and I'll leave you with that.

Notice that the light came in like this but it comes back like this.

So the direction has changed over the angle delta.

And the angle delta is very easy to calculate in terms of I and R.

Delta is 180 degrees + 2I - 4R.

I want you to check that at home.

The 4 Rs come in here.

One, two, three, four, and the 2 I's come in here and there.

If now I think of all possible narrow beams of light that can strike this raindrop, one that would strike it here would have an I of 0 degrees.

And then here would be 10 degrees and 20 degrees and 30 and 40.

And the largest value for I is when the light strikes here, would be 90 degrees.

And so I can calculate for all these values of I, which obviously all of them occur, sunlight strikes this raindrop, and all these angles for I are present.

So I can calculate now for all these angles of I what the value is for R and then I can calculate what delta is.

R follows from Snell's law and delta follows from this geometric relationship.

And what you will find now very much to your surprise, that there is a minimum value for delta which is about 138 degrees.

That means this angle phi here has a maximum value which is very roughly about 42 degrees.

And I will show you some numbers.

You can download this, by the way, this is on the Web, under lecture supplements.

Here all I have done I've taken I to be from 0 to 90 degrees, all these angles are possible, with Snell's law, using an index of refraction of 1.336, that you see at the bottom, I calculate R and then in the last column using that relationship I calculate delta.

And indeed you see that delta starts at 180 degrees when I is 0.

And then goes to a minimum of roughly 138, after which it increases again.

And this now is crucial, is key to an understanding of the rainbow.

Imagine now that I have one drop of water here.

And sunlight comes in at all angles of I, not just at one, but all angles of I.

Whatever you see here has of course axial symmetry.

It is a spherical drop.

And the light comes in like this.

So light can go this way but it can also go this way.

And it can also go this way and this way, so there's complete axial symmetry, so this whole drawing you can rotate about this line here.

And everything holds then in axial symmetry.

So therefore if phi maximum, if this angle phi maximum is 42 degrees, then the light that will go back in the direction of the sun, the light that goes through the journey A B C and then comes out of the raindrop, that's all I'm talking about, now, I'm not talking about this light that sneaks out here, it is this journey, A, refraction at A, reflection at B, and then coming out at C.

That light comes out in the form of a cone.

And the half -- top angle of the cone must be roughly 42 degrees.

And so I will go -- I'm going to draw that cone for you.

Like so.

And like so.

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