I’ve made a lot of videos, but a recent
one somehow engaged many viewers.
This video was about why light slows down
when it enters water, glass, or any transparent
medium.
For some reason, this video took off and the
comments were very interesting, covering the
entire gamut of claiming that the video was
insightful, to claiming that it was utter
rubbish.
But one particular question appeared over
and over again, and that comment was “But
why does light bend when it goes from air
to glass?”
And it’s true that I didn’t answer that
question because, well, it wasn’t what the
video was about.
But a lot of people asked, so, in this video,
I thought I’d tell you why.
Now, as it happens, there are lots of wrong
explanations out there.
Lots.
So the first thing I’m going to do is tell
you the wrong ones and why they’re wrong.
Don’t be surprised if I debunk one you learned.
So, let’s start with what we see when a
beam of light hits glass.
Light traveling through air will change its
direction when it hits the surface.
The amount that its direction will change
depends on the angle at which it hits the
surface.
Change the incident angle and the transmitted
angle will change correspondingly.
Those angles are related through a physics
formula called Snell’s Law, named after
Dutch astronomer Willebrord Snellius, who
lived in the 1600s.
I have his formula written over here.
If you’d like to learn how to use the formula,
there are many online tutorials.
But I’m not going to cover that here, because
I want to get into the question of what is
the physical process that causes the path
of the light to change as it enters the glass.
So let’s get into it.
There are three commonly suggested explanations
that are wrong or at least incomplete.
They are Fermat’s Principle, the analogy
of soldiers marching, and Huygens’s Principle.
Fermat’s principle says that light will
travel from one point to the other taking
the minimum amount of time.
It’s often explained in terms of a lifeguard
and a drowning swimmer.
In order to maximize the chances that the
swimmer will be saved, the lifeguard needs
to get to them as soon as possible.
So what path does the lifeguard take?
He could run directly to the water and start
swimming.
This means he would swim a long distance.
But because he could run so much faster than
he can swim, that'd be a bad choice.
So, you’d think that he might just run over
to the point where the swimming portion is
shortest.
But that turns out not to be the shortest
time either.
Then there’s the straight line path.
And because the shortest distance between
two points is a straight line, you might think
that this is the fastest path.
But because swimming is slower, it still isn’t
the shortest time.
It turns out that the shortest total time-
that is to say when you add both running and
swimming time- is exactly what you’d get
from Snell’s law for light.
You run a little longer than you would for
the straightest path, but that means you swim
a little less and the total time is shortest.
And that’s exactly what light does.
It follows the path of shortest time.
But, while that’s true, it doesn’t explain
anything.
It just says what it does, not why it does
it.
So that’s not an explanation and we need
to come up with a proper reason.
Let’s look at the second option on the list.
A very common explanation used to teach why
light bends involves soldiers marching over
firm ground but then who encounter mud, although
not all soldiers hit mud at the same time.
The idea is simple.
The first soldier who encounters the mud cannot
move as quickly as the rest of the soldiers,
so he slows down.
Each soldier hits the mud in turn, resulting
in a direction change.
Eventually, the lines of soldiers are moving
in a different direction at a different speed.
This is how it's often taught.
The problem is that this doesn’t work.
In fact, the soldiers will slow down, but
their direction doesn’t change.
They will continue to walk in the same direction,
but with marching lines that are slanted compared
to how they started.
This explanation gets the angle of the lines
of soldiers right, but not the direction of
travel.
In fact, the only way that this explanation
can be right is if the lines of soldiers are
rigid.
Then the first soldier hitting mud puts a
torque on the entire line and that would work.
The problem is that this torque means that
the soldiers on the top of the screen move
faster, which is to say that- if it were light-
this part of the beam of light would have
to move faster than light.
So this explanation doesn’t work either.
Okay, so far, we have a non-explanation and
a wrong explanation.
What about Huygens’s principle?
Huygens’s principle, named after Dutch physicist
Christian Huygens, depends crucially on the
wave nature of light.
We know that when a wave encounters a small
gap, the wave passes through the gap and spreads
out.
This principle is called diffraction and it's
universal to all waves.
If waves hit more than one gap, each gap will
allow waves to pass through and make ringlets.
The waves from each gap can interfere with
one another, with the waves either adding
to or subtracting from one another.
In this animation, the lines show the position
of the peak of the waves.
Showing how the waves add together is perhaps
easier looking at them from the side.
When the peak of two waves encounter one another,
the result is a single and bigger wave.
And, when a peak of one wave hits the trough
of another wave, they can cancel each other
out entirely.
What Huygens claimed is that there is no need
for gaps.
Each and every spot on a wave was acting like
it went through a gap and was spreading out.
Now we can take that idea and have waves moving
in one medium transition to another medium
where the speed is different.
So we can take light traveling through air
and shoot it through glass.
And, because we’re interested in explaining
the bending behavior, we'll have light hit
it at an angle.
Here’s how the Huygens explanation goes.
What happens is that the wave peaks hit the
glass at different times.
The waves then spread out and begin to interfere
with one another.
Notice that the wavelength is shorter in glass.
That’s because the wave is moving more slowly.
Where the lines accumulate, that’s where
the waves add to one another.
And they don’t just line up in one spot,
they line up again and again.
Remember that the lines denote the top of
the peaks of the waves.
We can see that there are many places where
the peaks line up.
We can overlay a couple of lines that show
the direction that the wave is traveling and
then remove the circular waves and we see
that this explanation seems to be doing a
good job of predicting what happens when light
goes from air to glass.
Looks good, right?
But not so fast.
What happens when we look at all of the waves
coming through the glass.
Things get a lot more complicated.
We do see that the waves add together to predict
the direction that light moves in glass.
On the other hand, that’s not the only alignment.
For instance, we see other geometries where
the waves line up.
There are the wave fronts that are going this
way.
And then there are the wave fronts going this
way.
So that’s a problem.
It seems that this approach doesn’t give
a unique prediction.
Okay, we have ruled out Fermat’s principle,
the soldier analogy and Huygens’s principle.
These are all often seen as explanations of
the cause of refraction, given by people who
really do know some physics.
So what's the real answer?
It turns out that the only way to really answer
the question of why light bends when it goes
from air to glass is to get serious about
the nature of light and to embrace the fact
that it is made of oscillating electromagnetic
fields and that means you need Maxwell’s
equations.
Because this isn’t a full-blown physics
class, I’m going to focus only on the electric
fields.
And I’m not going to do the derivation,
because that’s the fun part- I’ll leave
that to you.
But the big ideas are actually pretty straightforward.
You start with Maxwell’s equations, which
are always beautiful to see.
They’re a little scary looking, but the
question we are trying to answer is an easy
one, which makes the whole thing a lot simpler
than you’d imagine.
In fact, we’re going to need only the bottom
two equations.
So, let’s see what's going on.
We start with light going from air to glass,
hitting the surface at an angle.
In our figure, we can replace the waves with
the direction of motion.
Now it turns out that the electric field of
light is perpendicular to the direction that
light is traveling, and we can add that field
direction to the diagram.
And it's very important to remember that this
field has a component both parallel to the
surface of the glass and one that is perpendicular
to the glass.
And here is where Maxwell’s equations come
into play.
Two copies of the equations are written here,
one that covers when light is traveling in
air and one where it is traveling in glass.
So here is the key point.
The surface belongs to both the air region
and the glass region.
This means that at the surface, the equations
on the top and the equations at the bottom
have to apply.
And, with a little bit of calculus, you can
find two important restrictions.
The first is that the electric field parallel
to the surface of the glass has to be the
same in the air and glass.
And similarly, perpendicular to the surface,
what has to be the same is the electric field
times this epsilon, which is different for
each material and depends on its molecular
makeup.
We can then manipulate these equations to
see what the electric fields in glass should
look like.
Because epsilon is bigger in glass than in
air, that means that the perpendicular electric
field in glass has to be smaller than it is
in air.
Now we remember that the direction that light
travels is perpendicular to the electric field,
so we can put in an arrow to show the direction
light must travel in the glass.
And finally, we can see what light does when
it enters glass or water or any transparent
medium.
It bends.
And the reason that it bends is because the
epsilon in glass is bigger than in air.
Okay, what I just showed you is an equation
thing.
You're probably asking yourself what that
epsilon actually, physically means.
It's there because of how the electric fields
from the light interacts with matter in the
glass.
You start out with glass with no electric
field in it.
The glass has charges in it, but they're arranged
in a random way.
But when you send light in, you impose an
electric field on it.
That field makes the charges move around,
which sets up a counterbalancing electric
field from the charges.
The result is that the electric field in the
glass is lower than it is in air because of
how the electric field from the glass is in
the opposite direction.
And this is the reason that the perpendicular
electric field is lower in glass.
And that, my friends, is why light bends when
it goes from air to glass.
It’s not because of many of the ordinary
explanations.
It’s because of how light interacts with
glass and changes the glass’s properties.
And it’s because the electric field inside
the glass is affected by the arrangements
of atoms and molecules in the glass when light
hits it.
It’s the same reason why light slows down
in matter.
Inside matter, the electric field due to the
light and the electric field due to the matter
both have to be taken into account.
Outside they don’t.
And now, you know some serious physics.
Okay, so that was a very long video.
It kind of had to be to cover the many incomplete
or wrong explanations and then explain the
correct cause.
And, I imagine, some of you will ask about
the quantum explanation, which is similar
to the one I told you here using Maxwell’s
equations, except using the energy and momentum
of the photons instead.
There’s always more to learn in physics-
always more.
And you knew that, because, of course, physics
is everything.
