### Video Transcript

In this video, we’re talking about
the reflection of light. It’s not something we may think
about very often. But actually, the reflection of
light is responsible for most of what we see in our day-to-day life. For example, consider this room
with light coming in through the window and a lamp over there by the side of the
couch. As far as our eyes are concerned,
these are the only two light sources involved here. But nonetheless, we’re still able
to see everything else in the room, the end table, the couch, the carpet, the
picture on the wall. Each one of these objects is not
itself a source of light. But we see it because light
reflects off it to our eye. As we learn in this lesson how all
this reflection works, let’s start out by considering the light itself that is
reflected.

Light itself has always been a bit
of a mystery to us. In some ways, it seems that light
is a wave. One of the reasons for that is that
if we send two separate light waves towards one another, they interfere, either
constructively or destructively or somewhere in between, just like waves do. But at the same time, light also
behaves in some ways like a particle. We see this in particular in an
effect called the photoelectric effect. In this effect, we can send a
single packet of light, called a photon, into a metal surface. And if that single bit of light,
that particle, is energetic enough, when it hits the surface, it will cause the
ejection of an electron.

Since light clearly has both of
these properties, the ability to interfere with itself and the ability to eject
single electrons, maybe for now, instead of saying that light is a wave and a
particle, instead of saying that light is a wave as well as a particle, let’s put it
this way. In some ways, light acts like a
wave. And in some ways, it acts like a
particle. When it comes to the subject of the
reflection of light, in this case, it’s more helpful to think of light as a particle
than it is like a wave. Specifically, we’ll imagine that
light is a particle that moves in a straight line. That means whenever we see a bit of
light, we’ll draw it like a ray, a line that’s headed in some direction. If this ray of light never runs
into anything, say it’s just in our space, it will go on and on in that line
forever. But if there is some object for the
ray to run into, then all of that changes. Just like we would expect, if we
were, for example, throwing a ball at a wall, just like that ball might do, the ray
of light reflects. We could almost think of it as
bouncing off of this object.

It turns out that when the light
does reflect, it does so according to a very specific rule, known as the law of
reflection. Here’s what this law of reflection
tells us. It says that the angle at which a
ray of light is incident on a surface, and we’ll see what that means in a second, is
equal to the angle at which it’s reflected from that surface. To better understand this law,
let’s consider another light reflection example. Say that we have some surface. It could be a tabletop or a piece
of glass. It doesn’t really matter what the
material is for the purpose of figuring out the reflection though. And let’s say that, onto this
surface, we send a single ray of light. Now, the question is how will this
light reflect off of the surface that it’s running into. In other words, how will the light
bounce off of this surface on which we say it’s incident?

The law of reflection helps us
understand this reflected angle. As we saw, this law refers to two
specific angles. The first one is the angle at which
this ray of light is incident on the surface. Here’s how we figure out what that
angle is. First, what we do is we look at the
line of our surface or the plane of the surface. And then, at the location where our
ray of light hits or runs into that surface, we sketch in a line which is
perpendicular to the surface. This is sometimes called the normal
line, where here normal means perpendicular. This normal line is going to be
very important because it’s gonna help us understand what is the angle at which this
particular ray of light is incident on the surface. Looking at this sketch, two
separate angles may stand out to us, either one of which can be considered the angle
of incidence.

The first angle is from the surface
of a material up to the ray. We’ll call that angle 𝜃 one. And the second angle starts out at
the normal line and then goes down to our ray. And we’ll call this 𝜃 two. Now, either one of these angles,
what we’ve called 𝜃 one or 𝜃 two, could be the measure of the angle of incidence
of this ray of light. And notice, by the way, that if we
add together 𝜃 one and 𝜃 two, then we’ll get 90 degrees. But anyway, what’s helpful to know
at this point is that the angle that’s been chosen, by convention, to indicate
what’s called the angle of incidence is what we’ve labelled as 𝜃 two. In other words, to measure the
angle of incidence of a ray of light on a surface, we’ll need to measure the angle
between the normal line to that surface and the ray of light incident on it. Oftentimes in symbolic shorthand,
this angle is referred to as 𝜃 sub 𝑖. And that’s the symbol we’ll use to
indicate the angle of incidence.

Okay, we now know the angle of
incidence of this ray of light, in terms at least of what it looks like on our
diagram. But that still leaves the second
angle which our law of reflection talks about, the angle at which the ray is
reflected from the surface. And here’s where this law comes in
so handy because it tells us that the angle of incidence, what we’ve called 𝜃 sub
𝑖, is equal to the angle of reflection. In other words, if we were to start
at our normal line and measure an angle off that line to the right, which was equal
to 𝜃 sub 𝑖, the angle of incidence, then that will be the angle of reflection,
which we could call 𝜃 sub 𝑟. And it’s this angle that tells us
just how this incident ray of light is reflected from our surface.

So as a quick recap of this law of
reflection, when a ray of light is incident on a surface, that is, when it runs into
a surface, then the angle at which that ray hits the surface, called the angle of
incidence, which is measured off of the line at normal to or perpendicular to the
surface, is equal, the law of reflection says, to the angle at which that ray is
reflected from the surface. What we’ve called 𝜃 sub 𝑟, where
that angle, just like the angle of incidence, is again measured with respect to the
normal line. Using our symbols for the angle of
incidence and the angle of reflection, we can say that the law of reflection is this
mathematical statement. 𝜃 sub 𝑖 is equal to 𝜃 sub
𝑟.

One great thing about the law of
reflection is that it always applies no matter what kind of surface our light is
running into. Let’s consider what some of those
surfaces might be. Imagine that you have some very
smooth polished surface like a mirror. And you send a ray of light into
it. But then imagine that you have a
very different type of surface, an uneven one, say it’s rough sandpaper. For each one of these surfaces, the
smooth and the rough one, our question is the same. In what direction will the
reflected light travel? For both these types of surfaces,
the law of reflection applies.

Considering our smooth mirror, we
can sketch out a line in the plane of that mirror, where the ray hits the
mirror. And then, at 90 degrees or
perpendicular to that line, we can sketch in our normal line then measure the angle
between that normal line and our incident ray. And we’ll call that 𝜃 sub 𝑖. And then, to figure out the
reflected angle, we’ll measure the same angle off of the normal line, but to the
right, where this is 𝜃 sub 𝑟, the angle of reflection. And then, we can sketch in our
reflected ray according to this reflection angle. That’s the procedure for our smooth
mirror surface. And actually, it’s the same
procedure we’ll use for a rough sandpaper surface. For a rough surface, we’ll find out
very precisely where this ray is gonna hit that surface. Then what we do is we sketch out
our dashed line, which again is in line with the plane of the surface at that point,
where the ray is going to hit it.

Notice that this dashed line we’ve
drawn in at the point where the ray hits the surface is parallel to the surface. That’s important because we want
this line to show us just what the angle of the surface is at the point where the
ray hits it, not at any other point. With that line drawn in, we can
then sketch in our normal line, at 90 degrees to the one we’ve just made. Then, on this sketch, we can
identify our angle of incidence of our light ray, 𝜃 sub 𝑖. And then, to show the reflected ray
direction, we’ll create an angle of the same measure off to the opposite side of the
normal line. That’s 𝜃 sub 𝑟. We can then draw in our reflected
ray, which answers our question of which way the reflected light goes.

Now, considering these two
surfaces, the smooth and the rough ones, we can see that they’re very clearly
different. One way to see that difference is
to imagine sending in not one ray to these surfaces, but several parallel rays. That would look something like
this. And let’s consider just how these
rays would reflect off of these two different surfaces. In the case of our smooth mirror
surface, all these reflected rays, which came in parallel to one another, would exit
or reflect parallel with one another. They’d all be moving at the same
angle off of this smooth surface. There’s a special name for this
kind of reflection. It’s called specular
reflection. This describes reflection where
multiple rays come into a surface parallel to one another. And they leave that surface
parallel. So that’s what these reflected rays
would look like for our mirror. But now let’s consider our
sandpaper.

For this surface, even though the
rays did come in parallel, they definitely don’t leave that way. The first ray we had drawn in
reflected like this, recall. But then, our second ray bounces
off this surface here and then bounces off the surface again and goes off in this
direction whereas our third initially parallel ray bounces off the surface and goes
in this direction. So basically, they come in
parallel. But they leave pointed every which
way. This is known as diffuse
reflection. The name comes from the fact that
even if we shined a very bright highly directional light onto this rough surface,
the reflected light will be spread out in every direction from that surface. It would be diffused.

An important thing to realize about
both these types of reflection is that the law of reflection applies in both
cases. The only difference is that when
our surface is very smooth, like our mirror, then we don’t have to be very
particular about where we choose to draw our line parallel with the plane of that
surface because that plane is always pointed the same way. But on the other hand, with diffuse
reflection, that point we choose matters a lot. When we drew our initial line in
this case, we had to make sure we drew it at the exact point on that surface, where
the ray hit it. That way, we could reflect the
slope of the surface at that point. Of course, the slope of the surface
was very different for these other two parallel rays. In the case of the second ray, it
would look something like this and in the case of the third ray, something like
this. The slope changes when the surface
is rough.

We mentioned earlier that it’s the
reflection of light that lets us see everyday objects, which themselves are not
light sources. This means that whenever we’re
seeing something which is not a source of light, what we’re really seeing is light
that we could trace back to a light source that has bounced off that object. Let’s consider how this works when
our surface, like it is in this case, is a flat mirror. Say that we have a big plain mirror
on the wall. And we’re standing in front of it,
so that our eye is at this position. And say further that there’s a
flashlight above us shining a directed beam of light onto this mirror. The law of reflection tells us that
the reflected angled of this ray will equal its incident angle.

Now, like we mentioned, in this
case what we’re seeing is not actually the mirror. But it’s the light reflected by the
mirror. In other words, our eye is looking
at the source of this light, the flashlight. Interestingly, thanks to this
reflection, our eye doesn’t see the flashlight where it actually is though. It sees it behind the mirror, along
the line that reached our eye in the first place. So the image that our eye sees of
this flashlight isn’t real. It’s called a virtual image.

This type of image, not a real one
but a virtual one, is created whenever we see an object in a mirror. When we look in a mirror, the light
that reaches our eyes, just like in the case of this flashlight, seems to our eye to
be coming from a point behind the mirror. But we know that’s not where the
object actually is. Therefore, we call this image a
virtual image. Knowing all this about the
reflection of light, let’s get a bit of practice with these ideas through an
example.

What is the incident angle of the
light ray shown in the diagram?

We have our incident ray of light
here that reflects off of our flat surface here. And then, this ray drawn in is the
reflected ray. On our diagram, this 35-degree
angle represents the angle here between a reflected ray and the surface. We want to know what is the
incident angle of this light ray. Now, the first thing we can do is
figure out which part of our figure represents that incident angle.

At this point, it’s important to
remember that the incident angle as well as the reflected angle of a light ray, by
convention, are always measured with respect to the normal line to the surface the
ray reflects off of. That means that this dashed line
here, the normal line to the surface that’s perpendicular to it, is a reference
point for the incident as well as reflected angles of this ray. The ray’s incident angle is
measured from this normal line to the ray itself. And we can represent this angle
symbolically as 𝜃 sub 𝑖. And then, the reflected angle of
the ray, likewise, is measured from this normal line. We start at the normal line and go
to the reflected portion of that ray. And this angle we can represent
symbolically as 𝜃 sub 𝑟, the angle of reflection.

Now, there’s an important
relationship between 𝜃 sub 𝑖 and 𝜃 sub 𝑟 that’s worth recalling. The law of reflection tells us that
the angle of incidence of a light ray is equal to its angle of reflection. Written down as an equation, we say
that 𝜃 sub 𝑖 is equal to 𝜃 sub 𝑟. Looking back at our diagram then,
we can see that if we can solve for 𝜃 sub 𝑟, then we’ll also have solved, by the
law of reflection, for 𝜃 sub 𝑖. They’re equal. So what is 𝜃 sub 𝑟, the reflected
angle of this ray?

If we look at the plane of the
surface that the ray reflects off of, we know that this plane is at 90 degrees to
the line we’ve drawn, called the normal line. That means that this angle here
that we’ve drawn in must itself be equal to 90 degrees. And with that being the case, we
can now write an equation for 𝜃 sub 𝑟 in terms of these two angles, 90 degrees and
35 degrees. Looking at the diagram, we can see
that 𝜃 sub 𝑟, whatever that is, plus 35 degrees is equal to 90 degrees. We can see that this reflected
angle, 𝜃 sub 𝑟, plus this 35-degree angle must sum up to 90.

So to solve for 𝜃 sub 𝑟, let’s
subtract 35 degrees from both sides of this equation. When we do, that term cancels out
with the positive 35 on the left-hand side. And we see that 𝜃 sub 𝑟 is equal
to 90 degrees minus 35 degrees and that that is equal to 55 degrees. Now, remember that we wanted to
solve for the incident angle but that by the law of reflection, that incident angle
is equal to 𝜃 sub 𝑟. Therefore, we can write that 𝜃 sub
𝑖, the angle of incidence, is equal to 55 degrees. And that’s our final answer.

Let’s summarize now what we’ve
learned about the reflection of light. When it comes to the reflection of
light, the most important principle to recall is the law of reflection. This tells us that the angle of
incidence of a ray of light onto a surface equals its angle of reflection from that
surface. We saw that the way to figure out
these two angles, angle of incidence and angle of reflection, have to do with
drawing a normal line to the surface that the ray of light bounces off of. Both angles are drawn starting from
that normal line and then going until they reach the ray. Using these symbols, the law of
reflection can be summed up mathematically by this equation. The angle of incidence is equal to
the angle of reflection.

Furthermore, we saw that light can
experience specular as well as diffuse reflection from a surface. Specular reflection occurs when the
surface is smooth and diffuse when the surface is rough. And lastly, we learned that images
formed by flat mirrors are called virtual images. This is because the image that our
eye sees exists behind the mirror in virtual space.