### Video Transcript

In this video, we will learn how to
draw diagrams of light rays interacting with concave lenses. Let’s begin by reminding ourselves
what a concave lens actually is.

A lens is a piece of transparent
material, that is, material that lets light pass through it. Lenses are designed with particular
shapes so that they will change the direction of light rays that pass through
them. A concave lens is a type of lens
with a shape like this when we view it from the side. The lens is thinner in the middle
and thicker at the edges. When we draw diagrams of light rays
going through lenses, we typically draw them using this side view. So, we would have our lens like
this and the light rays coming in from the side like this.

This dashed line that we have drawn
is called the optical axis. This optical axis passes straight
through the lens’s center. So, we have drawn these light rays
approaching the lens, but we want to know what happens next. In other words, what effect does
the concave lens have on the direction of these rays? It turns out that there are a
couple of simple rules that tell us what happens. So, let’s clear some space and look
at what these rules are.

The first rule says that any light
ray that passes through the center of a concave lens does not change direction. When we talk about the center of
the lens, we mean this point here, which is on the optical axis and is also an equal
distance from the front surface and the back surface of the lens. This first rule is telling us that
whatever direction a light ray is initially traveling in, then if it goes through
this center of the lens, it keeps traveling in that same direction. So, a light ray traveling along the
optical axis passes through the center of the lens and keeps going in the same
direction along the axis. In the same way, a ray through the
center initially traveling in this direction will continue traveling in the same
direction.

It’s worth pointing out that what
we’ve just said is actually a slight simplification. A light ray going through a concave
lens experiences refraction when it enters the lens through the front surface and
again when it leaves the lens through the back surface. Refraction changes the direction of
a light ray. So, if we consider this light ray
here, then, in reality, its direction would change when it enters the lens at this
point and again when it leaves the lens at this point.

However, for any light ray with an
initial direction such that it’s headed toward the center of a concave lens, it
turns out that the resulting effect of these two refractions is that the light ray
leaves the lens with pretty much the exact same direction that it had before it
entered. So, for rays through the center of
a concave lens, the effect of these two refractions is so small that we can safely
ignore it and continue to use rule one.

Let’s now look at the second rule
for light rays through a concave lens. This rule is about light rays which
are parallel to the optical axis, for example, rays like these ones here. We can notice that the light ray
which is directly along the optical axis will go through the center of the lens. So, this ray is already covered by
rule one. And we know that it keeps traveling
in the same direction, which in this case is along the optical axis.

This means that this second rule is
going to be useful for rays that are parallel to the optical axis but not actually
on it. When rays like these which are
parallel to the optical axis pass through a concave lens, they get their directions
changed so that on the far side of the lens they look like they’re spreading out or
moving away from each other.

Now, the directions that these rays
get changed to are not just random. If we trace these rays back,
extending their new directions back before the lens, then these extended lines pass
through a common point. This point is on the optical axis
and is known as the focal point of the lens. This direction change is exactly
what this second rule describes. The second rule says that any light
ray which is parallel to the optical axis has its direction changed so that its new
direction is on a line that passes through the focal point on the near side of the
lens. So, that’s the focal point on the
side of the lens that the light rays entered from.

This second rule means that if
we’re shown a diagram like this, where we have light rays which are initially
parallel to the optical axis, we can use the new ray directions to work out the
position of the focal point of the lens. We simply need to trace the new ray
directions back before the lens. And the position where these two
extended rays cross over is the focal point of the lens. This point will always be on the
optical axis.

The other way of thinking about
this second rule is that if for a particular concave lens we know the position of
that lens’s focal point, then we can use this to work out the new direction of any
light ray which is initially parallel to the optical axis. We know that the new direction will
be on our line through this focal point. So, we draw the line through this
focal point that gets to the concave lens at the height of the incoming ray. This line then gives the direction
of the ray on the far side of the lens.

It’s briefly worth noticing that in
both of these diagrams, we have drawn the light rays with their directions changing
twice in order to give the total overall direction change. This is because the direction
changes are due to refraction at the lens surfaces. The first change in direction
happens at the front surface of the lens and the second change happens at the back
surface. The important thing, which is
summarized in this second rule, is that the overall effect of these two direction
changes is that if the ray is initially parallel to the optical axis, its final
direction is on a line through the focal point.

Okay, so we’ve now seen the two
important rules for light rays passing through a concave lens. Now, these rules can also be used
to help us work out what happens to light rays coming from an object. Let’s clear some space so we can
see how this works. We’ve kept the diagrams which
illustrate each of the rules on the screen as we’ll find that these will provide a
useful reference. Let’s now suppose that we have a
concave lens and that some distance away from this lens, we have an object that has
its base on the optical axis.

The object that we have drawn looks
like a vertical line. But remember that we are looking at
this situation from the side. The object could, for example, be a
sheet of paper like this with a drawing on it. And we just can’t see that drawing
because we’re looking at the paper from this edge. We want to know what effect this
concave lens has on the light rays coming from this object.

Let’s suppose that the focal point
of the lens on the side that the object is on is at this position here. We’ll consider some light rays
coming from the point on the top of the object. And we’ll use our two rules for
light rays passing through a concave lens to work out what happens to them. First, let’s think about the light
ray that goes through the center of the lens. We can recall that the first rule
tells us that any light ray through the center of a concave lens keeps going in the
same direction. So, this light ray here continues
on like this in a straight line. Now let’s think about the light ray
from the top of the object which is parallel to the optical axis.

We can recall that the second rule
tells us that if a ray is initially parallel to the axis, its direction gets changed
so that it leaves the lens along a line which passes through the focal point. In the case of this ray from the
top of the object, it must leave the lens with a direction along this line here. This is the line that goes through
the focal point and gets to the concave lens at the height of the incoming ray. And so, on the far side of the
lens, the ray looks like this. These two light rays come from the
same point on the top of the object. If the two rays then pass through
the same point as each other somewhere else, this will produce an image point
corresponding to the point on the object that they came from.

The concave lens causes the light
rays to spread out from each other. And we can see that on the far side
of the lens, the rays are getting further and further apart. So, on this far side of the lens,
the two rays are never going to cross each other, and so they cannot form an image
point. However, let’s now look on the same
side of the lens as the object. We can see that the light ray that
goes through the center of the lens does cross over the line that comes from
extending the new direction of the ray that was initially parallel to the optical
axis. The two lines cross at this
position here. This point where they cross is a
virtual image point corresponding to this point on the object. We can do exactly the same thing
for any other points on the object.

For example, we could consider this
point, which is halfway between the top and the bottom of the object. We can draw the light ray from this
point which goes through the center of the lens. Using the first rule, we know that
this light ray keeps traveling in the same direction. Using the second rule, we know that
the light ray which is parallel to the optical axis gets its direction changed to be
on a line that goes through the focal point.

This diagram is starting to get a
bit cluttered now. But hopefully we can see that
there’s a point just here where the ray that’s going through the center of the lens
crosses over the line that we get from extending the new direction of the ray that
was initially parallel to the optical axis. This point where they cross is a
virtual image point corresponding to this point in the middle of the object. In fact, every single point on this
object has a corresponding image point. And in this way, a virtual image of
the object is formed at this position here.

When we say that this image is a
virtual image, what we mean is that though it can be seen by eye, this image cannot
be projected onto a screen. We can see from our diagram that
the image is smaller than the object was. We can also see that the image and
the object are both the same way up. So, if the object looked like this,
then the image formed by this concave lens would look like this. The observations we have made about
this image are true in general for any image formed by a concave Lens. That is, the image of an object
formed by a concave lens is virtual, is smaller than the object, and is the same way
up as the object.

Let’s now have a look at an example
question.

Each of the following diagrams
shows a ray entering a thin concave lens. The point marked P is the focal
point of the lens. Before entering the lens, the ray
is parallel to the optical axis of the lens. Which diagram correctly shows the
path of the ray after it passes through the lens?

Okay, so this question shows us
five different diagrams in which a light ray passes through a concave lens. We’re told that the point marked P
in each of the diagrams is the focal point of the lens. And we’re also told that before it
enters the lens, the ray is parallel to the optical axis. We can recall that there is a rule
for concave lenses that says any light ray, which is initially parallel to the
optical axis, has its direction changed by the lens so that its new direction after
the lens is on a line that goes through the lens’s focal point.

If we look at diagram A, we can see
that the incoming ray does indeed have its direction changed by the lens. We can also see that when this new
ray direction is extended back before the lens, the extended line goes through the
point marked P, which we know is the focal point of the lens. This means that the ray’s path as
shown in diagram A does follow this rule. Let’s now check out the other
diagrams to make sure that we’ve got the right answer.

In the diagram labeled B, the
direction of the ray is completely unchanged by the concave lens. But we know that its direction
should be changed so that its new direction is on a line through this point marked
P. So, this diagram cannot be
correct. Looking at the diagrams labeled C,
D, and E, we see that in each case the ray’s direction is changed by the concave
lens. When we extend the new ray
direction in diagram C back before the lens, it’s clear that it doesn’t go through
the point marked P. The same thing is true for the rays
shown in diagrams D and E. So, in these three diagrams, the
new ray direction is not on a line that goes through the focal point. So, diagrams C, D, and E cannot be
correct.

This leaves us with diagram A,
which we already saw shows the ray’s direction getting changed so that it’s on a
line through the focal point. And so, our answer to the question
is that the diagram that correctly shows the path of the ray after it passes through
the lens is the diagram labeled A.

Let’s now finish up by summarizing
what we have learned in this video. First, we saw that a concave lens
is a lens that has a shape like this when we look at it from the side. We then saw that there are two
useful rules for what happens to light rays passing through a concave lens. The first rule is that any ray that
goes through the center of the lens continues traveling in the same direction.

The second rule is that any ray
which is parallel to the optical axis before it enters the lens has its direction
changed so that the new direction is on a line that passes through the lens’s focal
point. Finally, using these two rules, we
saw how a concave lens can be used to form an image of an object. We found that the image formed by a
concave lens is virtual, is smaller than the object, and is the same way up as the
object.