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# Parabolic mirrors and real images

## Video transcript

In this video, I
want to expose you to a special class of mirrors
called parabolic mirrors. Or sometimes called
parabolic reflectors. And what's neat about
parabolic mirrors-- and I'll draw a cross
section of one right here. And if you're familiar
with the algebra, they are essentially-- the
cross section, especially, is in the shape of a parabola. So let me draw a
parabola right here. So it's in the
shape of a parabola. Just like that. And what's neat about
a parabolic mirror-- and I'm not going to go
into the math right here. I just want to give
you the general idea. And let me just draw
its principal axis. So this is the line of
symmetry of the parabola. So this is its principal
axis right over here. It divides it in two. This is just a cross section. You could imagine
if this was spun around that principal
axis, you would get something that
would look like this. You would get something
that would look like a bowl. But it's actually the
shape of a parabola. It's not an actual sphere shape. So if you rotate
this around, you would get a circle
around the edge. So this would be a
circle right over here. But this shape down here
is not a hemisphere. It's not spherical. It's actually a parabola. And the reason why we
care about a parabola, or what's neat about
parabolic mirrors, is if I have parallel
light rays coming into a parabolic
mirror-- I'll do my best to draw a
parallel light ray. So parallel to its central axis. So if I have a light ray
that comes like that, it will reflect off
of the-- it's parallel to this principal axis--
it will reflect like that. And I'll tell you what's
neat about this in a second. Now let me draw
another parallel ray. Let's say I have a
parallel ray that's coming in right over there. So it hits the parabolic
mirror at that point. It's going to reflect--
so it comes in like that. And if I have another ray
that comes in like this, it will reflect so that
the reflection goes right over there. So what's neat about this? Well, what's neat is
any light ray that comes in parallel-- any incident
light ray that's parallel to the principal axis of
this parabolic mirror-- the reflected ray is going
to go through the same point. I don't care where
you hit the mirror. As long as it was parallel
to the principal axis, the reflected ray is
going to hit this point. And this point right
here is the focus. This is the focus of
the parabolic mirror. Now, what's neat about this? Well, let's say
that you were trying to capture heat from the sun. You were trying to concentrate
the electromagnetic radiation from the sun. So what you could
imagine-- you could go to the middle of the
desert-- and people do do this-- and you set up of
parabolic mirrors like this that are
pointed at the sun. And the sun's rays come in. And the sun is so far away,
they're essentially just coming in parallel because they
are radiating from the sun. But the sun is 93
million miles away. So the rays for our purposes are
essentially coming in parallel. And what's neat
about them is, is when they hit the surface
of the parabolic mirror, they all get reflected
to one point. So if you have a
ray coming in there, it's going to get
reflected there. If you have a ray
coming in like that, it's going to get
reflected like that. And so all of the energy can be
focused on a point like that. And so could imagine you might
have a water pipe running into the screen here. And so all of that
light energy would be used to heat up
that water pipe. So it's a pretty neat way
to concentrate energy. Another thing you
might want, maybe instead of taking
in energy, maybe you want to give out energy so
that all the beams of light are parallel. For example, let's say you
have a light for a car. If you have a light,
you could imagine if car headlights were just--
if I drew a car like this-- let me scroll down a little bit. If I drew a car like
this-- let me draw-- have a reasonable
attempt at a car. So let's say this is
a car right over here. I think you get the idea. This is the wheel housing. That's the wheel. So forth and so on. This isn't about the
drawing of the car. But you could imagine if
we just stuck light bulbs at the front of cars. So you could imagine
just a light bulb sitting at the front of a car. So that's a light bulb. And that would provide light
but it would provide light in all directions
radially outward. And it would be kind of useless. First of all, the
way I drew it here, it would probably show
up in the dude's eye who's trying to drive the car. But it's a lot of wasted energy. A lot of the light is
coming back onto the car. And it's pointing in all
sorts of random directions. It's not so useful. When you are driving
a car, you want all of the light pointed at the
road or maybe the stuff that's directly above the road. So how could you
point the light? Well, you could use
a parabolic mirror. And any car you
look at will have a light inside of
a parabolic mirror. And what does that do? Let's say instead
of this situation that I just drew--
let me clear this out. And I'll draw it
on a larger scale. Let's say I had a
parabolic mirror here. So I have a parabolic mirror. Obviously, this looks more like
a snow shovel or something. But I'm drawing it way huge just
so you get the general idea. So this is a parabolic mirror. And let's say we put the light
bulb now at the focal point. At the focus. At the focus of this
parabolic mirror. Now what's going to happen? Well, light that has to
go in this direction, that comes radially
outward, that's good. Because that's light that's
being useful to the driver. It's actually
illuminating the road. But light that's going
backwards-- light that's radiating outward from that
focus of the parabola-- it's going to do the exact
opposite of that solar energy collector. It's going to be
reflected out parallely. Or a parallel way. And so all of the
light-- because of this parabolic reflector,
or parabolic mirror-- all of the light that this
light source is generating, or most of it, is going
to be emitted parallel to the principal
axis of the parabola. And actually you
could point the light. If you actually moved
this parabola around, you can point which
direction the light's in. So it's actually a pretty
useful thing to have. Now the other thing
about parabolic mirrors is that they actually
form real images. In the last video, we
talked about the notion of a virtual image. You think something
is there because it looks like the light is
converging at some point. But that point isn't even there. It's actually from some other
point getting reflected. But a real image-- let
me draw it over here. So let me draw a
parabolic mirror. Let me draw big
parabolic mirrors to make the diagram clear. And let me draw
its principal axis. This is a side profile of it. Let me draw its principal
axis, just like that. And let's put an object. So I'm going to define a couple
of interesting points here. So first of all, we
have our focal point. I'll call that F. And then there's something
called the center of curvature. And the curvature I always
imagine as a sphere. But for the center of curvature
of a parabolic mirror, it's actually going to be
two times the focal length of this distance right here. Let me make it clear. I'll call that-- this
distance right here is F. Then this
distance right here, to the center of curvature,
we'll just call that point C. But this distance over here
is going to be F as well. Or it's going to be 2F
from-- you could imagine that vertex, or that minimum
point of the parabola, depending on how
you want to view it. Now, what I want to do is
put a couple of objects in front of this
parabolic mirror. And just think
about what happens to the light rays
of that object. So let's first put
an object here. So I'm just going to draw
the object as an arrow. And maybe some light
is shining on it from who knows what direction. But it's going to reflect
that light diffusely. Assuming it's not shiny. And I'm just going to
pick points on this object to radially emit
light outward from. Or reflect light outward from. And see what happens
to those light rays. And for the sake of simplicity,
whenever you do something with a parabolic mirror,
it's good to emit one radial ray that's parallel
and one that goes to the focus. Because we know what they're
going to do after that. So let's do one that's parallel. And of course,
these are just two of the gazillions
of light rays that are being emitted from
every point of this object. But we're just doing
this to understand what will the image of this
object actually look like. So let's do one parallel. It hits the surface of
the parabolic mirror. And then it reflects and
goes through the focus. We know that already. And then let's make another
light ray go through the focus. Let me draw it a little
bit better than that. Another light ray going
through the focal point. Just like that. And then it reflects. And it'll be reflected
in a parallel way. So what just happened here? Those two rays that were emitted
by the same point on this arrow object, they radially
emit outward. They reflect on this parabolic
mirror at two different points, but then they converge again. They converge right over there. And actually if you put-- and we
could do that with every point. If you did the stuff that
leaves that point-- actually both of those are going
to go and come back-- go through the focal point and
then come back right over here. They'll keep going. But you could
imagine, you could use with every point on this arrow. And what you're
going to do is get an image that looks like this. This point up here
corresponds to that point. This point corresponds
to that point. And so if you were to put
a screen right over here-- this is a screen. It could just be a, I don't
know, white tablecloth. Or if there was a
wall right over here. Then it would actually
show the image. You would actually be
projecting the image onto this wall right over here. It would actually be
a projected image. And that projected image
that we're talking about, where the light is converging--
so the light comes radially outward from each
point of this arrow. And then it converges on
a point on the screen. That image that gets formed,
we call that a real image. It's real image. This is a real image. And you might want
to compare that to what we call a virtual image. A virtual image is
an image that looks like it's coming from someplace. Because it looks like things
are diverging from some point. But they've really been
reflected off of some surface. So what we think is
there, really isn't there. A real image is an image
that's actually projectable. We could put a screen
right over here and then these guys are going
to be hitting the screen and essentially defusing
the exact same light as this point of
the actual object. And because of that, the screen
will look just like the object. This is a projectable image. Anyway, hopefully you
found that useful. I realize I've gone
longer than I like to with some of these videos. We'll talk a little bit
more about parabolic mirrors in the next video.