Parabolic mirrors 2 Video Lecture - JEE

Let's draw a bunch
of parabolic mirrors.
And what I want to
do in this video is,
do a bunch of
examples of objects
in front of parabolic mirrors.
And think about what the
images of those objects
will be based on how
far those objects are.
And besides just giving
us a better understanding
of parabolic mirrors, this will
also, it 'll hopefully also
give us a sense of how do
we manipulate or how can we
conceptualize these light rays?
Which will be a
pretty useful tool
when we tackle other types
of reflective or refractive
devices like lenses.
So that's a parabolic mirror.
I've drawn its principal
axis right over here.
And let me just copy and paste
this just so that I-- actually,
let me draw the
focal point, too.
So this is the focus
right over here.
So let me just draw the
focus right over there.
And this is the
center of curvature.
It's twice the distance from
this point as the focus.
So that is the-- let me make
it as close as possible.
So that is the center of
curvature right over there.
And let me copy and paste this.
So we can reuse this
later on in the video.
I don't have to keep drawing it.
So copy it.
So I've copied that.
Now, let's put an
object-- and I think
we did this in the last video.
Let's put an object beyond
the center of curvature.
So let's put an object here.
And the convention is to use
an upward pointing arrow.
This isn't a light ray.
It's used to show an object.
And we use the tip of
the arrow to really show
the top of the object.
And that's usually where we
trace our light rays from.
But it doesn't have to be there.
We could do the middle.
You could do the bottom.
And you could figure out what
the image of the object's
going to be.
So let's do that.
And when you're dealing
with parabolic mirrors,
it's easiest to have to
show just two light rays.
One that goes parallel
to the principal axis.
And one that goes
through the focus.
Because you know what's going
to happen to each of those
when they reflect.
And you don't have
to do any math there.
So let's have the parallel ray.
That's the parallel ray.
When it reflects-- the parallel
incident ray, when it reflects,
it will go through the focus.
And then, let's
have an incident ray
that goes through the focus.
And when it reflects,
it will be parallel.
And this is the example we
actually saw in the last video.
And so whatever light
is being emitted
from this point over
here in that direction,
it will come back and
converge again at this point.
And we could actually do it
at every point along the--
we could do it halfway.
This halfway point of the
arrow, just to make it clear.
This halfway point of
the arrow, same thing.
You have a parallel.
Something parallel will go,
will reflect a parallel incident
ray will reflect
through the focus.
And I'll just do it for
this one right over here.
It will reflect
through the focus.
And then, if you
have something that
goes through the focus,
an incident ray that
goes through a focus, it
will reflect parallel.
So this point will correspond
to this point over here.
I think that makes it clearer,
that this image, the image
of this object
when it's reflected
by this parabolic mirror
will look just like that.
So it will actually
form a real image.
It'll form a real
image that is smaller
than this original image.
It's not so clear, the
way I did it over here.
But you could push this
even further back out.
And it'll be clear
that this is going
to be a smaller real image
than that right over there.
Now, let's do a
couple more examples.
So let me just paste my drawing.
So I don't have to redraw it.
Let's see what happens.
Let me write it.
So here the image, just so
we can keep track of things.
Here, the image is real and
smaller than the actual object
when the actual object is
beyond the center of curvature.
And actually, let me make it a
little bit clearer by drawing.
Let me do another example
like that where I do something
big way out here just
to make it clear.
So once again, we go parallel,
reflect through the focus.
And then, we can go
through the focus.
We go through the focus and
then reflect out like that.
And there you see,
now, it's much clearer
that the image is going
to be much smaller.
And, of course, inverted
relative to the actual object.
Now, let's do this again.
But this time, let's
place the object
at the center of curvature.
So let's place the
object right over here.
So right at that distance
that twice the distance
from the vertex of the
parabola to the focus.
So we do a parallel line.
These lines are the
hardest thing to do.
So parallel incident ray,
parallel to the principal axis.
This is the principal
axis right here.
Principal axis-- that's what
this line is right over here.
It's kind of the line of
symmetry of the parabola.
When it reflects, it will
reflect through the focus.
And then, let's take
another ray that goes,
the incident ray goes
through the focus.
And when it reflects, it
will reflect parallel.
And my drawing isn't the
neatest drawing on the planet.
And actually, let me draw it
a little bit better than that.
Well, that's pretty good.
Let me just.
So that's the incident
ray that was parallel.
And then, an incident
ray that goes through--
let me-- I'm having
trouble drawing these.
An incident ray that
goes through the focus
will then come out and
reflect right over there.
And they'll converge.
And the way I've drawn it,
my drawing isn't ideal.
But the way, but
the reality is, is
that they'll converge so
that the image will just
be in an inverted same size
version as this thing up here.
Because it's symmetric.
Let me see if I can
redraw this whole thing,
so it comes out neater.
So far, that looks good.
So then, you want to
reflect like that.
You got-- you're coming
through the focus.
And then, you have another ray
that goes through the focus.
This whole thing
should be symmetric.
And then, when it reflects,
it comes back out like that.
So that makes a
little bit clearer.
So this is the object.
And now, its image is
just an inverted version
of this object.
The image comes into
focus, or the rays
converge at the same distance
from the actual mirror
as the actual object.
And it's going to be the
same size, just inverted.
So that's-- so here the image
is real and the same size
as the object.
Let's do a couple more of these.
I think you get the hang of it.
And you might want
to try them out.
You might want to
pause the video
and try it on paper because
really nothing beats practice.
So let's stick our object
between the center of curvature
and the focus and
the focal point.
So if we put our
object there, we
could have a light ray that goes
parallel to the principal axis.
And then, it will reflect
out through the focus.
And then, you could
have another point.
It goes through another ray.
It goes through the focus.
And then, it reflects.
And then, it will reflect out.
And let me draw it
better than that.
Actually, this is--
I should probably
have used a more precise
tool when I did all of this.
So let me draw it
right over here.
So I have the parallel one.
And then, it goes
through the focus
just like that when
it gets reflected.
I really should have
had a line tool for this
to have neater drawings.
And then, a ray that
goes through the focus
will be reflected out parallel.
It would be reflected
out parallel.
And at least for the light
that comes from that tip,
they will re-converge
at that tip.
And if you did it for
every point on this arrow,
the image would be
an inverted arrow
that is bigger
than the original.
And it is beyond.
It's almost the opposite of the
first example that we showed.
And so now the image is
bigger than the original.
So the image is real.
And it is bigger.
And the image will,
where it converges
is going to be beyond
the center of curvature.
And you could imagine.
If this was the
object right here,
then this would be
the image over here.
If you just trace
the lines backwards.
So there's a symmetry
here between this example
and the first one
we did up here.
Now, let's just
do a couple more.
Let's imagine if the object is
actually at the focal point.
It's actually at the focus.
So let's draw an object there.
Let's think about
what would happen.
So if we're at the focus, a
ray that comes out parallel
will go through the focal point
and come out just like that.
And then, here we're going to--
you can't-- we can't have a ray
that just goes.
Well, actually we could have
a ray that goes through.
Well, that-- you can't
go into the object.
So here, I'll do a
slightly different ray.
I'll do a ray that intersects
the parabolic mirror
right over there.
And the reason why
I want to do there
is because there
the parabolic mirror
is essentially flat and
essentially vertical.
So you can imagine
that the incident
ray is going to be the same
thing as the reflected ray.
So you could draw a ray
that comes in like that.
So this is a departure
from what we did before.
And then the reflected ray
will come out like that.
So what happens
when an object is
at the focal point is that all
of the light that's coming off
of this object in any
direction will all be made,
it will all be made parallel.
And so it won't converge.
So it won't converge.
So it won't be able
to form a real image.
And it's not, it
doesn't look like it's
diverging from some
point in the mirror.
So it won't even
form a virtual image.
So here, there will
actually be no image
when the object is actually
at the focal point.
And then, the last case,
as you can imagine,
is if an object is closer
than the focal point.
So let's draw that.
So let's put an object
at the focal point.
So right over here.
And here, just for the sake
of argument, one I can draw.
I can always draw a parallel.
And anything that, any
light that goes parallel
will then look to come
out in a direction that
would go through
the focal point.
So it would go through
the focal point.
It would come out
in that direction.
Although the object
itself is blocking.
But it would look to
go in that direction.
It would be reflected
in that direction.
And then, you could
imagine a ray of light
that would have been coming
from the focal point,
or would have been coming
from the same direction.
So you could imagine coming
from the same direction
as the focal point
would be reflected
parallel, in a
parallel direction
to the principal axis.
Now, these two light
rays are not converging.
But they look like
they're diverging
from some point
behind the mirror.
They'll look like
they're diverging
from some point
behind the mirror.
So in this case, we are
forming a virtual image.
And the virtual
image will actually
look something like this.
And so it'll be larger than
the original virtual image.
So it's kind of a magnifying.
If you were to go
to-- what are called?
--a fun house at the circus or
the amusement park, whatever.
And if you were to get close
enough to parabolas mirror,
it would show a
magnified version of you.
A virtual version of you.
And actually, let me draw
that a little bit bigger
just because it
might not be clear.
So let me draw.
So if this is the mirror,
that is the focal point.
This right here is
the principal axis.
This is maybe you,
maybe whatever object.
You could draw a ray
that goes parallel.
It will reflect in the
direction of the focal point.
So it'll reflect out like that.
It'll be blocked by
the object, though.
And then, something
that looks like it
would have come from
the focal point,
from the same direction
as the focal point
would then be reflected
parallel to the principal axis.
So these two rays, once
again, there are diverging.
But they look like, to the
human brain, to the human eye,
they look like they came
from that point over there.
And so this would
correspond to that point
on the virtual image.
So hopefully, that
gives you some practice.
But the most important
thing, it gives you
some practice dealing
with these arbitrary rays
that we're showing emanating
from the tip of this arrow.
And we could do it
for the whole arrow.
But the reason why we're picking
these rays in these directions
is that they're
easy to work with.
They go through the focus,
they'll come out parallel.
If the incident ray
is parallel, it'll
come out through the focus.