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- [Voiceover] When you're
dealing with these thin lenses,

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you're going to have to use
this formula right here,

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one over f equals one over
d-o plus one over d-i.

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Not too bad, except when are
these positive or negative?

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Let's find out.

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F is the focal length.

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The focal length, when
you've got a thin lens,

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there's a focal point on
each side of the lens.

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The focal length is the distance from

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the center of the lens to
one of these focal points.

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Which one, it's doesn't actually matter,

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because if you want to know
whether the focal length

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is positive or negative,
all you have to look at

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is what type of lens you have.

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In this case, we've got a convex lens,

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also known as a converging lens.

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It turns out, for these types of lenses,

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the focal length is always,
always, going to be positive.

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If this focal length right here
was, say, eight centimeters,

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we would plug in positive
eight centimeters.

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It doesn't matter, we could
have measured on this side.

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This side will be eight centimeters.

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We still plug in positive
eight centimeters

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into this focal length

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if it is a converging, or a convex, lens.

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If you had the other type of lens ...

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Here's the other kind.

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This one is either diverging

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or it's going to be concave.

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If you have a concave or diverging lens,

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it also will have two focal points

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typically drawn on either side.

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These will be a certain distance along

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that principal axis to
the center of the lens.

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If you measured this, by
definition for a concave

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or a diverging lens, the
focal length is always

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going to be a negative focal length.

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So, if this distance here
was eight centimeters,

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you'd have to plug in
negative eight centimeters

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up here into the focal length.

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All you need to look at is
what type of lens you have.

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D-o, d-i, doesn't matter.

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D-o and d-i could be big, small,

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positive, negative.

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You could have a real
image, a virtual image.

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It doesn't matter.

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All you have to look at is
what type of lens you have.

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That will tell you
whether you should plug in

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a positive focal length,
or a negative focal length.

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All right, so focal length isn't too bad.

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How about d-o?

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D-o represents the object distance.

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If I had an object over here,

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and we always draw objects as arrows.

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That lets us know whether
they're right-side up

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or upside down.

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Here's my object.

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The object distance refers to the distance

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from, always measured from
the center of the lens

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to where the thing is,

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and in this case the thing is the object,

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so here's my d-o.

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This object distance ...

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this one's even easier ...

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object distance, just always positive.

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So my object distance, I'm just always

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going to make that positive.

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If this is 30 centimeters, I'm plugging in

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positive 30 centimeters over there.

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If it's 40 centimeters,
positive 40 centimeters.

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Always going to be positive unless ...

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there is one exception.

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If you had multiple lenses it's possible

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you might have to deal with
a negative object distance,

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but, if you're dealing with a single lens,

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whether it's concave or convex,

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I don't care what kind of lens it is,

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if it's a single lens,
your object distance

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is going to be a positive distance

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if you only have one lens.

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Okay, so object distance is even easier.

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Always positive, no
matter what the case is,

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if you have a single lens.

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How about image distance?

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Image distance is the tricky one.

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This refers to the distance from the lens

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to where the image is,

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but your image can be on
one side or the other.

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Let's see here, let's say
for this case over here

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I ended up with an image
upside down over here,

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something like this.

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Say this is my image that
was formed by this object

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in this converging, convex, lens.

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Image distance is defined
to be from the center

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of the lens to where my image is,

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always measured parallel
to this principal axis.

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Sometimes people get confused.

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They think, well, am I supposed to measure

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from the center here
on this diagonal line?

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No, you never do that!

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You always go from the center,

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parallel to the principal axis,

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to where the image is.

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This is defined to be the image distance.

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When will this be positive and negative?

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Here's the tricky one, so be careful.

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Image distance will be
positive if the image distance

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is on this other side of
the lens than the object.

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One way to remember it is image distance

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will be positive if it's
on the opposite side

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of the lens as the object,

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or, the way I like to remember it,

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if you're using this lens
right, you should be looking,

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your eye should be looking
through the lens at the object.

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Putting your eye over
here does no good at all.

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Really, your lens is
kind of pointless now.

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If my eye's over here,
I'm looking at my object,

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and I'm just holding
a lens in front of it.

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This is really doing no good.

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So I don't want my eye over there.

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If I'm using this lens right,

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my eye would be over on this side,

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and I'd be looking at this object,

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I'd be looking through.

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I'm not shooting light
rays out of my eyes,

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but I'm looking in this direction

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through the lens at my object.

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I wouldn't see the object.

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What I would actually see
is an image of the object,

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I'd see this image right here,

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but still, I'm trying to
look through the lens.

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A way to remember if the
image distance is positive,

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if this image distance
has been brought closer

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to your eye than the object was,

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if it's on the side of this
lens that your eye is on,

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that will be a positive image distance.

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So if it's on this, in
this case, the right side,

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but what's important is it's on

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the opposite side of the object,

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and the same side as your eye,

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that's when image
distance will be positive.

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That'll be true regardless,

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whether you've got a concave,
convex, converging, diverging.

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If the image is on the same
side as your eye over here,

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then it should be a
positive image distance.

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Now, for this diverging case,

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maybe the image ended
up over here somewhere.

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I'm going to draw an image over here.

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Again, image distance from the lens,

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center of the lens, to
where your image is,

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so I'm going to draw that line.

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This would be my image distance.

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In this case, my eye still
should be on this side.

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My eye's on this side because I should be

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looking through my lens at my object.

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I'm looking through
the lens at the object.

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I'd see this image
because this image is on

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the opposite side of the lens as my eye,

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or, another way to think about it,

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it's on the same side of the object.

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This would be a negative image distance.

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I'd have to plug in a negative number,

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or if I got a negative
number out of this formula

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for d-i, I would know that that image

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is formed on the opposite
side of the lens as my eye.

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Those are the sign conventions for using

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this thin lens formula.

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But notice something.

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This formula's only giving you
these horizontal distances.

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It tells you nothing about
how tall the image should be,

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or how tall the object is.

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It only tells you these
horizontal distances.

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To know about the height,

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you'd have to use a different formula.

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That other formula was
this magnification formula.

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It said the magnification, M, equals

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negative the image distance.

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If you took the image distance

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and then divided by the object distance

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you'd get the magnification.

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So we notice something.

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We notice something important here.

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If the image distance comes out negative,

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we'd have magnification as negative

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of another negative number,

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object distance always positive,

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so we'd have a negative of a negative,

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that would give us a positive.

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If our image distance comes out negative

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like it did down here, then we'd get

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a positive magnification
and positive magnification

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means you've got a right-side
up image, if it's positive.

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If our image distance
came out to be positive,

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like on this side, if we had
a positive image distance,

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we'd have a negative of a positive number,

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that would give us a
negative magnification.

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That means it's upside down.

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So it's important to note
if our image distance

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comes out negative,
negative image distance

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means not inverted, and
positive image distance

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means that it is inverted from
whatever it was originally.

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Let's look at a few examples.

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Say you got this example.

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It said find the image distance,

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and it just gave you this diagram.

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We're going to have to use
this thin lens formula.

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We'll have to figure out what
f is, f, the focal length.

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00:09:01,484 --> 00:09:03,415
We've got these two focal lengths, here,

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eight centimeters on both sides.

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Should I make it a
positive eight centimeters

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00:09:06,929 --> 00:09:08,595
or a positive eight centimeters?

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00:09:08,595 --> 00:09:11,411
Remember, the rule is
that you just look at

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00:09:11,411 --> 00:09:12,867
what type of lens you have.

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In this case, I have a concave lens,

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or another way of saying
that is a diverging lens.

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Because I have that type
of lens it doesn't matter.

217
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I don't have to look at anything else.

218
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I automatically know my focal length

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is going to be one over
negative eight centimeters.

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One over negative eight centimeters equals

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one over the object distance, here we go,

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object over here, 24 centimeters away.

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Should I make it positive or negative?

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I've only got one lens here.

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That means object distance is
always going to be positive.

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00:09:45,649 --> 00:09:49,336
So that's one over
positive 24 centimeters.

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00:09:49,336 --> 00:09:51,768
Now we can solve for our image distance.

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One over d-i.

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If I use algebra to solve here I'll have

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one over negative eight centimeters

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minus one over 24 centimeters,

232
00:10:02,904 --> 00:10:06,072
and note, I can put this
all in terms of centimeters,

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I can put it all in terms of meters.

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00:10:07,677 --> 00:10:09,706
It doesn't matter what units I use here.

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Those are the units I'll get out.

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I just have to make sure I'm consistent.

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So if I solve this on the left-hand side,

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turns out you'll get negative
one over six centimeters

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equals, well, that's not what d-i equals.

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That's what one over d-i equals,

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so don't forget at the very end

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you have to take one over both sides.

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00:10:29,194 --> 00:10:30,951
If you take one over both sides,

244
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my d-i turns out to be
negative six centimeters.

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00:10:34,914 --> 00:10:36,486
What does that mean?

246
00:10:36,486 --> 00:10:38,616
D-i of negative six centimeters.

247
00:10:39,431 --> 00:10:40,845
That means my image is
going to be six centimeters

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00:10:40,845 --> 00:10:44,131
away from the lens, and the negative means

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it's going to be on the
opposite side as my eye

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00:10:46,875 --> 00:10:48,741
or the same side as my object.

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My eye's going to be over here.

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If I'm using this lens right,
I've got my eye right here

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looking for the image.

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The negative image distance means

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it's going to be over on
the left-hand side, where?

256
00:11:00,771 --> 00:11:03,361
Six means six centimeters
and away from what?

257
00:11:03,361 --> 00:11:07,817
Everything's measured from
the center of the lens,

258
00:11:07,817 --> 00:11:12,215
and so from here to there
would be six centimeters.

259
00:11:13,276 --> 00:11:14,831
This tells me on my principal axis,

260
00:11:14,831 --> 00:11:17,118
my image is going to be right around here,

261
00:11:17,118 --> 00:11:18,760
six centimeters away from the lens,

262
00:11:18,760 --> 00:11:19,947
but it doesn't tell me,

263
00:11:19,947 --> 00:11:22,188
note, this does not tell me how high

264
00:11:22,188 --> 00:11:24,113
the image is going to be, how tall,

265
00:11:24,113 --> 00:11:25,539
whether it's right-side up ...

266
00:11:25,900 --> 00:11:27,474
Actually, hold on.

267
00:11:27,474 --> 00:11:29,244
It does tell us whether
it's right-side up.

268
00:11:29,244 --> 00:11:30,850
This came out to be negative.

269
00:11:30,850 --> 00:11:32,116
Remember our rule?

270
00:11:32,116 --> 00:11:35,717
Negative image distances means
it's got to be right-side up.

271
00:11:35,717 --> 00:11:37,549
I'm going to have a right-side up image,

272
00:11:37,549 --> 00:11:39,024
but I don't know how tall yet.

273
00:11:39,024 --> 00:11:40,955
I'm going to have to use
the magnification equation

274
00:11:40,955 --> 00:11:42,384
to figure that out.

275
00:11:42,384 --> 00:11:43,431
I'll come over here.

276
00:11:43,431 --> 00:11:48,389
Magnification is negative d-i over d-o.

277
00:11:49,235 --> 00:11:50,329
What was my d-i?

278
00:11:50,329 --> 00:11:53,714
Negative of d-i was negative six,

279
00:11:53,714 --> 00:11:56,251
so I'm going to plug in
negative six centimeters.

280
00:11:58,077 --> 00:12:01,024
On the bottom, I'm going to plug in,

281
00:12:01,024 --> 00:12:05,119
let's see, it was 24 centimeters
was my object distance.

282
00:12:05,119 --> 00:12:06,064
What does that give me?

283
00:12:06,064 --> 00:12:08,545
Negative cancels the
negative, I get positive,

284
00:12:08,545 --> 00:12:11,246
and I get positive one-fourth.

285
00:12:11,246 --> 00:12:12,693
Positive one-fourth.

286
00:12:12,693 --> 00:12:16,320
Remember, here, positive
magnification means right-side up.

287
00:12:16,320 --> 00:12:19,495
One-fourth means that
my image is going to be

288
00:12:19,495 --> 00:12:21,899
a fourth the size of my object.

289
00:12:21,899 --> 00:12:26,221
If my object were, say,
eight centimeters tall,

290
00:12:26,221 --> 00:12:29,477
my image would only be
two centimeters tall.

291
00:12:29,477 --> 00:12:33,053
I'm going to draw an image
here that's right-side up,

292
00:12:33,053 --> 00:12:35,222
right-side up because I got a positive,

293
00:12:35,222 --> 00:12:39,025
and it's got to be a
fourth as big as my object,

294
00:12:39,025 --> 00:12:41,730
so let's see, one-fourth
might be around here,

295
00:12:41,730 --> 00:12:43,325
so it's got to be right-side up

296
00:12:43,325 --> 00:12:45,388
and about a fourth as big.

297
00:12:45,388 --> 00:12:46,863
I'd get a really little image.

298
00:12:46,863 --> 00:12:48,266
It'd be right around there.

299
00:12:48,266 --> 00:12:50,994
That's what I would see when
I looked through this lens.

300
00:12:50,994 --> 00:12:53,528
That's an example of using
the thin lens equation

301
00:12:53,528 --> 00:00:00,000
and the magnification equation.