# Thin lens equation and problem solving | Geometric optics | Physics | Khan Academy

– [Voiceover] When you’re
dealing with these thin lenses, you’re going to have to use
this formula right here, one over f equals one over
d-o plus one over d-i. Not too bad, except when are
these positive or negative? Let’s find out. F is the focal length. The focal length, when
you’ve got a thin lens, there’s a focal point on
each side of the lens. The focal length is the distance from the center of the lens to
one of these focal points. Which one, it’s doesn’t actually matter, because if you want to know
whether the focal length is positive or negative,
all you have to look at is what type of lens you have. In this case, we’ve got a convex lens, also known as a converging lens. It turns out, for these types of lenses, the focal length is always,
always, going to be positive. If this focal length right here
was, say, eight centimeters, we would plug in positive
eight centimeters. It doesn’t matter, we could
have measured on this side. This side will be eight centimeters. We still plug in positive
eight centimeters into this focal length if it is a converging, or a convex, lens. If you had the other type of lens … Here’s the other kind. This one is either diverging or it’s going to be concave. If you have a concave or diverging lens, it also will have two focal points typically drawn on either side. These will be a certain distance along that principal axis to
the center of the lens. If you measured this, by
definition for a concave or a diverging lens, the
focal length is always going to be a negative focal length. So, if this distance here
was eight centimeters, you’d have to plug in
negative eight centimeters up here into the focal length. All you need to look at is
what type of lens you have. D-o, d-i, doesn’t matter. D-o and d-i could be big, small, positive, negative. You could have a real
image, a virtual image. It doesn’t matter. All you have to look at is
what type of lens you have. That will tell you
whether you should plug in a positive focal length,
or a negative focal length. All right, so focal length isn’t too bad. How about d-o? D-o represents the object distance. If I had an object over here, and we always draw objects as arrows. That lets us know whether
they’re right-side up or upside down. Here’s my object. The object distance refers to the distance from, always measured from
the center of the lens to where the thing is, and in this case the thing is the object, so here’s my d-o. This object distance … this one’s even easier … object distance, just always positive. So my object distance, I’m just always going to make that positive. If this is 30 centimeters, I’m plugging in positive 30 centimeters over there. If it’s 40 centimeters,
positive 40 centimeters. Always going to be positive unless … there is one exception. If you had multiple lenses it’s possible you might have to deal with
a negative object distance, but, if you’re dealing with a single lens, whether it’s concave or convex, I don’t care what kind of lens it is, if it’s a single lens,
your object distance is going to be a positive distance if you only have one lens. Okay, so object distance is even easier. Always positive, no
matter what the case is, if you have a single lens. How about image distance? Image distance is the tricky one. This refers to the distance from the lens to where the image is, but your image can be on
one side or the other. Let’s see here, let’s say
for this case over here I ended up with an image
upside down over here, something like this. Say this is my image that
was formed by this object in this converging, convex, lens. Image distance is defined
to be from the center of the lens to where my image is, always measured parallel
to this principal axis. Sometimes people get confused. They think, well, am I supposed to measure from the center here
on this diagonal line? No, you never do that! You always go from the center, parallel to the principal axis, to where the image is. This is defined to be the image distance. When will this be positive and negative? Here’s the tricky one, so be careful. Image distance will be
positive if the image distance is on this other side of
the lens than the object. One way to remember it is image distance will be positive if it’s
on the opposite side of the lens as the object, or, the way I like to remember it, if you’re using this lens
right, you should be looking, your eye should be looking
through the lens at the object. Putting your eye over
here does no good at all. Really, your lens is
kind of pointless now. If my eye’s over here,
I’m looking at my object, and I’m just holding
a lens in front of it. This is really doing no good. So I don’t want my eye over there. If I’m using this lens right, my eye would be over on this side, and I’d be looking at this object, I’d be looking through. I’m not shooting light
rays out of my eyes, but I’m looking in this direction through the lens at my object. I wouldn’t see the object. What I would actually see
is an image of the object, I’d see this image right here, but still, I’m trying to
look through the lens. A way to remember if the
image distance is positive, if this image distance
has been brought closer to your eye than the object was, if it’s on the side of this
lens that your eye is on, that will be a positive image distance. So if it’s on this, in
this case, the right side, but what’s important is it’s on the opposite side of the object, and the same side as your eye, that’s when image
distance will be positive. That’ll be true regardless, whether you’ve got a concave,
convex, converging, diverging. If the image is on the same
side as your eye over here, then it should be a
positive image distance. Now, for this diverging case, maybe the image ended
up over here somewhere. I’m going to draw an image over here. Again, image distance from the lens, center of the lens, to
where your image is, so I’m going to draw that line. This would be my image distance. In this case, my eye still
should be on this side. My eye’s on this side because I should be looking through my lens at my object. I’m looking through
the lens at the object. I’d see this image
because this image is on the opposite side of the lens as my eye, or, another way to think about it, it’s on the same side of the object. This would be a negative image distance. I’d have to plug in a negative number, or if I got a negative
number out of this formula for d-i, I would know that that image is formed on the opposite
side of the lens as my eye. Those are the sign conventions for using this thin lens formula. But notice something. This formula’s only giving you
these horizontal distances. It tells you nothing about
how tall the image should be, or how tall the object is. It only tells you these
horizontal distances. To know about the height, you’d have to use a different formula. That other formula was
this magnification formula. It said the magnification, M, equals negative the image distance. If you took the image distance and then divided by the object distance you’d get the magnification. So we notice something. We notice something important here. If the image distance comes out negative, we’d have magnification as negative of another negative number, object distance always positive, so we’d have a negative of a negative, that would give us a positive. If our image distance comes out negative like it did down here, then we’d get a positive magnification
and positive magnification means you’ve got a right-side
up image, if it’s positive. If our image distance
came out to be positive, like on this side, if we had
a positive image distance, we’d have a negative of a positive number, that would give us a
negative magnification. That means it’s upside down. So it’s important to note
if our image distance comes out negative,
negative image distance means not inverted, and
positive image distance means that it is inverted from
whatever it was originally. Let’s look at a few examples. Say you got this example. It said find the image distance, and it just gave you this diagram. We’re going to have to use
this thin lens formula. We’ll have to figure out what
f is, f, the focal length. We’ve got these two focal lengths, here, eight centimeters on both sides. Should I make it a
positive eight centimeters or a positive eight centimeters? Remember, the rule is
that you just look at what type of lens you have. In this case, I have a concave lens, or another way of saying
that is a diverging lens. Because I have that type
of lens it doesn’t matter. I don’t have to look at anything else. I automatically know my focal length is going to be one over
negative eight centimeters. One over negative eight centimeters equals one over the object distance, here we go, object over here, 24 centimeters away. Should I make it positive or negative? I’ve only got one lens here. That means object distance is
always going to be positive. So that’s one over
positive 24 centimeters. Now we can solve for our image distance. One over d-i. If I use algebra to solve here I’ll have one over negative eight centimeters minus one over 24 centimeters, and note, I can put this
all in terms of centimeters, I can put it all in terms of meters. It doesn’t matter what units I use here. Those are the units I’ll get out. I just have to make sure I’m consistent. So if I solve this on the left-hand side, turns out you’ll get negative
one over six centimeters equals, well, that’s not what d-i equals. That’s what one over d-i equals, so don’t forget at the very end you have to take one over both sides. If you take one over both sides, my d-i turns out to be
negative six centimeters. What does that mean? D-i of negative six centimeters. That means my image is
going to be six centimeters away from the lens, and the negative means it’s going to be on the
opposite side as my eye or the same side as my object. My eye’s going to be over here. If I’m using this lens right,
I’ve got my eye right here looking for the image. The negative image distance means it’s going to be over on
the left-hand side, where? Six means six centimeters
and away from what? Everything’s measured from
the center of the lens, and so from here to there
would be six centimeters. This tells me on my principal axis, my image is going to be right around here, six centimeters away from the lens, but it doesn’t tell me, note, this does not tell me how high the image is going to be, how tall, whether it’s right-side up … Actually, hold on. It does tell us whether
it’s right-side up. This came out to be negative. Remember our rule? Negative image distances means
it’s got to be right-side up. I’m going to have a right-side up image, but I don’t know how tall yet. I’m going to have to use
the magnification equation to figure that out. I’ll come over here. Magnification is negative d-i over d-o. What was my d-i? Negative of d-i was negative six, so I’m going to plug in
negative six centimeters. On the bottom, I’m going to plug in, let’s see, it was 24 centimeters
was my object distance. What does that give me? Negative cancels the
negative, I get positive, and I get positive one-fourth. Positive one-fourth. Remember, here, positive
magnification means right-side up. One-fourth means that
my image is going to be a fourth the size of my object. If my object were, say,
eight centimeters tall, my image would only be
two centimeters tall. I’m going to draw an image
here that’s right-side up, right-side up because I got a positive, and it’s got to be a
fourth as big as my object, so let’s see, one-fourth
might be around here, so it’s got to be right-side up and about a fourth as big. I’d get a really little image. It’d be right around there. That’s what I would see when
I looked through this lens. That’s an example of using
the thin lens equation and the magnification equation.

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