# 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.

Sal, please put David's videos into the main list with physics and math videos, especially those about waves. These are the best lessons I've ever heard.

Yes, i think sal must add these videos in the playlist, it would be very convinient

Best Khan instructor I've watched, thank you for the help.

How did you get from 1/-8cm – 1/24 to -1/6cm ?

love you

Why some other videos concace is positive and convex is negative. So which is the correct

This is remarkably articulated. Thank you.

Can someone answer this plz.. so, if the distance of image is positive= the image is inverted and real..?

If the distance of image is negative= the image is not inverted and virtual…?

can you explain why the image is not inverted? im really confused @[email protected]

cofusing video

nicely present. thanks

This all makes so much sense.

BRAVO!!!!!!!

Does it matter where your eye is located? Does the image stay in focus whether your eye is 10cm away, 50cm away etc?

I'm trying to determine if there is a good position to place your eye so that the object is in focus or if focus matters? Referring to this product: http://www.shapeways.com/product/67MLBFUU4/vr-one?li=search-results-1&optionId=57570022

AUEEESOMMMMEEEEEEE!

Love this guy

THANK YOU! I am curious though. How do you know if an image is real or virtual?

ayy, that's a pretty confusing video you got yourself there

Thank you so much for this explanation!

I usually don't have any problems with physics, but this issue was just too confusing. I have a book that mentions how to determine the signs, but it just made matters worse. I'd take my semi-understanding and attempt to answer an equation with it, and I just keep getting it wrong. This video makes it so amazingly simple. Again, thank you so much, you've been a great help.

im not even studdying for the mcat but this summed up 2 weeks of lecture in 10 minutes you are a fucking boss thanks

You save so many lives everyday!

5:24 its SUPERMAN's EYE

isnt the image of concave lens always on the same side as the object? meaning the image distance will be negative??????

thank fuck for u

Soooo how do you figure out the heights of the object and image……?

would b better if u could explain sign convention rules der like d graphs…stuff formed on d right side is +ve and stuff on d left side -ve and so on…

Amazing! Thank you!

Where is Sal, dammit?

The app you uses please it will help me make exersices and solve things

we have been taught that object distance should always be negative?

khan academy is doing god's work

upload more and more lectures on every small or big topic☺ your doing greaaat work🙋🙋

I think you made a mistake regarding the sign of magnification and it indicating inverted/non-inverted image.

I was so lost, thanks for the help!

on one of the practice problems amcas gave the equation /0 + 1/i = s and asked to solve for the strength of the person's eye. Looking back it was probably a plug and chug especially since they gave the equation but it was the first time I have seen this equation. I'm only familiar with the solving for focal length and magnification. Are there any videos on lens strength anywhere?

Man talking so fast thinks a gun is at his head

i like this guy more than the other one. he just seems less boring but helpful none the less

how to get the size of object?

how did you get 1/6th? -8-24 =32? right? or are we subtracting fractions? someone please explain!

youre the hero i needed since last week

excellent

how one man can so clearly explain in 12 minutes what my professor fails to explain in 2 hours i will never understand

Can anyone help me with this ? The image of an object formed on the screen by a convex lens has height a. By moving the lens towards the screen , it is found that there is a second lens position at which another image of height b is formed on the screen. Prove that the height of the object is (ab)^0.5 .

Crazy shit being taught in Grade 10 nowadays

anyone with exams tomorrow trying to study on youtube

thanks that helped

please add a option in this app to ask questions

whether mirror and lens equations are same? for lens it should be (1/f = 1/v – 1/u) right?

You've literally saved my ass, it all makes sense now

this is actually wow

Isn't the sign of the focal lengths opposite? + for concave and – for convex

for a concave mirror: f = +0.5R

for a convex mirror: f = -0.5 R

you need a pay raise

These comments are so helpful

What about calculating strength of lens?

I'm eight and what I am doing here

thanks

Good!!

we are taught that distance in front of the lens is negative and behind the lens is positive

How did this guy get 1/-6 from the equation like what did he do???

I was so depressed not understanding this for weeks and you just solved it out in 10 mins…

lmao you mest up the answer for di is -12

this is on my exam tomorrow and my teacher didn’t even teach this 😭 gr 10 btw

how did you go from -8cm-24cm equals 6 😭

Watching these the day before the exam like a crazed hobo frantically searching for cigarette butts on the street to get his kick

Thanks you for this video it help me very much 😍

Great examples!

U r so clutch

What about when object distance is less than the focal length for a convex lens? Doesn't that make S' negative, but it's on the opposite side of the lens than the object?

Mm our equations are different,

Hi -Di

— = —

Ho Do

Those lines are straighter than the lines on the lgbt flag

Thank u 👍

when i saw 6 instead of 12…..you had me in the first half not gone lie

This is the absolute BEST video I've watched for thin lens, thank you so much

am I the only one who's not a student here and is studying for Cock and ball torture society instead?

I’m so gonna fail

Learning this in grade 7, some really crazy shit

Thought you had to convert to meters at the end when using magnification equation?

THANK YOU.

I never quite got around to learning this. Now I'm studying for the PGRE and optics is like my weakest area.

Which softeare do you use sir !

Fuck my life. My Physics teacher asked me to watch your video due to the class suspenion. Well , your viedo is so long and made me suffer a lot. Life is always hard… LMAO>>>

This is my University.