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# Thin lens equation and problem solving

## Video transcript

when you're dealing with these thin lenses you're gonna have to use this formula right here 1 over F equals 1 over D o plus 1 over D I not too bad except when are these positive or negative let's find out F is the focal length so the focal length you know 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 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 and in this case we've got a convex lens also known as a converging lens and it turns out for these types of lenses the focal length is always always going to be positive so if this focal length right here was say 8 centimeters we would plug in positive 8 centimeters doesn't matter we could have measured on this side the side will be 8 centimeters we still plug in positive 8 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 so 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 and 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 8 centimeters you'd have to plug in negative 8 centimeters up here into the focal length so all you need to look at is what type of lens you have do di doesn't matter do and di could be big small positive negative you could have a real image of 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 do-do represents the object distance so 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 do 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 so 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 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 gonna be a positive distance if you only have one lens okay so object distance is even easier always positive no matter what what the case is if you have a single lens how about image distance image distance is the tricky one so this refers to the distance from the lens to where the images but your image can be on one side or the other and so 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 well image distance is defined to be from the center of the lens to where my images always measured parallel to this principle axis sometimes people get confused they think well am I supposed to measure from the center here on this like diagonal line no you never do that you always go from the center parallel to the principal axis to where the images 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 so one way to remember it is image distance would 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 I should be looking through the lens at the object putting your eye over here does no good at all really I mean your lens is kind of pointless now if my eyes 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 want my eye over there and this 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'd 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 and a way to remember if the image distance is positive if this image distance has been brought closer to your eye and 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 of your eye that's when image distance will be positive if you ended up and that will be true regardless whether you've got concave convex converging diverging if the image is on the same side is 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 so again image distance from the lens center of the lens to where your images so I'm going to draw that line this would be my image distance in this case look at my eye still should be on this side my eyes on the 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's on the same side as 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 di I would know that that image is formed on the opposite side of the lens as my eye so those are the sign conventions for using this thin lens formula but notice something this formula is 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 and so we notice something we notice something important here look it if the image distance comes out negative we have magnification is negative of another negative number object distance always positive so we'd have a negative of a negative that give us a positive if our image distance comes out negative like it did down here well then we get a positive magnification and positive magnification means you've got a right-side up image if it's positive and if our image distance came out to be positive like on this side if we had a positive image distance well 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 okay so let's look at a few examples so say you got this example instead find the image distance and just gave you this diagram well we're going to use this thin lens formula so 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 negative 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 and 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 1 over negative 8 centimeters okay so 1 over negative 8 centimeters equals 1 over the object distance here we go object over here 24 centimeters away should I make it positive or negative I only got one lens here that means object distance is always going to be positive so that's 1 over positive 24 centimeters now we can solve for our image distance so 1 over di if I use algebra to solve here I'll have 1 over negative 8 centimeters minus 1 over 24 centimeters and note I can put this all in terms of centimeters I could 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 okay so if I solve this on the left hand side turns out you'll get negative 1 over 6 centimeters equals well that's not what di equals that's what 1 over di equals so don't forget at the very end you have to take 1 over both sides so take 1 over both sides my di turns out to be negative 6 centimeters what does that mean di of negative 6 centimeters well that means my image is going to be 6 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 so my eyes going to be over here if I'm using this lens right I've got my eye right here looking for the image and a negative image distance means it's going to be over on the left hand side where 6 means 6 centimeters and away from what everything is measured from the center of the lenz and so from here to there would be six centimeters so my this tells me on my principle axis my image is going to be right around here six centimeters away from the lens but it doesn't tell me nope this does not tell me how high the image is going to be how tall whether it's right side up I'll 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 so I'll come over here magnification is negative di over do well what was my di so negative of di was negative six so I'm gonna plug in negative six centimeters and 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 1/4 positive 1/4 and remember here positive magnification means right side up 1/4 means that my image is going to be 1/4 the size of my object so if my object were say 8 centimeters tall my image would only be 2 centimeters tall and so I'm going to draw I've got to draw an image here that's right side up right side up because I got a positive and it's got to be 1/4 as big as my object so let's see 1/4 might be around here so it's got to be right side up and about 1/4 is big I'd get a really little image it'd be right around there that's what I would see when I look through this lens so that's an example of using the thin lens equation and the magnification equation