The Lens Maker's Formula

Transcript

0:00Introduction

Hello everyone! Here is a small puzzle. Take a convex glass lens, the kind that focuses sunlight to a bright spot, with a focal length of ten centimetres in air. Drop it into a special liquid. Same glass, same shape. And suddenly, it spreads light out, like a concave lens! So what changed? The lens maker's formula answers this, and it comes up often in JEE, so let's begin.

0:27Sign convention: follow the light

So, first the signs, because that is where most mistakes happen. Every distance is measured from the pole, the centre of the surface, marked P in the figure. Distances along the incident light are positive, the plus sign, and those against it are negative. Okay? In the figure, light goes from the object O, in a medium of index n one, into one of index n two; the small numbers below are labels. The u, the v, the image I and the formula on the right are from the previous lesson. C is the centre of curvature, the dashed line through it is the normal, and capital R, the radius, is the bottom arrow. So, is this radius positive or negative? Positive, yes, exactly, because C lies on the right, where the light is going! So R is positive when the centre of curvature lies on the side the light travels to; that is the greater than sign. The sign comes from the light, not the shape. Right? Now, a lens has two surfaces: R one, with a small one below, is the one the light meets first, and R two, with a small two below, is the second. An equiconvex lens has both faces bulging out, with the same radius, capital R, okay? With light from the left, what are the signs of R one and R two? The first face has its centre on the right, where the light goes, so R one is plus R, positive. The second face bulges the other way, its centre is on the left, so R two is minus R, negative. Same shape, opposite signs! Clear?

2:02The lens maker's formula

Now the lens maker's formula. Here f is the focal length, n L, with a small L below, is the refractive index of the lens, and n m, with a small m below, that of the medium. One over f equals n L over n m, minus one, times one over R one, minus one over R two. It needs a thin lens, the same medium on both sides, and rays close to the axis. With me? Usually the lens is in air, so n m is one, and n is the index of the glass. Then one over f equals n minus one, times the same bracket. Okay? Now put in our equiconvex signs, plus R and then minus R, as we just found. What do you get for f? One over R, minus one over minus R, gives two over R, okay? So f equals R over two times the bracket, n minus one. Right? Now a plano-convex lens: the flat face has an infinite radius, so its term is zero, and f equals R over the bracket, n minus one, which is twice the equiconvex value. Clear?

3:10Power, and cutting a lens

Next, the power P of a lens is one over f, with f in metres; its unit is the dioptre, the capital D in brackets. Okay? In the figure, the convex lens bends a ray parallel to the axis through the focus F, so it converges, and its power is positive. The concave lens spreads that ray out, as if from F along the dashed line, so its power is negative, the less than sign. The other ray, the arrows, the points marked two F and the captions are for the next lesson. Right? Now a common JEE trap. The axis is the horizontal line through the lens. Cut an equiconvex lens of focal length f perpendicular to the axis, into two plano-convex pieces. What is the focal length of each? Many students say f over two, but no, no, wait! Each piece keeps only one curved face, so it has half the power, and its focal length is two f, like our plano-convex result. With me? And if you cut along the axis, into upper and lower halves? Each half still has both curved faces, so its focal length stays f. It just collects half the light. Got it?

4:19The same lens in a liquid

Now back to our puzzle, a lens in a liquid. The radii do not change, so when we take the formula in air over the one in the liquid, the second bracket cancels. So f m over f air, the small labels meaning medium and air, equals n L minus one, over n L over n m, minus one; the slash also means over. Okay? If the liquid has the same index as the glass, the bottom is zero, and the focal length is infinite. So the lens has no effect at all. Right? Now tell me this. If the liquid has a higher refractive index than the glass, what happens to the sign of f m? Then n L over n m is less than one, the bottom is negative, and the focal length flips sign! So a convex lens diverges, and a concave lens converges. Clear?

5:07Example 1: biconcave lens, radii 10 cm and 15 cm, n = 1.5

Alright, example one. Here, a biconcave lens of index one point five, with both faces curving inwards, has radii ten centimetres and fifteen centimetres; c m means centimetres. Find its focal length. Pause the video and try it first. Done? Okay, let's check. Many students take R one as ten and R two as fifteen. Then one over ten minus one over fifteen gives one over thirty, and half of that is one over sixty. So f is sixty centimetres, converging, right? No, no, wait, that is the trap, we forgot the signs! The first face curves inwards, so its centre lies on the left, against the light: R one is minus ten centimetres. The second face has its centre on the right, so R two is plus fifteen centimetres. With me? Now minus one over ten, minus one over fifteen, gives minus one over six. Then zero point five times minus one over six, the cross sign means times, gives minus one over twelve. Okay? So f equals minus twelve centimetres, and the minus sign means a diverging lens. The signs changed the answer from sixty to minus twelve, so put them in first. Right?

6:26Example 2: convex lens (f = 10 cm) in a liquid of index 1.6

Now example two, our opening puzzle. Here, a convex glass lens, with n equal to one point five, has focal length ten centimetres in air. Describe it when immersed in a liquid of index one point six. Pause the video and try it first. Done? Okay, let's check with the ratio formula. On top, one point five minus one is zero point five. At the bottom, one point five over one point six gives zero point nine three seven five, then minus one. With me? So the bottom is minus zero point zero six two five, and zero point five over that gives minus eight. Clear? So f m equals minus eight times ten centimetres, which is minus eighty centimetres. Okay? Many students write eighty centimetres and stop. But what does that minus sign tell you? Yes, exactly, the lens has changed its nature! It is now a diverging lens, of focal length eighty centimetres. Puzzle solved: the liquid, at one point six, is optically denser than the glass. Got it?

7:36Quick revision: formula, signs of R, lens in a denser liquid

So, quick revision. For a thin lens, one over f equals n L over n m, minus one, times one over R one, minus one over R two. R is positive when the centre lies on the side the light goes to. And in a liquid of higher index than the lens, convex diverges and concave converges. Now try the practice problems on this page. All the best!