AIEEE and Earlier JEE Main Questions: Lenses

Transcript

0:00Introduction

Hello everyone! Here is a small puzzle. One glass lens has a power of minus five dioptres in air, so it spreads light out; it is a diverging lens. Now we dip the same lens into a liquid. Will its power become more negative, less negative, or could it even turn positive? Hold that thought. Today we solve two past questions on lenses, from AIEEE, the exam that later became JEE Main, and the second one is exactly this puzzle. Questions on lens power come up often, so let's begin.

0:36Power of a lens, in dioptres

So, first our tools. In the figure, the left side shows a convex lens and the right side a concave lens. The black arrow is the object, the brown arrows are its images, capital F marks a focus, and two F marks the point at twice the focal length. Now the sign convention from our course: we measure every distance from the lens. Distances along the incident light get the plus sign you see on screen, and distances against the light get the minus sign. Okay? Now the focal length, f. The convex lens bends the top ray through a focus on the far side, along the light, so f is greater than zero: a converging lens. The concave lens spreads that ray out, and the dashed line goes back to a focus on the incoming side, against the light. So f is less than zero: a diverging lens. With me? Now the star of today, the power of a lens. Capital P equals one over f, with f measured in metres. Remember that, in metres. Right? Its unit is the dioptre, the capital D on screen. And P has the same sign as f, so a converging lens has positive power, and a diverging lens has negative power. Clear? Quick check, from earlier in this chapter. One convex lens has a focal length of twenty five centimetres; the c m means centimetres. So its power is one over twenty five, right? No, no, careful, that is the trap! Twenty five centimetres is zero point two five metres. So P equals one over zero point two five, which gives plus four dioptres. Put in centimetres instead, and your answer comes out a hundred times too small. Got it? Now, two thin lenses are placed in contact, touching each other. What happens to their powers? They simply add, yes, exactly! P equals P one, plus P two, and so on; the small one and two below are just labels, and the dots mean more lenses, if there are any. Add the powers with their signs, not the focal lengths. Okay?

2:43Lens maker's formula in any medium

Next, where does f come from? From the lens maker's formula. One over f equals n L over n m, minus one, times one over R one, minus one over R two. Here n L, with a small L below, is the refractive index of the lens, and n m, with a small m below, is the index of the medium around it. R one and R two are the radii of curvature of the first and the second face. It needs a thin lens and rays close to the axis. With me? Usually the lens sits in air, so n m equals one. Then one over f equals n L minus one, times the same bracket of radii. Right? Now we dip the same lens into a liquid. Tell me, which bracket changes, and which one stays? Only the first one changes, very good! The radii come from the shape of the glass, so the second bracket stays the same in every medium. Let us call it capital K, the shape bracket. This one idea cracks our puzzle. Okay? So, what if the liquid has exactly the same refractive index as the glass? Then n L over n m is one, the first bracket is zero, and one over f is zero. So the focal length is infinite, and the lens has no effect at all. Clear? And now the last case: the liquid is optically denser, so n m is greater than n L; that is the greater than sign on screen. Then n L over n m is less than one, the first bracket turns negative, and the sign of f flips. So a convex lens diverges, and a concave lens converges. Remember this one. Got it?

4:21Example 1: lenses of -15 D and +5 D in contact, find f

Now example one, from AIEEE two thousand seven. Two lenses, of power minus fifteen dioptres and plus five dioptres, are placed in contact. Find the focal length of the combination. Pause the video and try it first. Done? Okay, let's check. The lenses touch, so the powers add, with their signs. Minus fifteen plus five gives minus ten dioptres. Okay? Now f is one over P, so one over minus ten. So the focal length is minus one over ten centimetres, right? No, no, wait, that is the trap from before! Power in dioptres gives the focal length in metres, not in centimetres. So f is minus one over ten metres. With me? And minus one over ten metres is minus zero point one metres, which is minus ten centimetres. The minus sign tells us the combination behaves as a diverging lens. Clear?

5:23Example 2: -5 D glass lens in a liquid of index 1.6

Now example two, our opening puzzle, from AIEEE two thousand five. Here, a thin glass lens, with n equal to one point five, has power minus five dioptres in air. Find its power in a liquid of index one point six. Pause the video and try it first. Done? Okay, let's check. In air, the power, one over f, is n minus one, times capital K, the shape bracket. So zero point five times K equals minus five. With me? So K equals minus five over zero point five, which is minus ten per metre; the small minus one up on the m means per metre. Okay? Now the liquid. Same glass, same shape, so K stays minus ten. Only the first bracket changes: one point five over one point six, minus one. Right? Now, before you calculate, tell me this. Will the power still be negative in the liquid? No, it will not, and here is why. The liquid, at one point six, is denser than the glass, at one point five, so this bracket is negative. One point five over one point six is zero point nine three seven five, and minus one gives minus zero point zero six two five. Clear? So one over f is minus zero point zero six two five, times minus ten. Minus times minus is plus, so the power is plus zero point six two five dioptres. Got it? Positive power means a converging lens. Puzzle solved: the diverging lens in air has become a converging lens in the liquid, because the liquid is optically denser than the glass. So never carry the old sign over by habit; check the sign of that first bracket every time. Okay? One more thing, for when you practise the real paper. Some books define the power of a lens in a medium as n m over f, instead of one over f. Then the power is one point six times zero point six two five, which is plus one dioptre, and that is the option most answer keys give for this question. Either way, the power is positive, and the lens converges. Okay?

7:32Quick revision: P = 1/f in metres, powers add, nL/nm rule

So, quick revision. Power is one over f, with f in metres, and for thin lenses in contact, the powers add. For a thin lens in any medium, one over f equals n L over n m, minus one, times the shape bracket, and the shape bracket never changes. And in a medium denser than the lens, convex diverges and concave converges. Now try the practice problems on this page. All the best!