Spherical Mirrors and the Mirror Formula

Transcript

0:00Introduction

Hello everyone! Look into the inside of a shiny steel spoon. Your face is upside down, and small. Now turn it around. On the back, you are upright, but still small. So how can one spoon give two such different images? Let us find out exactly why. Spherical mirrors come up often in JEE, and most mistakes here are sign mistakes. So, let's begin.

0:26Pole, centre of curvature and focus

First, let us learn the names. In the left half of the figure, the thick dark blue curve is a concave mirror, like the inside of the spoon. P is the pole, the centre of the mirror surface, and C is the centre of curvature. The horizontal line through P and C is the principal axis. Okay? Now let me ask you this. The orange ray comes in parallel to the axis. After reflection, which point does it pass through? It passes through the focus F, yes, exactly! F sits midway between the pole and C, so the focal length, small f, equals capital R over two, where R is the radius of curvature. Clear? But this needs rays close to the axis, called paraxial rays, and so does the mirror formula, okay? The blue grey ray from B, the tip of the object, the black arrow A B, goes through C and comes straight back. Where the two rays cross, the dark red arrow is the image, upside down. On the right is a convex mirror. Its reflected rays seem to come from behind the mirror, along the dashed lines, the orange one from F, and where they meet we get a small, upright image. With me?

1:36Sign convention: measure from the pole

Now comes the most important part, the signs. Rule one: every distance is measured from the pole, nowhere else. Okay? Rule two: distances along the incident light, the light coming in, are positive, and distances against it are negative. Right? We draw the object on the left, so the light travels left to right. The object distance is called u. Is u positive or negative? That is correct, it is negative! From the pole, we measure back to the object, against the light. So on screen, u is less than zero, with the less than sign: for a real object, u is always negative. Got it? Now, for a concave mirror, the focus lies in front, on the object's side. So is f positive or negative? Yes, it is negative again! For a concave mirror, f and R are both less than zero. For a convex mirror, F and C lie behind the mirror, along the light, so f and R are greater than zero. Taking f positive for a concave mirror is a very common sign mistake, okay?

2:41Mirror formula and magnification

Now the star of this chapter, the mirror formula. The image distance is called v. One over v, plus one over u, equals one over f, and that equals two over capital R as well. It works for both mirrors, signs included. Clear? Next, the magnification, m, is h prime over h. Here h is the object's height, and h prime, h with the little dash, is the image's height, and heights above the axis are positive. It also equals minus v over u. Notice that minus sign, okay? Suppose v comes out negative. Where is the image, and is it real or virtual? Negative v means the image is in front of the mirror, where the light really goes: a real image! Positive v means behind the mirror: a virtual image. Right? Now the sign of m: negative means inverted, and positive means erect. The two straight bars mean the modulus, the size of m without its sign. If it is more than one, the image is magnified. Got it?

3:43Where a concave mirror forms the image

So, where does a concave mirror form the image? The arrow on screen means gives. Object beyond C gives an image between F and C: real, inverted, smaller. That is your face in the spoon, far beyond the centre of curvature of that tiny bowl! Right? Object between C and F: the image goes beyond C, still real and inverted, but now larger. Okay? Object between F and the pole: now the image is behind the mirror, virtual, erect and larger. Clear? And a convex mirror always gives a virtual, erect, smaller image, between P and F, wherever you place a real object in front of it. That is the back of the spoon, so our spoon mystery is solved!

4:27Topper's trick, and one trap

Now a trick that toppers use. Combine the mirror formula with m equals minus v over u, and you get m equals f over, f minus u. You do not even need v! Just put f negative for concave, and u negative for a real object. Okay? Here is a trap for you. You dip a concave mirror in water. Does its focal length change? No, it does not change at all! The focal length of a mirror depends only on its shape, so it is the same in any medium. Careful, this is unlike a lens. Got it?

5:01Example 1: object 30 cm from a concave mirror of f = 20 cm

Now our first example. An object is thirty centimetres in front of a concave mirror of focal length twenty centimetres. Find the position, nature and magnification of the image. Pause the video and try it first. Are you done? Okay, let's check. Signs go in first: the object is in front, so u equals minus thirty centimetres, and the mirror is concave, so f equals minus twenty centimetres. Right? From the mirror formula, v equals u times f, over u minus f; the brackets hold the negative numbers. Minus thirty times minus twenty is six hundred, and minus thirty plus twenty is minus ten. Six hundred over minus ten gives minus sixty centimetres. Clear? Then m equals minus v over u. Minus sixty over minus thirty is two, and the minus sign in front makes it minus two. Okay? So v is negative: the image is sixty centimetres in front of the mirror, and real. And m is minus two: inverted, and twice the size, just as our table said. With me?

6:07Example 2: two object positions for an image 3 times as big

Now a slightly harder one. At what distances from a concave mirror of focal length twelve centimetres must an object be placed, to give an image three times its size? Pause the video and try it first. Here is a hint: use the topper's trick. Are you done? Okay, let's check. The mirror is concave, so f equals minus twelve centimetres. We know m and want u, so we use m equals f over, f minus u. Right? The image is three times its size, so m equals minus three, and that is the only answer, isn't it? Well, minus three is one answer, for a real, inverted image. Minus twelve equals minus three times, minus twelve minus u. Dividing both sides by minus three gives four equals minus twelve minus u. The double arrow then gives u equals minus sixteen centimetres. Clear? But no, no, wait, the question never said real! An erect, virtual image, with m equals plus three, is also possible. Then minus twelve equals three times, minus twelve minus u. Dividing both sides by three gives minus four equals minus twelve minus u, so u equals minus eight centimetres. Okay? So there are two answers: sixteen centimetres for a real image, or eight for a virtual one. Our table agrees: sixteen is between C and F, eight between F and the pole. Missing the second one is the trap. Got it?

7:38Quick revision: signs, 1/v + 1/u = 1/f, m = -v/u = f/(f-u)

So, quick revision. First, measure from the pole: u is negative for a real object, f is negative for a concave mirror and positive for a convex one. Second, one over v plus one over u equals one over f, and m equals minus v over u, or f over, f minus u. And third, negative v means a real image, and negative m an inverted one. Now try the practice problems on this page, and put the signs in first. All the best!