Curved Mirror Notes

Our activity today brings together two things they have worked with before: using mirrors to reflect light in the direction we want, and producing images by ensuring that light reaching our eye from a particular direction is necessarily light that came from a particular point in space – which will cause us to see an image.  You may want to review both of these to start with. Have the kids describe how light moves in straight lines, bring out the fact that our brain uses this to produce a picture of the world around us: the eyes detect what color light comes from which direction, and the brain puts this together into an image of what lies where in front of us. The brain adds a 3D component using parallax; we do not need to dwell on details, though they are cool, but we should note that we do have a way of estimating distances and that how light beams from an object diverge as they approach us has something to do with it (this determines the difference between what our two eyes see).  If we want to form an image, as we did with the pinhole viewer, we need to set things up so that light reaching a particular point on the screen (which when we look at it corresponds to a particular angle from which our eye will see it) should be light coming from a particular point in the object we are trying to image.  When light hits a mirror, it reflects at an angle equal to the angle at which it was incident. This complicated-sounding Law of Reflection basically tells us light bounces off mirrors the same way tennis balls bounce off walls. It may be useful to remind them that light hitting other objects bounces off in all directions: that is why when we look at most things we see…the thing we are looking at, not an image of something else! In a mirror, this reflection means when we are looking at it, each point on the mirror (corresponding to a particular angle to our eye) can reflect light at us only from one point in front of the mirror. This leads our brain to form an image of what is in front of the mirror, which we see as being behind the mirror.  Because a flat mirror does not change the angle between two rays, rays reflected in the mirror appear to diverge in the same way they diverged before. This means the image of an object we see in the mirror appears to us to be a distance from us equal to the distance the light from the object has traveled to get to our eyes. This means objects farther from the mirror (in front of it) will appear to be farther behind it.  Now, we can combine these ideas by using mirrors that are curved to produce images. If the mirror is curved light hitting different points on the mirror will be making different angles to the mirror surface so will bounce off differently. We can show this either using the blackboard optics kit (of which we have one) that lets you see the light rays, or by drawing on the board and asking kids to help predict how parallel rays will be reflected off a convex and a concave mirror. You should get something like this

 Parallel light rays hitting a concave mirror are bent towards each other so they all intersect at the focal point. If we are looking at this from a point to the left of the focal point, we will see rays that come from a very distant object as though they are all coming from the focal point. A distant object produces an image at the focal point (if the object is not directly in front of the mirror, rays will converge to a different point in the focal plane) so we really do see an image of distant things in the focal plane – this image is what we look at (using our eyepiece) when we look at stars through our telescope! The point is that if you look to the left of the focal point, where our eyes are, it is impossible to tell the difference between the diverging rays that would come from an object sitting at the focal point and the left-moving rays in the top figure. Our eyes don’t think mirror, they see what the light tells them, which is that there is an object there in the focal plane. This image is called a real image and is different from the image we see in a flat mirror. It is a lot more like the image we saw in the pinhole. If we put a screen or a film in the focal plane light would hit it and we could see the image on the screen or capture it on the film.  Parallel rays hitting a convex mirror are bent away from each other and appear to diverge. Is there an image here? Not in the same sense, but there is. Our eyes register the diverging rays, which are exactly what we would see from an object placed at the focal point, which here lies to the right of the mirror, behind it. So it is there that we see images of distant objects. They seem to be behind the mirror. This is a virtual image like the one we see in a flat mirror, because no light is actually getting behind the mirror. We can’t capture this image on screen or film. One important difference between a convex mirror and a flat one is that distances are distorted. No matter how far an object is in front of the mirror, its image is never farther behind the mirror than the focal point – large distances get squeezed. This is the reason for the warning on cars’ side-view mirrors, which are convex, about objects in mirror being closer than they appear. The warning is a bit confusing – in fact objects appear closer in the mirror than they are, but the distortions of distance make it very hard to gauge distances. Things get “close to infinity” really fast.  Curved mirrors also generate images of things closer in. We can predict where the images will be and what they will look like by ray tracing using just the facts we know about parallel rays and the idea we introduced last time – the reversibility of light motion. If the mirrors in the maze were set up so light from the flashlight hit the target, then when we put our eye where the flashlight had been light from the target could reach us, retracing the steps of the previous beam. Convex mirrors are easier, let’s start with that: We use a parallel ray leaving the object as well as one that aims for the focal point. By reversibility, the latter is reflected parallel to the axis. (There is a third ray in the diagram, we can ignore it.) The two rays then appear to emanate from a point behind the mirror, closer than the focal point; this is where the virtual image will appear. Note that the image is smaller than the object – a convex mirror demagnifies. As the object gets farther from the mirror on the left, the image will get closer to the focal point on the right – and smaller.

Convex mirrors are useful because by demagnifying they squeeze a lot of the world into a small image, so they allow us to look in lots of directions at once. This is why they show up in sideview mirrors on cars and in security mirrors on roads and in stores.

 With a concave mirror things are more interesting. For starters, though we did not emphasize this before, the image is inverted – upside down and eastside west. I am using the funny language there because, as we saw with flat mirrors, left and right are very relative concepts and confusing here.

Once more we can find the image using two rays: One goes parallel to the axis and thus is reflected so it goes through the focal point. The other goes through the focal point and is reflected parallel. The two rays intersect (for real!) to the left of the focal point, so to our eye, looking from the left, they appear to have come from this point. Thus the image will be inverted and form to the left of the focal point. Note that this image too is demagnified. (If this confuses you regarding telescopes, recall that magnification there is determined by the eyepiece. The mirror produces a real image, the eyepiece magnifies it.) This is a real image in that it not only seems to be there but could be captured on screen or film, the light really is there. As the object comes closer, the image moves farther to the left and grows. In fact, when the object is twice the focal length to the left of the mirror, the image appears at the same distance and size (still inverted)

As the object gets closer still, the image moves farther to the left and becomes larger (finally we see magnification)

As the object gets close to the focal plane, the image moves off to infinity but also gets very large. If we look into a concave mirror, starting from a great distance away, we will see our inverted image grow larger as we draw closer to the mirror. As we approach the focal plane, our image will grow to fill the entire mirror. What happens as we get closer still? What happens is that as we get closer the mirror forms an upright virtual image. This appears behind the mirror, at great distance and magnification, and as object moves closer the image comes closer to the mirror and the magnification decreases. As we move closer to the convex mirror, our image returns, large and right-side-up, and gets smaller as we get closer.

 There is a lot of stuff here and you will not likely manage to get it all across, so you get to choose what you can present. We have a number of toys for demonstrating these ideas, will split them up before leaving. They include: o Blackboard optics set that allows you to show them the light rays o Large convex and concave mirrors for them to look at themselves in o The magical real image of a pig reflected by two concave mirrors – so real light bounces off it!