Mirrors and Lenses

by Jared Rovny

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    00:01 We now have a sense of what both reflection and refraction are. So, we're going to discuss these in the context of mirrors and lenses for those two exact ideas will be very important. Again to contextualize, we’ve already talked about what reflection and refraction are. Now after we've discussed mirrors and lenses, we’ll apply these mirrors and lenses to optical instruments. But first, let’s go through mirrors and lenses. This might be a bit of a difficult topic especially the first time through. So, I highly recommend working through some example problems because the more example problems you do with something like this which can be very common in terms of example problems, you’ll see the pattern.

    00:37 You’ll start seeing exactly how to solve these kinds of problems because the same sorts of logic will be applied again and again. It goes like this. We’re first going to start with mirrors.

    00:47 Suppose we have a mirror like this one. We assume that this mirror is part of a circle. So, we think of this as a circular mirror. If it is a circular mirror, we can describe it as either being concave or convex.

    01:00 A concave mirror has sort of the sides pointing inwards. A convex mirror has the sides pointing outwards.

    01:07 If you find this hard to remember, you can some time maybe think about the word concave in terms of the cave part of that word as though it’s sort of creating a cave by being sloped inwards.

    01:18 But keep in mind the different definitions between these two, concave and convex. Since it is going to be considered as part of some broader circle, we could think of where the center of that circle might be.

    01:30 So, in a dotted line here, we’ve sort of completed the circle for this mirror. We talk about the center point for this circle. We label it with a capital letter C. Then we can talk about the radius of this sort of hypothetical circle. We can also define one of the most important properties that will be related to both mirrors and lenses when we get there as well. We define it by asking ourselves, "What happens if we send a ray of light in towards this mirror? Where will it be reflected?" So, we first define as a horizontal dotted line through the center of the mirror that we just discussed. Then we send a beam parallel to this horizontal line and then find out where this beam reflects. The point that this beam goes through relative to the center of the mirror is called the focal point of this mirror.

    02:17 This definition is very, very important so certainly be aware of the focal point as we introduce it and talk about it more in the context of both mirrors and lenses. For a circular mirror like this one, it turns out the focal distance, the distance that the focal point is away from the mirror itself will be half the radius of this mirror. Looking again at this image, we can see the center point of our mirror.

    02:40 We see the curved mirror with a particular radius. The distance of that focal point from the center of the mirror will be 1/2 of the distance from the mirror itself to the center point which we again labeled as C. When we’re talking about light from distant object which will often be the case, even if you’re just a person looking at something that’s relatively far away or even more than 20 or 30 feet away, we can think of most of the light beams that are heading towards us as being parallel to each other as well as parallel to the center of the mirror, in this case, that we’re using to capture or reflect that light. This is because we could imagine any beam that’s coming from this distant source, imagine it’s slightly not parallel, slightly too far off from its angle. The further it is away, the further this small angle will diverge. So that as these two light beams are heading towards our mirror, they’re going to get further and further apart. Even a small divergence of the angle, if the angle is slightly wrong will grow to be a bigger and bigger distance over time. Those rays that aren’t very parallel are going to miss our mirror. Again, the only light rays coming from something that is very distant will be primarily and mostly parallel rays, parallel to each other. Here, we’re going to introduce the key idea of how we can find an image that’s created by either a mirror or lens.

    04:01 We all know that if we look in a mirror or we see some object in a mirror, we’re looking at an image of that object and not the actual object itself. What we’re going to ask now is what are the properties of this image, the image that the real object is creating when it’s reflected in the mirror? What we need to understand first of all is that any object, look at anything in your room right now. This is going to be an important idea. That object is emitting light in all directions and this is very easy to test. Just walk to the other side of whatever object you picked and you can still see it, which means that that object is emitting light in all directions all the time.

    04:38 What we can do is take a few of the rays of that light which is being emitted in all directions and we’ll pick the ones, the rays coming from that object that are very easy to analyze in terms of the mirror as we just introduced it. For example, one of the rays will be going parallel to the mirror and will bounce through the focal point. We could take a different ray which will go through the focal point and we’ll talk about how that ray will be reflected to be parallel to the axis that we defined.

    05:05 But the important point for this slide is that if we look at these two rays, both of them coming from the tip of this object. So, we’re using this green arrow as a hypothetical object with light coming from the tip of this arrow. At this point where these two bounced beams are intersecting with each other, if we looked at that point with our eyes, our eyes would think that that point was where the light was originally coming from. In other words, that’s where we will see the image. A different way to say this is that when light comes to your eyes, there’s no way for your eyes to know that that light was bouncing before it reached your eyes. So, your eyes will always assume and your brain will always assume that that light was coming from that source as the originating point for that source of light. So, this point where this two light rays that we’ve drawn intersect and where many of the other light rays if we drew them would also intersect will be the point that we perceive the tip of that arrow. We could also ask the same question, first with the tip of the arrow right there being the tip of the object rather than the base of the object. We can do the same thing with the base of the object. But here’s how we go through the entire procedure to find this image.

    06:15 First, from the tip of the arrow, we send a parallel beam to this axis that we defined for our mirror.

    06:21 Then we also send a beam through the focal point of our particular mirror. So, we pick these two beams out of the many beams of light that are coming from the object. We then send the parallel beam through the focal point of the mirror as we already discussed. This, in fact, defines the focal point. We need to know now that when we send a beam through the focal point first and it hits the mirror, it will bounce parallel to the axis after leaving the mirror.

    06:46 So, you can see this as sort of an opposite’s thing. If it’s going in parallel, it will go through the focal point.

    06:52 If it starts through the focal point, it will leave by moving parallel. These are two of the kinds of beams we’ll be using over and over again. The third step is, as we discussed, that this point will seem like it was the new source of the light coming from the arrow. As we started to mention, the base of this arrow, we could follow the exact same procedure but it would be a very boring procedure because all the light coming from the base of the arrow heading towards the mirror would come right back to the base of the arrow. Now that we know where we would perceive the base of the arrow and the tip of the arrow, we can see where the arrow itself would look like it was coming from. So now, with this particular mirror and with the object placed where it is, it will appear as though there’s a tiny inverted version of that object below the axis.

    07:35 So, we know if we’re looking at an object this close to a spherical mirror and looked at that object in the mirror, it would in fact look upside down and much smaller than it actually was.

    About the Lecture

    The lecture Mirrors and Lenses by Jared Rovny is from the course Geometrical Optics.

    Included Quiz Questions

    1. A ray of light incoming to a circular mirror parallel to the mirror axis
    2. A ray of light incoming to a circular mirror perpendicular to the mirror axis
    3. Two perpendicular rays of light passing through the center C
    4. Two parallel rays of light passing through the center C
    5. A ray of light passing through the mirror axis at twice the mirror radius
    1. Any light at a slight angle will diverge more and more over longer distances, missing the mirror
    2. Any light that is not parallel will destructively interfere over long distances
    3. A non-parallel ray will destructively interfere with other non-parallel rays upon arrival at the mirror
    4. Parallel rays always constructively interfere, such that at long distances they dominate the amount of light
    5. Mirrors act as focusers of light, causing the beams to become parallel over long distances
    1. A source emits light in all directions, which is the same behavior as an effective “source” where beams intersect
    2. Whenever two beams intersect, they form an image at the point of intersection
    3. When two beams intersect, the focal point and the center of a mirror coincide for the image to come into focus for an observer
    4. The point of intersection occurs at the focal point of a mirror, causing an image to form
    5. The point of intersection of two beams is always equidistant from the observer as the original source
    1. Parallel rays go through the focus; rays through the focus reflect parallel
    2. Parallel rays go through the center; rays through the focus reflect parallel
    3. Parallel rays go through the focus; rays through the center reflect parallel
    4. Parallel rays go through the center; rays through the center reflect parallel
    5. None of these is accurate

    Author of lecture Mirrors and Lenses

     Jared Rovny

    Jared Rovny

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