Both a converging lens and a concave mirror can produce virtua...

Question

Both a converging lens and a concave mirror can produce virtual images that are larger than the object. Concave mirrors can be used as magnifying makeup or shaving mirrors, but converging lenses cannot be. (a) Draw ray diagrams to explain why not. (b) If a concave mirror has the same focal length as a converging lens, and an object is placed first at a distance of (1/2)ƒ from the lens and then at a distance of (1/2)ƒ from the mirror, how will the magnification of the object compare in the two cases?

Accepted Answer

Hello, fellow physicists today, we're gonna solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use. In order to solve this problem, a student wants to form a magnified image of a small object using a single optical device. He has a converging lens and a concave mirror, both of the same focal length F I which device should he use? I I, if the object distance is F divided by three, what magnifications can be produced by both devices individually? Awesome. So it appears for this particular problem we're asked to solve for two separate answers. So ultimately, we'll, we'll know that we've solved for this problem correctly when we solve for part I, which asks what device should he use to accomplish? His goal of creating a magnified image of a small object using a single optical device. And our second answer for I I, we need to figure out if the object distance is F divided by three, what magnifications can be produced by both devices individually. Awesome. So let's read off our multiple choice answers to see what our final answer pair might be or answer set might be A and I'm gonna read the answers for I first and then for I I second. So A is converging lens and magnification for converging lens is equal to 1.5. The magnification for the concave mirror is 1.5 B is concave mirror magnification for the converging lens is equal to 3.0 magnification for the concave mirror is 2.0 C is either the converging lens or the concave mirror. And magnification for the converging lens equals 1.5 magnification for the concave mirror is equal to 1.5 and either the converging lens or the concave mirror. Awesome. And then magnification for the converging lens is equal to 3.0. And the magnification for the concave mirror is equal to 2.0. Awesome. So first off, in order to help us better visualize this problem, let's create a bunch of like let's create a figure or a diagram for the converging lens. And for the concave mirror to help us better visualize what is going on. So I've gone ahead and found a handy dandy diagram that we can refer to. Awesome. So at the very top, so going from top to bottom, we have our concave mirror now as you could see it curves, so it's like a little bit of a curve. So it looks like almost like a bowl got cut in half. So we have our concave mirror up top. And then we have two examples of converging lenses. And as we could see the red arrow or specifically the red dotted arrow for each of these cases, because we have two converging mirror cases. So once again to recap at the very top of our figures, we have our concave mirror and then second image, we have our converging lens. First example of what could happen for a one style of converging lens. And then finally, we have our third converging lens situation. Awesome. So it's important to note that our second converging lens situation focuses on part I. So that's when it want, when the student wants to create a magnified image of a small object using the converging mirror. And then our third converging, sorry I A converging mirror by main converging lens. And then for our third image here, we have our converging lens that's focusing on the eye eye condition. And so we don't get confused. I'll write that down for each objects. So for the eye, eye condition where the object distance is F divided by three OK. And also for both cases, the dotted red line for all of these different figures denotes the image and the solid red arrow denotes the object orientation where and also for the, it's the same for the image, the dotted red arrow indicates the object. So the orientation of the image. So the object and the image image is dotted arrow an object is solid red arrow. Awesome. And then we have our rays which are denoted by solid black lines with arrows. So those are our rays that strike the concave mirror or the lens and they will show the path of the rays are in the. So the path of the rays are indicated with arrows. Awesome. So now that we have a good visualization of what's happening for this particular problem, we must note that from these ray diagrams that we created, it can be seen that the magnified image can be formed by both a converging lens and a concave mirror. So therefore, for this particular problem and for the particular conditions set in this prom, the student can use either the converging lens or the concave mirror to accomplish this goal of magnifying an image of a small object. Awesome. So with that in mind, we've already, so for the first answer, so let's make a note here up top that we can use both both the concave mirror and the convex or the concave. So we can use the concave mirror and the converging lens. Awesome. So moving right along here. So that's for part I, so we solve for part I, we can use both. Now we could start solving for part I I which is our second and last answer. So therefore, since we know that we can use either the concave mirror or the converging lens, we need to recall a note that the equations for the lens and the mirror are, are similar. Meaning we should recall a note that one divided by F and let's write this in black. So one divided by F where F is our focal length is equal to one divided by do where do denotes the object distance plus one divided by D I where D I denotes the image distance. So quick recap here F is our focal length of the lens or the concave mirror do is the object distance and D I is the image distance. So now we need to take this equation and rearrange it to isolate and solve for D I. That is what we're ultimately going to be doing. And then we need to plug in all of our known variables since we know that. So we need to know so that D so D by itself is equal to. And it's important to note actually to make things simple. So we don't get too confused. We need to note that for this case, since we're dealing with I I, our focal length is different. We need to note that our focal length for this case is gonna be or we need to note not the focal length. So it says focal length divided by three, but that's the object distance. So we need to note that do is equal to F divided by three. So that's so don't get confused. But it's important to note that F is the focal length. OK. So considering that, so plugging in our known variables, we need to rearrange to start solving for D I or object distance. So D I is equal to one divided by F minus one, divided by F divided by three. So let me get D I all by itself. I'm gonna skip some steps here, but you'll know you've done your algebra correctly when D I is equal to negative F divided by two. So that's our image distances for D I Awesome. So now we need to take our D I our object or sorry, our image distance. And now we need to figure out the magnification because that's what we're ultimately trying to solve for. So the magnification equation as we should recall is equal to negative D I divided by do. And when we plug in our known variables, that would mean that it's negative negative F divided by two, all divided by. So it's F divided by two. So F divided by two all divided by F, divided by three. Awesome. Which let me plug that into our calculator. We'll get three has which is equal to 1.5. And that is our final answer. Hooray, we did it. So therefore, without a doubt, our final answer has to be looking at our multiple choice answers, multiple choice answers letter C which states that for I either the converging lens or the concave mirror and I I is magnification for the converging lens is equal to 1.5 and the magnification for the concave mirror is also 1.5. Thank you so much for watching. Hopefully.

Key Concepts

Ray Diagrams

Magnification

Virtual Images

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