(III) An object is placed a distance r in front of a wall, whe...

Question

(III) An object is placed a distance r in front of a wall, where r exactly equals the radius of curvature of a certain concave mirror. At what distance from the wall should this mirror be placed so that a real image of the object is formed on the wall? What is the lateral magnification of the image?

Accepted Answer

Hello, fellow physicists today, we're gonna solve the following practice prog 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 projector needs to cast an image of a slide onto a screen that is located 5 m away. If the focal length of the projector's lens is 2.5 m, how far should the slide be placed from the lens? In order to ensure that the image is formed clearly on the screen, also determine the magnification of the image. Awesome. So it appears for this particular problem we're asked to solve for two separate answers. Our first answer is we're trying to figure out how far from the slide should. So how far should the slide be placed from the lens which referring to the lens on the projector? In order to ensure that the image is formed clearly on the screen. So we're trying to figure out how far away from the lens our slide needs to be located. And our second answer is we're trying to figure out what the magnification of this particular image will be awesome. So now that we have an idea of what we're solving for, let's read off our multiple choice answers to see what our final answer might be. So A is the slide should be placed 5 m from the lens with a magnification of negative 1.0. Meaning the image is inverted B is the slide should be placed 10 m from the lens with a magnification of positive 1.0. Meaning that the image is upright C is the slide should be placed 5 m from the lens with a magnification of negative 1.5. Meaning the image is inverted. And finally D is the slide should be placed 10 m from the lens with a magnification of positive 1.5 meaning the image is upright. Awesome. So first off, let us write down all of our known variables. So we're told that the focal length F of the lens is equal to 2.5 m. And once again, this is the focal length, we're also told the distance of the image from the lens. So this is the distance to the screen. Let us call this value D I which is the image distance is equal to 5 m. Fantastic. However, the only value that we do not know we do not know the value of do we do not know the object distance? So we're trying to figure out what do is so and once again, D is the distance from the slide to the lens. So in order to help us solve for, do we need to recall and use the lens equation. So as we should recall, the lens equation states that one divided by F. So one divided by the focal length is equal to one divided by the object distance plus one divided by the image distance. So now we can plug in all of our known variables and start to solve for do which is our unknown variable. So we need take one divided by 2.5 m and we need to set it equal to one divided by do we do not know what do is plus one divided by 5 m. Awesome. So ultimately, we are trying to rearrange this equation to isolate and sulfur do. So when we get do all by itself, we will find that do is equal to 5 m. So I've gone ahead and done the solving for you. But all you need to do is use algebra to rearrange this equation to get do by itself. So ultimately, you're gonna have to take the inverse in order to get do by itself. And when you take the inverse you will get that do is equal to 5 m. Awesome. So that means we've solved for our first answer. Hooray, we did it. So now our next step is we need to calculate the magnification. So we need to recall and use our magnification formula. So our magnification M is equal to negative D I divided by Do Awesome. And as we can see, we can plug in our known variables. So we know that D I is equal to 5 m and we know that do is also equal to 5 m. So when we plug that into our calculator or you could just do the math in your head, you will find that our final answer is equal to negative one and that's it we've officially solved for this problem. So therefore, the slide should be placed 5 m from the lens with a magnification of negative one, which means that the image is inverted and is the same size as the object. So this means that our final answer has to be multiple choice answer letter A which once again is the slide should be placed 5 m from the lens with a magnification of negative 1.0 meaning the image is inverted. Thank you so much for watching. Hopefully, that helped and I can't wait to see you in the next video. Bye.

(III) An object is placed a distance r in front of a wall, where r exactly equals the radius of curvature of a certain concave mirror. At what distance from the wall should this mirror be placed so that a real image of the object is formed on the wall? What is the lateral magnification of the image?

Key Concepts

Radius of Curvature

Mirror Formula

Lateral Magnification

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