Solve Reflection Problem: Convex & Plane Mirrors 28cm Apart

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In summary, the problem involves a convex mirror and plane mirror placed 28 cm apart with a small luminous object placed midway between them. The distance between the two images observed on the plane mirror is 24 cm, and using the lens/mirror equation, we can calculate the radius of curvature of the convex mirror to be 35 cm.
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Hyperreality
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A convex mirror and a plane mirror are placed facing each other and 28cm apart. Midway between them and on the principal axis of the convex mirror a small luminous object is placed. If the distance between the two images observed on looking into the plane mirror is 24 cm, calculate the radius of curvature of the convex.

I have been stuck on this problem for quite a while now.

First, theere is a image on the plane mirror, so that means the virtual image produced by the convex mirror has to be at the position of the object right?

Please give me some hints on how to tacke this problem...
 
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A complete solution is offered.

The mirrors are 28 cm apart so the object being halfway between them is 14 cm from the plane mirror. Its image in the plane mirror will be 14 cm beyond the mirror. We want the the second image to be 24 cm beyond that, for a total of 38 cm behind the plane mirror.

For this to occur, its virtual image in the convex mirror must be 38 cm away from the plane mirror. The convex mirror itself already 28 cm from the plane mirror, so the virtual image must be 38 - 28 = 10 cm beyond the convex mirror.
upload_2016-2-9_11-8-30.png

Invoke the lens / mirror equation:

##\frac{1}{O} + \frac{1}{f} = \frac{1}{i}##

Since the is a convex mirror the image is virtual so i is negative. O is negative, too, by convention. So we set:

##O = -14~cm##
##i = -10~cm##

Solve for f:

##f = \frac{O⋅i}{O - i} = -35~cm##

So the radius of curvature of the convex mirror is 35 cm.
 
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1. What is the reflection problem for convex and plane mirrors?

The reflection problem for convex and plane mirrors refers to the issue of determining the location and size of an object's reflected image when placed in front of a convex or plane mirror. This problem involves understanding the principles of reflection and using mathematical equations to calculate the position and size of the image.

2. How can I solve the reflection problem for convex and plane mirrors?

To solve the reflection problem for convex and plane mirrors, you will need to use the laws of reflection and the mirror equation. These equations involve understanding the angle of incidence and the angle of reflection, as well as the distance between the object and the mirror. By plugging in the given values into the equations, you can solve for the position and size of the reflected image.

3. What are the key differences between convex and plane mirrors in terms of solving the reflection problem?

The main difference between convex and plane mirrors when solving the reflection problem is the way in which the equations are applied. For convex mirrors, the equations involve negative values due to the curvature of the mirror, while for plane mirrors, the equations are simpler as the mirror has no curvature. Additionally, the location and size of the reflected image will differ between convex and plane mirrors due to their different shapes.

4. Are there any common mistakes when solving the reflection problem for convex and plane mirrors?

Yes, some common mistakes when solving the reflection problem for convex and plane mirrors include using incorrect values for the distance between the object and the mirror, misinterpreting the angles of incidence and reflection, and forgetting to account for the curvature of a convex mirror. It is important to carefully follow the equations and double-check your work to avoid these mistakes.

5. Can the reflection problem be solved using other methods besides equations?

Yes, besides using equations, there are other methods that can be used to solve the reflection problem for convex and plane mirrors. These include using ray diagrams, which involve drawing the path of light rays from the object to the mirror and to the image. Another method is using the mirror formula, which involves using the magnification equation to calculate the size of the reflected image. However, equations are generally the most accurate and efficient method for solving the reflection problem.

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