Refer to Interactive Solution 25.19 at for help with problems like this one. A concave mirror (R

Refer to Interactive Solution 25.19 at for help with...

Refer to Interactive Solution $\underline{25.19}$ at for help with problems like this one. A concave mirror $(R=56.0 \mathrm{~cm})$ is used to project a transparent slide onto a wall. The slide is located at a distance of $31.0 \mathrm{~cm}$ from the mirror, and a small flashlight shines light through the slide and onto the mirror. The setup is similar to that in Figure $25-19 a$. (a) How far from the wall should the mirror be located? (b) The height of the object on the slide is $0.95 \mathrm{~cm}$. What is the height of the image? (c) How should the slide be oriented, so that the picture on the wall looks normal?

Key Concepts

Mirror Equation

The mirror equation relates the object distance, the image distance, and the focal length of a mirror. It is a fundamental equation in geometric optics which, when applied to spherical mirrors like concave mirrors, allows one to determine any of these distances provided the other two are known. This equation is essential for analyzing how images are formed, including whether they are real or virtual.

Focal Length and Radius of Curvature

For spherical mirrors, the focal length is directly related to the radius of curvature by the relation f = R/2. Understanding this relationship is key because it simplifies the process of finding the focal length when the radius of curvature is provided, and it plays an important role in applying the mirror equation to determine object and image distances.

Magnification

Magnification in optics is defined as the ratio of the height of the image to the height of the object, and is also related to the ratio of the image distance to the object distance. It is a crucial concept because it not only helps determine the size of the image relative to the object but also indicates the orientation of the image; a negative magnification indicates that the image is inverted relative to the object.

Image Orientation and Projection

In the context of projecting images using concave mirrors, image orientation is important to ensure that the final, observed image appears normal to the viewer. This concept involves understanding that the real images formed by concave mirrors are typically inverted, and compensatory adjustments (such as reorienting the source object) may be necessary so that the image appears correctly oriented on a projection surface like a wall.

Transcript

00:01 High in the given problem radius of curvature of a concave spherical mirror is given as r is equal to plus 56 .0 centimeter.

00:26 So its focal length will be equal to half of the radius of curvature means it will be 28 .0 centimeter.

00:39 Now the distance of slide in front of the mirror means the distance of object means this is do which will be given as plus 31 .0 centimeter as it is in front of the mirror.

01:08 Now we have to find the distance of image means the distance of wall in front of this concave mirror again.

01:20 So to find it in this first part of the problem we use mirror equation which is a relation among the object distance do the image distance di and the focal length of this spherical mirror.

01:43 So plugging in all known values we get for so this is 31 cm.

01:52 In the age distance that is missing means 1 by d .i is equal to 1 bifocal length means 28 centimeters.

02:01 So it is rearranged to give 1 by d .i is equal to 1 by 28 minus 1 by 31.

02:11 And this is solved to get the answer for the image distance, di.

02:16 Which comes out to be 289 .3 centimeter or approximately we can say this d .i is equal to plus 290 centimeter here this positive sign says that represents that the image is also being formed in front of the concave mirror means the distance of screen on the wall is at a distance of 290 cm from this mirror.

02:53 Now in the second part of the problem we have to find the height of this image on the wall and it's in nature.

03:02 So the height of the object is given as 0 .95 cm which is put in the slide...

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