Question

A. You wish to create an image that is 10 meters from an object. This image is to be inverted and half the height of the object. You wish to accomplish this using one spherical mirror.

What is the focal length f of the mirror that would accomplish this?

Express your answer in meters, as a fraction or to three significant figures.

B. You wish to create an image that is 10 meters from an object. This image is to be upright and half the height of the object. You wish to accomplish this using one spherical mirror.

What is the focal length f of the mirror that would accomplishthis?

Express your answer in meters, as a fraction or to threesignificant figures.

Answer 1

Concepts and reason

The concept required to solve the given problem is focal length of the mirror.

First, find the object distance and the image distance from the expression for the magnification of the mirror. Then, apply the mirror equation to find the focal length of the mirror.

In the next part, find the object distance and the image distance from the magnification equation. After that, use mirror equation to calculate the focal length of the mirror.

Fundamentals

Mirror equation gives the relation between the object distance, image distance and the focal length of the mirror.

Mirror equation can be expressed as,

Here, is the object distance, is the image distance, and is the focal length of the mirror.

Magnification is the ratio of image distance to the object distance. It can also be defined as the ratio of the image’s height to the height of the object.

Here, is the magnification of the mirror.

Magnification can also be expressed as,

Here, is the height of the object and is the height of the image.

(A)

Expression for magnification of the mirror is,

…… (1)

The image height should be half of the object.

Substitute for .

Since the image is inverted, the magnification has a negative value. In this case, the image is half the size of the object.

Substitute for .

Rearrange the above expression for image distance of the mirror.

…… (2)

The distance between the object and the image is,

Here, is the distance between the object and the image.

Substitute for ; the distance between the object and image is,

Rearrange the above expression; the image distance of the mirror is,

…… (3)

Compare the equations (3) and (4).

Rearrange the above expression and solve for the object distance,

Substitute for in the equation (4); the image distance is,

Apply the mirror equation to find the focal length of the mirror,

Substitute for and for .

Focal length of the mirror is,

(B)

Expression for the magnification of the spherical mirror is,

…… (4)

Here, the image is upright and the height of the image is half of the height of the object.

Magnification can be expressed as,

Substitute for .

Since the image is upright, the magnification has a positive value.

Substitute for in equation (4).

Rearrange the above expression; image distance is,

The expression can also be represented as,

…… (5)

Here, the positive magnification value indicates that the object and image are on the same side of the mirror.

Let be the object distance from the mirror. Both the object and image are on the same side of the mirror, and the object and image are separated by a distance of .

Substitute for and for in equation (5).

Rearrange the above expression,

Image distance is given by,

Substitute for ; image distance is,

Mirror equation can be expressed as,

Substitute for and for .

Focal length of the spherical mirror is,

Ans: Part A

The focal length of the mirror that produces inverted image is .