Question

A mirror creates an image that has a height of 2.50 cm that is behind the mirror and 167 cm from the object.The image was created from a 5.00 cm tall object. What is the focal length of the mirror?

Answer 1

We are told that the image is behind the mirror. This means that the image is virtual.

We know that a virtual image can be a convex mirror, a concave mirror (or a plane mirror). In contrast, a real image can only be formed by a concave mirror. How do we know which type of mirror we are dealing with?

Remember that a concave mirror only forms a virtual image when the object is nearer than the focal length and the resulting image is enlarged. In contrast, a convex mirror creates a diminished image (and a plane mirror creates an image of the same size as the object).

We have a 5cm object creating a 2.5cm image which means that the image is diminished. So, the mirror is convex.

Now for some equations. Remember that there are multiple sign conventions that are followed in optics (though of course, they all describe the same physics and get the same results). So, be mindful that your textbook might put a - sign at a different place than I do. In my convention, distances along the direction of incident light are positive.

Our given quantities are

because this is the distance between the object and mirror.

Since we are dealing with a convex mirror. the object distance is negative and the image distance positive. So,

For a spherical mirror, the magnification is given by

Substituting,

Rearranging,

Substitute this in the relation we have from the distance between the object and image

Using the mirror equation

for the focal length

Substitute

Taking the reciprocal

The focal length of the mirror is 111.35cm