Note:  Please read part one, http://physicspost.com/articles.php?articleId=149
 

An algebraic method of determining the location and characteristics of images gives accurate results with no reliance on the ability to draw a straight line.

This is the equation for use with any curved mirror, converging or diverging.

                                  where:

f = focal length
do  = object distance
di  =  Image distance

Notice that the locations of the object and image are inversely related.  As the object moves closer to the mirror the image must move farther away from the mirror.

Each variable could have a positive or negative value:

• do  is always positive
• di  is positive for real images and negative for virtual images
• f is positive for real focal points and negative for virtual focal points.

The image is located 15 cm from the vertex.  Since di is positive, the image is real.

What about the size of the image?  Magnification is the term used for relating the size of the image to the size of the object.  All images are not larger than the object, so magnification refers to the relative size of the image to the object.

A magnification greater than one indicates that the image is larger than the object while a magnification less than one indicates a smaller image.

A negative magnification indicates that the image is inverted, which will happen for all real images.  A positive magnification is for images which are up right, which is the case for all virtual images.

          EX.  A magnification of -2 means the image is twice as big as the object, inverted, and real
               (since all real images are inverted)

        EX.  A magnification of 0.5 means the image is half the size of the object, upright, and virtual
                (since all virtual images are upright)
 

The following equation is used to determine magnification or other size related questions.

The negative sign is important to get the orientation (right side up or up side down) correct.

EX:    For the previous example if the object is 20 cm tall,
              a)  what is the magnification of the mirror?

              b)  how large is the image?

The negative signs indicate that the image is inverted.

Any questions involving a diverging mirror are solved the same way with the exception that the focal length is a negative number due to the virtual nature of the focal point.

By using the equations you would find that if the object is located at a distance of twice the focal length (2f) then the image is also located at 2f.  They would also be the same size with the image inverted.  You might recall that the position of 2f is also the location of the center of curvature of the mirror.

The following table summarizes the image characteristics for various locations of the object.  The information is organized by the object moving closer to the mirror from very far away.  Notice that the image moves away from the mirror until the object is located closer than the focal point creating virtual images behind the mirror.