# The convex mirror equation

First, understand the sign rules of the convex mirror.

**The sign rules for the convex mirror**

– __Object distance (do)__

If an object is in the front of a mirror surface which reflecting light, where the light passes through the object, then *the object distance (do)* is positive.

– __Image distance (di)__

If the image is in the front of a mirror surface which reflecting light, where light passes through the image, then *the image distance (di)* is positive (real image). If the image is behind the mirror surface that reflecting light, where light does not pass through the image, then *the image distance* is negative (virtual image).

– __The radius of curvature (R)__

The center of the curvature of the convex mirror is behind the mirror surface which reflects light, where the light does not pass through it so that the radius of curvature of the convex mirror is negative. The radius of curvature is negative, so the focal length (f) is also negative.

– __Object height (h)__

If the object is above the principal axis of the convex mirror, *the object height (h)* is positive (object is upright). Conversely, if the object is below the principal axis of the convex mirror, *the object height* is negative (object is inverted).

– __Image height (h’)__

If the image is above the principal axis of the convex mirror, the image height (h ‘) is positive (image is upright). If the image is below the principal axis of the convex mirror, the image height is negative (image is inverted).

– __Magnification of image (m)__

If the magnification of image > 1 then the size of the image is greater than the size of the object. If the magnification of image = 1, the size of the image is the same as the size of the object. If the magnification of the image is < 1, the size of the image is smaller than the size of the object.

**The equation of convex mirror**

Based on the figure below, there are two beams of light to a convex mirror, and the convex mirror reflects the beam of light.

do = object distance, di = image distance, h = P P’ = object height, h’ = Q Q’ = image height, F = the focal point of the convex mirror.

The P’AP triangle is similar to the Q’AQ triangle. Therefore :

The BFA triangle is similar to the Q’FQ triangle where the distance of AB = the height of the object (h) and the distance of FA = the focal length (f) of the convex mirror. Therefore :

Based on the sign rules of the convex mirror, this equation can be changed to the equation of the concave mirror, if the image distance (di) is given a negative sign because the beam of light does not pass the image and focal length (f) is also given a negative sign because the focal point of the convex mirror is not passed by light (see the figure of the image formation above). According to this statement, the equation of the convex mirror changes to:

do = the object distance, di = the image distance, f = the focal length

Always remember the sign rules of the convex mirror when using this equation to solve the problems of the convex mirrors.

**Magnification of image (m)**

Observe the figure of the image formation above. The P’AP and Q’AQ triangles are similar so that we can derive the relationship between the object distance and the image distance with the object height and the image height:

This equation is written again as below by adding m:

m = Magnification of the image

h = the object height (positive if the object is above the principal axis of the convex mirror or the object is upright. Negative if the object is inverted)

h ‘= the image height (positive if the image is above the principal axis of the convex mirror or the image is upright. Negative if the image is inverted)

do = the object distance (positive if the light beam passes through the object)

di = the image distance (positive if the light beam passes through the image and negative if the image is not passed through by the light beam)