# Concave Mirror

A spherical mirror is polished on both sides. When the convex side is used as a mirror, the magnification is +1/4. What is the magnification when the concave side is used as a mirror, the object remaining the same distance from the mirror?

In the case of the convex mirror one has the equation

$(1/p_1)-(1/p_2) = -1/f$,

where $p_1$ is the position of the object, positive if the object is to the left of the mirror

$p_2$ is the position of the image, positive if the image is to the right of the mirror,

The magnification is $Y_2/Y_1 =p_2/p_1 =1/4$, hence $p_1 =4*p_2$

$1/(4*p_2)-1/p_2 = -1/f$

$p_2 =(3*f)/4$

$p_1 =4*p_2 =3*f$

In the case of a concave mirror one has

$1/p_1 +1/p_2 =1/f$

where $p_1$ is the position of the object, positive to the left of the mirror

$p_2$ is the position of the image, positive to the right of the mirror

$1/p_2 = (1/f )- (1/p_1)$

$p_2 = (f*p_1)/(p_1-f) > 0$

the magnification is

$Y_2/Y_1 = -P_2/P_1 =-f/(p_1-f) = -f/(3f-f) = -1/2$

the image is half of the object size and inverted (see the diagram).