When an object is placed normally on the principal axis of a spherial mirror at a distance 'a' from its pole, its image is formed at a distance 'v' from the pole of mirror such that `(1)/(v)+(1)/(u)=(1)/(f)`, where f = focal length of given mirror.
The relation is called mirror formula and is true for all types of mirrors under all conditions. However, values of u, v and f must be put with proper signs are per the sign convention followed. If 'h' be the height of the linear object and h' the height of image, then ratio `(h')/(h)` is called linear magnification or lateral magnification and its value m is given as :
`m=(h')/(h)=-(v)/(u)`
(e) Find an expression for longitudinal magnification of a small object placed linearly along the axis of a spherical mirror.

Explanation

Let linear size of object be du andlinear size of its image be dv.
As per mirror formula, we have
`(1)/(v)+(1)/(u)=(1)/(f)`
On differentiation, we get
`(-dv)/(v^(2))-(du)/(u^(2))=0`
`rArr" "(dv)/(v^(2))=-(du)/(u^(2))`
`therefore" Longitudinal magnification "=(dv)/(dy)=-(v^(2))/(u^(2))`