Obtain the equation for lateral magnification for thin lens.

Explanation

(i) The mirror equation establishes a relation among object distance u, image distance v and focal length f for a spherical mirror. An object AB is considered on the principal axis of a concave mirror beyond the center of curvature C.
(ii) Let us consider three paraxial rays from point B on the object.
(iii) The first paraxial ray BD travelling parallel to principal axis is incident on the concave mirror at D. close to the pole P. After reflection the ray passes through the focus F. The second paraxial ray BP incident at the pole P is reflected along PB.. The third paraxial ray BC passing through centre of curvature C. falls normally on the mirror at E is reflected back along the same path.
(iii) The three reflected rays intersect at the point B.. A perpendicular draw as A.B to the principal axis is the real, inverted image of the object AB.
As per law of reflection, the angle of incidence `angleBAP` is equal to the angle of reflection `angleBPA.`. The triangles `DeltaBPA and DeltaBPA.` are similar. Thus, from the rule of similar triangles. `(A.B)/(AB)=(PA.)/(PA)`
The order set of similar triangles are, `DeltaDPF and Delta B.A.F.` (PD is almost a striaght vertical line)
`(A.B)/(PA)=(PD.)/(PF)`
As, the distances PD = AB the above eqution becomes,
`(A.B.)/(AB)=(A.F)/(PF)`
From equations (1) and (2) we can write,
`(PA.)/(PA)=(A.F)/(PF)`
As, A.F = PA. - PF, the above equation becomes,
`(PA.)/(PA)=(PA.-PF)/(PF)`
We can apply the sign conventions for the various distance in the above equation.
`PA=u,PA.=-v,PF=-f`
All the three distances are negative as per sign covention, because they are measured to the left of the pole. Now, the euqation (3) becomes,
`(-v)/(-u)=(-v-(f))/(-f)`
On further simplification,
`(v)/(u)=(v-f)/(f),(v)/(u)=(v)/(f)=1` Dividing either side with v,
`(1)/(u)=(1)/(f)-(1)/(v)` The above equation (4) is called mirror equation.
Lateral magnification in spherical mirros:
The lateral or transverse magnification is defined as the ratio of the heigh of the image to the height of the object. The height of the object and image are measured perpendicular to the principal axis.
`"magnification" (m) = ("heigh of the image(h)")/("height of the object(h)")`
Applying proper sign conventions for equation (1),
`(A.B.)/(AB)=(PA.)/(PA)`
`A.B. = h,Ah,PA.=-v,PA=-u`
`(-h.)/(h)=(-v)/(-u)`
On simplifying we get,
`m=(h.)/(h)=-(v)/(u)`
Using mirror equation, we can further write the magnification as,
`m=(h.)/(h)-(f-v)/(f)=(f)/(f-u)`