1.

Obtain the equation for lateral magnification for thin lens.

Answer»

Solution :The mirror equation establishes a relation among object distance u, image distance v and focal length f for a spherical mirror.
(i) An object AB is considered on the principal axis of a concave mirror beyond the center of curvature C.
(ii) 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.
(IV) After reflection, the ray passes through the focus F. The second paraxial ray BP incident at the pole P is reflected along
(v) 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. The three reflected rays intersect at the point B'.
(vi) A perpendicular drawn as A' B' to the principal axis is the real, inverted image of the object AB.
(vii) As per law of reflection, the angle of incidence `angleBPA` is equal to the angle of reflection `angleB'PA' `.
(viii) The triangles `DeltaBPA` and `DeltaBʻPA'` are similar. Thus, from the rule of similar triangles,
` (A' B')/(AB) = (PA')/(PA) "" ....(1)`
(ix) The other set of similar triangles are, `DeltaDPF` and `DeltaB'A' F.` (PD is almost a straight vertical line)
` (A'B')/(PD) = (A'F)/(PF)`
As, the distances PD = AB the above equation becomes,
` (A'B')/(AB) = (PA')/(PA) "" ...(2)`
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) "" ....(3)`
Apply the sign conventions for the various distances in the above equation.
` PA = - u , PA' = -v , PF = - f`
Negative sign is because they are measured to the left of the pole. Now, the equation (3) becomes,
` (-v)/(-u) = (-u - (-f))/(-f)`
On further simplification,
`1/u = 1/f - 1/v , v/u = v/f -1`
Dividing either side with v,
` 1/u = 1/f - 1/v`
After rearranging,
` 1/v + 1/u = 1/f "" ....(4)`
The above equation (4) is called mirror equation.


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