(a) Obtain lens makers formula using the expression

 

Here the ray of light propagating from a rarer medium of refractive index (n₁) to a denser medium of refractive index (n₂) is incident on the convex side of spherical refracting surface of radius of curvature R.

(b) Draw a ray diagram to show the image formation by a concave mirror when the object is kept between its focus and the pole. Using this diagram, derive the magnification formula for the image formed.

Len’s Makers formula:

Consider a thick lens L. The refractive index of the material of lens is n_2. The lens is placed in a medium of refractive index n₁. R₁ and R₂ is the radius of curvature of the lens and poles are represented by P₁ and P₂. Thickness of the lens is given by t. O is a point object which is placed on the principal axis of the lens.

Object distance from pole P₁ is u.

Image formed is at a distance of v’ from P₁.
Now, using the refraction formula at spherical surface, we have 
                                                 ... (1)

Image I’ acts as a virtual object for second surface and after refraction at second surface, final image is formed at I.

Distance of I from pole P₂ of second surface is v.

Distance of virtual object I’ from pole P₂ is (v’ – t).
For refraction at second surface, the ray is going from second medium (refractive index n₂) to first medium (refractive index n₁), therefore from refraction formula at spherical surface.
i.e.,  

Thickness of the lens is negligibly small as compared to v’. 
Therefore, 
                              ... (2)

Now, adding equations (1) and (2), we have 

Now, using the lens formula, we get 

 is the required Lens maker’s formula. 

b) Ray diagram for the image formation of an object between focus (F) and pole (P) of a concave mirror is given by,

Magnification, m = 
From fig. 
Also,