Question

a) using the relation for the refraction at a single spherical refracting surface, derive lens marker's formula for a thin convex lens.
b) the radius of curvature of either face of a convex lens is equal to its focal length. What is the refractive index of its material

Answer 1

a)

The relation for refraction of light rays (travelling from medium 1 to medium 2) at a single spherical surface (concave or convex) :

     μ₂ / v - μ₁ / u = (μ₂ - μ₁) / R

μ₁, μ₂ are the refractive indices of the media 1 and 2 respectively.

When there are two surfaces,
     μ₂ / v₁  - μ₁ / u  = (μ₂ - μ₁) / R₁ 
     μ₁ / v  - μ₂ / v₁  = (μ₁ - μ₂) / R₂  ,    as v₁ is the object distance for the 2nd surface

  Adding them,   
       μ₁ (1/v  - 1/u) = (μ₂ - μ₁) * [1/ R₁ - 1 / R₂)
       1/v  - 1/u = (μ₂/μ₁ - 1) * [1/ R₁ - 1 / R₂)

    for a thin bi-convex lens,  R₂ =  - R₁ = R

       1/v - 1/u = (μ₂ /μ₁ - 1) * 2 / R  = (μ₂₁ -1) 2 / R

       μ₂₁ = μ  is the refractive index of the medium of lens wrt to the medium in which the lens is kept.

As,  1/ v - 1/ u =  1/f      is the thin lens formula.

Thus the lens makers formula is :
          1/f = (μ -1) * 2 / R
   f = focal length of the bi convex lens..

======================
b)
           R = f
   Then substituting this in the lens makers formula above, we get
                 μ = 1.5