Question

Obtain an expression for refraction at a single convex spherical surface, i.e., the relation between μ₁ (rarer medium), μ₂ (denser medium), object distance u, image distance v and the radius of curvature R.

Answer 1

In ΔCOA,

i = α + γ and r = γ - β

γ = r + β

Let O be a point object in the rarer-medium of refractive index μ₁ and lying on the principle axis. The image of this object, O is formed by refraction at the convex spherical surface of radius at the point 'I' into the medium 'B' of refractive index μ₂ of curvature, as shown in the figure.

The convex surface has a small aperture and the angles of incidence ‘i’ and refraction ‘r’ are small.

Let ∠AOP = α, ∠AIP = β, ∠ACP = γ from A, and draw AN ⊥^er to the principle axis for refraction at point A₁. By snell's law,

`(sin i)/(sin r) = mu_2/mu_1`

∵ i and r are small.

then `i/r = mu_2/mu_1`

`=> mu_1i = mu_2r`

Substituting the value of i and r.

μ₁(α + γ) = μ₂(γ - β)

∵ α, β, γ are small, they can be replaced.

∴ μ₁(tan α + tan γ) = μ₂(tan γ - tan β) 

or `mu_1 ("AN"/"NO" + "AN"/"NC") = mu_2 ("AN"/"NC" + "AN"/"NI")`

or `mu_1/"NO" + mu_1/"NC" = mu_2/"NC" - mu_2/"NI"`

Due to small aperture, the point N lies close to P. Also, applying sign convention, we get

NO ~ PO = - u, NC ~ PC = R

NI ~ PI = v

Putting the value in the above expression

`mu_1/(- u) + mu_1/"R" = mu_2/"R" - mu_2/v`

OR

`mu_2/v - mu_1/u = (mu_2 - mu_1)/"R"`