Question

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

Answer 1

Let O be a point object in the rarer medium of refractiive index μ₁ and lying on the principal 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 fig. The convex surface has a small aperture and the angles of incidence z and refraction r are small. Let ∠AOP = α, ∠AIP = β and ∠ACP = γ

From A, draw AN perpendicular to principal axis. For refraction at the point A, by Snell’s law, we have

\(\frac{sinr}{sinr}=\frac{\mu_2}{\mu_1}\)

Since i and r are small, sin i = i and sin r = r

∴  \(\frac{i}{r}\) = \(\frac {\mu_2}{\mu_1}\)

μ₁i = μ₂r

Now, in ΔAOC, we have

i = α + γ

and in ΔIAC, γ = r + β

r = γ - β

Substituting the value of i and r from (ii) and (iii) into (i) , we have

μ₁(α + γ) = μ₂ (γ - β)

Since α, β and γ are  small, they can be replaced by their tangents

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 these values in (iv), we get