Derive the relation between object distance, image distance, refractive indices and radius of curvature for refraction at a convex spherical surface.
How to lay out the answer. Follow the frame, then write it in your own words.
1 · State what you assume. A point object on the axis in medium n₁; a convex surface leading into a denser medium n₂; paraxial rays, so all angles are small.
2 · Draw the ray diagram. [Convex surface; O in n₁; C and I in n₂; ray ON refracting towards the normal CN and meeting the axis at I; NM perpendicular to the axis; angles α, β, γ, i and r]
3 · Write the angles.
α ≈ MN/OM, β ≈ MN/MC, γ ≈ MN/MI
i = α + β (exterior angle of triangle NOC)
r = β − γ (from triangle NCI)
4 · Use Snell's law for small angles. n₁ i = n₂ r, so n₁/OM + n₂/MI = (n₂ − n₁)/MC.
5 · Apply the sign convention. OM = −u, MI = +v and MC = +R.
6 · Write the result. n₂/v − n₁/u = (n₂ − n₁)/R.
7 · Close in one line. With signs, the same formula works for concave surfaces and for objects in the denser medium.
Scoring tip: Draw C between the surface and I so that the exterior-angle steps are clear. Do not forget to state the paraxial assumption.
