Question

Explain refraction at spherical surfaces. Derivation of the relation between u, v, n and R. 

Answer 1

Figure shows the geometry of formation of image I of an object O on the principal axis of a spherical surface with centre of curvature C, and radius of curvature R. The rays are incident from a medium of refractive index n₁, to another of refractive index n₂. As before, we take the aperture (or the lateral size) of the surface to be small compared to other distances involved, so that small angle approximation can be made. In particular, NM will be taken to be nearly equal to the length of the perpendicular from the point N on the principal axis. We have, for small angles,

Now, for ∆NOC, i is the exterior angle. Therefore, i = ∠NOM + ∠NCM 

Now, by Snell’s law 

n₁ sin i = n₂ sin r 

or for small angles n₁i = n₂r 

Substituting i and r from Eqs. (1) and (2), we get

Here, OM, MI and MC represent magnitudes of distances. Applying the Cartesian sign convention, 

OM = –u, MI = +v, MC = +R 

Substituting these in Eq. (4), we get

Equation (4) gives us a relation between object and image distance in terms of refractive index of the medium and the radius of curvature of the curved spherical surface. It holds for any curved spherical surface.