Refraction at a Spherical Surfaces

A spherical refracting surface is a curved boundary separating two transparent media, where one medium has refractive index n₁​ and the other has refractive index n₂.

When light travels from one medium to another through this curved surface, the rays bend due to refraction, and an image is formed depending on the curvature of the surface and the refractive indices of the two media.

This concept forms the basis for understanding lenses, because a thin lens may be treated as a combination of two spherical refracting surfaces.

For refraction at a spherical surface, the relation is:

\[\frac{n_2}{v}-\frac{n_1}{u}=\frac{n_2-n_1}{R}\]

Consider an object point O placed on the principal axis of a spherical surface separating medium 1 and medium 2.

• Medium 1 has a refractive index n₁.
• Medium 2 has a refractive index n₂.
• The centre of curvature is C.
• The object distance is u.
• The image distance is v.
• The radius of curvature is R.

• All distances are measured from the pole.
• Distances measured in the direction of incident light are taken as positive.
• Distances measured opposite to the direction of incident light are taken as negative.
• Heights above the principal axis are positive.
• Heights below the principal axis are negative.

When Light Goes from a Rarer to a Denser Medium

The refracted ray bends towards the normal, and the image position depends on the curvature of the spherical surface and the object location.

When Light Goes from a Denser to a Rarer Medium

The refracted ray bends away from the normal, and the image may shift differently for the same object distance because the refractive index contrast changes.

Real-Life Analogy

Looking through a curved glass bowl or a transparent marble gives a distorted view of an object because light is refracted at a curved surface rather than a flat one.

Stepwise Logic

1. Consider a paraxial ray from object O incident on the spherical surface.
2. Draw the normal at the point of incidence by joining that point to the centre of curvature C.
3. Use the small-angle approximation for the angles made by the incident ray, refracted ray, and normal.
4. Apply the refraction relation in its small-angle form.
5. Rearrange terms in terms of u, v, and R.
6. Obtain the standard formula for refraction at a spherical surface.

An object is placed at a distance of 100 cm in air from a convex spherical surface of glass of refractive index 1.5. The radius of curvature is 20 cm. Find the image distance.

Given

• n₁ = 1 for air.
• n₂ = 1.5 for glass.
• u = −100 cm.
• R = +20 cm.

Formula

Substitution

Result

The image is formed 100 cm from the spherical surface inside the glass medium.