Refraction at a Spherical Surface

What the Figure Shows

The figure depicts a convex spherical refracting surface — a single curved arc — that separates two transparent media. The medium to the left has refractive index $n_1$ (optically rarer), and the medium to the right has refractive index $n_2$ (optically denser, $n_2 > n_1$). The principal axis is a horizontal straight line passing through the centre of curvature $C$ and the pole $M$ of the spherical surface. The radius of curvature $R$ is the distance $MC$ along the axis.

A point object $O$ is placed on the principal axis in medium $1$. One ray from $O$ travels along the axis (the axial ray) and strikes the surface normally at $M$, continuing undeviated. Another ray from $O$ strikes the surface at point $N$ (not at the pole). At $N$, the ray refracts and bends toward the normal (since $n_2 > n_1$), and after refraction it meets the axial ray at point $I$ on the principal axis in medium $2$. Thus $I$ is the real image of $O$ formed by the spherical surface.

Key Geometrical Elements and Labels

• $O$: object point on the principal axis in medium $n_1$.
• $I$: image point on the principal axis in medium $n_2$.
• $C$: centre of curvature of the spherical surface, on the principal axis.
• $M$: pole of the spherical surface (the point where the principal axis meets the surface).
• $N$: point of incidence of the non-axial ray on the spherical surface.
• $R$: radius of curvature, $R = MC$.
• $n_1$, $n_2$: refractive indices of the two media.
• Angles marked: $\angle NOM$, $\angle NCM$, $\angle NIM$ — these are the angles that the incident ray, the radius, and the refracted ray make with the principal axis, respectively. The angle of incidence $i$ is the exterior angle of triangle $NOC$, and the angle of refraction $r$ is the difference $\angle NCM - \angle NIM$.

Physical Idea Taught

The figure illustrates how a single spherical surface can form an image by refraction. The derivation uses the small-angle approximation (aperture of the surface is small compared to distances $OM$, $MI$, $MC$), so that the perpendicular $NM$ is nearly equal to the arc length, and $\tan \theta \approx \theta$. By applying Snell's law ($n_1 \sin i = n_2 \sin r$) in the small-angle limit ($n_1 i = n_2 r$), and expressing $i$ and $r$ in terms of the distances $OM$, $MI$, $MC$, the textbook arrives at the fundamental relation for refraction at a spherical surface.

Key Formula Derived

The final result, after applying the Cartesian sign convention, is:

where: …