Q.A spherical surface separates two transparent media. Derive an expression that relates object and image distances with the radius of curvature for a point object. Clearly state the assumptions, if any.

Refraction at a Spherical Surface

Imagine you are looking at a fish in a pond. The fish appears closer to the surface than it actually is. That is refraction — light bends when it moves from water into air. Now imagine the boundary between the two media is not flat, but curved like the surface of a lens or a glass marble. That is a spherical refracting surface.

The Intuition

When light hits a flat surface (like a glass slab), it bends once and travels straight. But when the surface is curved, something more interesting happens. Each ray of light strikes the curve at a slightly different angle, so each ray bends by a different amount. The result is that all rays leaving one point can be made to converge to (or diverge from) another point — forming an image.

Think of a spherical surface as a tiny piece of a sphere. The centre of that sphere is called the centre of curvature $C$, and the distance from the surface to $C$ is the radius of curvature $R$. The line joining the centre of curvature to the centre of the surface is the principal axis.

The Sign Convention

Before we write the formula, we need a consistent way to measure distances. The standard convention (Cartesian sign convention) is:

• Distances measured against the direction of incident light are negative.
• Distances measured along the direction of incident light are positive.
• The pole $P$ (the vertex of the spherical surface) is the origin.

So for a convex surface (bulging toward the incident light), $R$ is positive. For a concave surface (curving away), $R$ is negative. Object distance $u$ is always negative (object is on the incident side). Image distance $v$ can be positive or negative depending on where the image forms.

The Derivation in One Paragraph

Consider a point object $O$ on the principal axis. A ray from $O$ strikes the spherical surface at point $A$ and bends according to Snell's law: $n_1 \sin i = n_2 \sin r$. For small angles (paraxial approximation), $\sin\theta \approx \theta$, so $n_1 i = n_2 r$. Using geometry, $i = \alpha + \gamma$ and $r = \gamma - \beta$, where $\alpha$, $\beta$, $\gamma$ are the angles the ray makes with the principal axis at $O$, $I$, and $C$ respectively. Substituting and using $\alpha \approx \tan\alpha = \frac{AP}{PO}$, $\beta \approx \frac{AP}{PI}$, $\gamma \approx \frac{AP}{PC}$, and cancelling $AP$, you get the relation.

Here:

• $n_1$ = refractive index of the medium where the object lies
• $n_2$ = refractive index of the medium where the image forms
• $u$ = object distance from pole (negative)
• $v$ = image distance from pole (positive if real image on the opposite side)
• $R$ = radius of curvature (positive if centre of curvature is on the image side)

What This Formula Tells You

The formula is a single equation that connects four things: where the object is, where the image forms, how curved the surface is, and what the two media are. If you know any three, you can find the fourth.

For a convex surface (say, light going from air into glass through a convex surface), $R > 0$ and $n_2 > n_1$, so the right side is positive. This means $v$ is positive — the image forms on the other side. The surface converges light.

For a concave surface (light going from air into glass through a concave surface), $R < 0$, so the right side is negative. The surface diverges light. …