Q.Derive the equation for refraction at a single spherical surface.

Refraction at a Spherical Surface

Imagine you're looking at a fish in a pond. The fish appears closer to the surface than it actually is. That's refraction — light bends when it moves from water to air. Now take that idea and replace the flat water surface with a curved one, like a glass lens or a drop of water. That's refraction at a spherical surface.

The core intuition

When light hits a flat surface (like a glass slab), it bends once and travels straight. But when the surface is curved — part of a sphere — the angle at which light hits changes depending on where on the surface it strikes. A ray hitting near the centre meets the surface almost head-on; a ray hitting near the edge meets it at a steep slant. This variation in incidence angle is what makes spherical surfaces focus or diverge light.

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

The precise geometry

We need to track what happens to a ray from an object point $O$ on the principal axis. The ray travels in medium 1 (refractive index $n_1$), hits the spherical surface at point $A$, and enters medium 2 (refractive index $n_2$). The surface has radius $R$, with centre $C$.

The key is Snell's law at point $A$:

But $i$ and $r$ are measured from the normal at $A$. For a spherical surface, the normal at any point is the line joining that point to $C$. So the normal is $AC$.

For small angles (paraxial rays — rays close to the axis), $\sin \theta \approx \theta$ in radians. This approximation is the backbone of all standard lens and mirror formulas. It lets us replace Snell's law with:

Now look at the geometry. Let the object distance from the pole be $u$ (negative by sign convention — object on left), and the image distance be $v$ (positive if image is on the right, in medium 2). The angle the incident ray makes with the axis is $\alpha$, the refracted ray makes $\beta$, and the normal makes $\theta$ with the axis.

From the triangles:

• In $\triangle OAC$: $i = \alpha + \theta$
• In $\triangle AIC$: $r = \theta - \beta$ (for a convex surface towards the object)

Substitute into Snell's law:

For small angles, $\alpha \approx \frac{AP}{PO} \approx \frac{h}{-u}$ (since $u$ is negative), $\beta \approx \frac{h}{v}$, and $\theta \approx \frac{h}{R}$.

Plugging these in:

Cancel $h$ (non-zero) and rearrange:

This is the refraction at a spherical surface formula. It relates object distance $u$, image distance $v$, radii $R$, and the two refractive indices.

Sign convention (crucial for exams)

Use the Cartesian sign convention (the one used in NCERT and most Indian boards):

• Distances measured from the pole $P$ along the principal axis.
• Positive in the direction of incident light (usually left to right).
• Negative opposite to incident light.
• $R$ is positive if the centre of curvature $C$ is on the right (convex surface towards object), negative if $C$ is on the left (concave surface towards object).

What the formula tells you

• If $n_2 > n_1$ (going from rarer to denser), the right side $\frac{n_2 - n_1}{R}$ is positive for a convex surface. This means $v$ is positive — the image forms on the other side (real image). …