What a Spherical Refracting Surface Is
A spherical refracting surface is the curved boundary that separates two transparent media of different refractive indices, where at least one medium is bounded by a portion of a sphere. Glass marbles, water drops and the curved face of a glass paperweight are everyday examples; NIOS §20.5 lists exactly such objects. NCERT §9.5 notes that an infinitesimal part of a spherical surface can be regarded as planar, so the ordinary laws of refraction apply at every point on it.
The key geometric fact is that the normal at the point of incidence on a spherical surface is perpendicular to the tangent plane there and therefore passes through the centre of curvature $C$. This is the same property used for spherical mirrors. Because the centre of curvature lies on the principal axis, the normal at any near-axis point is essentially a radial line from $C$.
Refraction at one such surface is studied first, in isolation. A thin lens, by contrast, is bounded by two surfaces; applying the single-surface result successively at both of them yields the lens maker's formula. The single surface is the irreducible building block, and that is the scope of this page.
Light travels from medium $n_1$ into medium $n_2$ across a convex surface. The normal at $N$ passes through the centre of curvature $C$; the refracted ray bends toward this normal because $n_2 > n_1$.
The Governing Relation
For a single spherical refracting surface, NCERT Eq. (9.16) gives the relation between the object distance $u$ and the image distance $v$:
$$\frac{n_2}{v} - \frac{n_1}{u} = \frac{n_2 - n_1}{R}$$
Here $n_1$ is the refractive index of the medium from which light is incident, $n_2$ is the index of the medium into which it refracts, and $R$ is the radius of curvature of the surface. NCERT states explicitly that this relation holds for any curved spherical surface, convex or concave. The two indices enter asymmetrically: $n_2$ multiplies the image term and $n_1$ multiplies the object term, so the order of the media can never be swapped.
| Symbol | Meaning | Sign in standard figure |
|---|---|---|
n₁ | Index of medium of incidence (where the object sits) | Always positive |
n₂ | Index of medium of refraction (beyond the surface) | Always positive |
u | Object distance from pole P | Negative (object left of P) |
v | Image distance from pole P | Sign from calculation |
R | Radius of curvature (pole to centre C) | + if C right of P, − if left |
Deriving the Formula
NCERT considers an object $O$ on the principal axis and a ray meeting the surface at a near-axis point $N$ (Fig. 9.15). With $M$ the foot of the perpendicular from $N$ to the axis, the small angles satisfy $\tan\angle NOM = \tfrac{MN}{OM}$, $\tan\angle NCM = \tfrac{MN}{MC}$ and $\tan\angle NIM = \tfrac{MN}{MI}$. Treating $i$ as the exterior angle of triangle $NOC$ gives $i = \angle NOM + \angle NCM$, and $r = \angle NCM - \angle NIM$ for the refracted ray.
Snell's law at the surface reads $n_1 \sin i = n_2 \sin r$. For paraxial rays the angles are small, so $\sin i \approx i$ and $\sin r \approx r$, giving the linearised form $n_1 i = n_2 r$. NIOS §20.5 obtains the identical relation $i = \mu r$ for an air–glass surface, with the same small-angle reasoning.
Substituting the angle expressions and writing each $\tan\theta \approx \theta$ produces $\dfrac{n_1}{OM} + \dfrac{n_2}{MI} = \dfrac{n_2 - n_1}{MC}$, where $OM$, $MI$ and $MC$ are magnitudes. Applying the Cartesian sign convention $OM = -u$, $MI = +v$, $MC = +R$ converts these magnitudes into signed coordinates and delivers Eq. (9.16).
Exterior-angle geometry: $i = \angle NOM + \angle NCM$, $r = \angle NCM - \angle NIM$.
Paraxial Snell's law: $n_1 i = n_2 r$.
Magnitude relation: $\dfrac{n_1}{OM} + \dfrac{n_2}{MI} = \dfrac{n_2 - n_1}{MC}$.
Apply $OM = -u,\; MI = +v,\; MC = +R$ ⟶ $\dfrac{n_2}{v} - \dfrac{n_1}{u} = \dfrac{n_2 - n_1}{R}$.
The paraxial approximation is not optional
The clean relation exists only because $\sin i \approx i$ and $\tan\theta \approx \theta$ were used. These hold solely for rays close to the axis making small angles. A question that places the object far off-axis, or asks about marginal (wide-angle) rays, is signalling that the simple formula and a single sharp image no longer apply.
Use $\dfrac{n_2}{v} - \dfrac{n_1}{u} = \dfrac{n_2 - n_1}{R}$ only for paraxial rays.
Following NIOS Fig. 20.11: angles $\alpha,\beta,\gamma$ are subtended at $O$, $I$ and $C$. With $i = \alpha + \gamma$ and $r = \beta + \gamma$, the paraxial Snell relation reduces to the single-surface formula.
Sign Convention for a Single Surface
Distances are measured from the pole $P$, with the direction of incident light taken as positive, following the Cartesian convention shared with spherical mirrors (NIOS §20.5). In the standard NCERT figure the object lies to the left, so $u$ is negative. The radius $R$ is positive when the centre of curvature $C$ lies on the outgoing side (the same side as the refracted light) and negative when $C$ lies on the incoming side.
| Surface shape (light incident from left) | Centre of curvature C | Sign of R |
|---|---|---|
| Convex toward the incoming light | On the outgoing (right) side | + R |
| Concave toward the incoming light | On the incoming (left) side | − R |
Identify n₁ and n₂ by the direction of travel
Light always travels from $n_1$ into $n_2$. The medium containing the object is $n_1$; the medium on the far side is $n_2$. Reversing them silently flips the right-hand side and the image position. If a problem sends light from glass into air, then $n_1 = 1.5$ and $n_2 = 1$, not the other way round. Fix the direction of incidence first, then label the indices.
$n_1 \to n_2$ follows the light. Object medium is always $n_1$.
Applying this surface relation twice gives the Lens Maker's Formula.
Worked Example from NCERT
NCERT Example 9.5 applies Eq. (9.16) directly to a single glass surface. It illustrates how the signed substitution produces a real image even when the image lies inside the denser medium.
Light from a point source in air falls on a spherical glass surface ($n = 1.5$, radius of curvature $20$ cm). The source is $100$ cm from the surface. Where is the image formed?
Using $\dfrac{n_2}{v} - \dfrac{n_1}{u} = \dfrac{n_2 - n_1}{R}$ with $u = -100$ cm, $R = +20$ cm, $n_1 = 1$, $n_2 = 1.5$:
$$\frac{1.5}{v} - \frac{1}{-100} = \frac{1.5 - 1}{20} = \frac{0.5}{20}$$
$$\frac{1.5}{v} = \frac{0.5}{20} - \frac{1}{100} = \frac{2.5}{100} - \frac{1}{100} = \frac{1.5}{100}$$
Hence $v = +100$ cm. The image is formed $100$ cm from the glass surface, in the direction of the incident light. The positive sign marks it as a real image inside the glass, exactly as NCERT states.
Why It Matters for Lenses
NCERT §9.5.2 builds the lens directly on this surface relation. A double convex lens is treated as two refractions in succession: the first surface forms an intermediate image $I_1$ of the object $O$, and $I_1$ then acts as the object for the second surface, which forms the final image $I$. Equation (9.16) is written for each interface in turn.
Adding the two equations and using the thin-lens assumption eliminates the intermediate image distance, after which the two radii $R_1$ and $R_2$ combine into the lens maker's formula. The single focal length of a thin lens emerges only from this combination; a lone refracting surface, with different media on its two sides, does not possess one symmetric focal length.
One surface has no single focal length
Because the medium differs on the two sides of a single surface, its first and second focal distances are unequal, and the mirror-style $\tfrac{1}{v}-\tfrac{1}{u}=\tfrac{1}{f}$ does not apply. The relation must stay in its $n$-and-$R$ form. A single focal length appears only after two surfaces are combined into a thin lens.
Keep the relation as $\dfrac{n_2}{v}-\dfrac{n_1}{u}=\dfrac{n_2-n_1}{R}$; reduce to a lens only with two surfaces.
Refraction at a Spherical Surface in One Glance
- Single-surface relation: $\dfrac{n_2}{v} - \dfrac{n_1}{u} = \dfrac{n_2 - n_1}{R}$ (NCERT Eq. 9.16).
- $n_1$ is the object-side medium, $n_2$ the far-side medium; light travels $n_1 \to n_2$.
- Derived from paraxial Snell's law $n_1 i = n_2 r$ with $\tan\theta \approx \theta$.
- Cartesian signs: $u$ negative for a left-side object; $R$ positive if $C$ is on the outgoing side.
- NCERT Example 9.5: $u=-100$, $R=+20$, $n_1=1$, $n_2=1.5$ give $v=+100$ cm, a real image.
- Applied at both surfaces of a thin lens, it yields the lens maker's formula.