Refraction at a Spherical Surface

Lenses are built from spherical refracting surfaces. To derive the lens formula we first need the relation between object and image distances for a single curved interface separating two media.

Concept

Setup: a spherical surface of centre $C$ and radius $R$ separates medium 1 (refractive index $n_1$) from medium 2 (refractive index $n_2$). A point object $O$ lies on the axis in medium 1; its image $I$ forms in medium 2. All distances are measured from the pole $P$ with the Cartesian sign convention.

For a surface convex towards the incident light, $R > 0$.

Derivation

Let a paraxial ray from $O$ strike the surface at $B$. Let

• $\angle BOP = \alpha$ (small)
• $\angle BCP = \beta$
• $\angle BIP = \gamma$

so the incidence angle $i = \alpha + \beta$ (exterior angle of $\triangle OBC$), and the refraction angle $r = \beta - \gamma$.

For paraxial (small) angles, $\sin\theta \approx \tan\theta \approx \theta$ in radians, and

$\alpha \approx \frac{h}{-u},\quad \beta \approx \frac{h}{R},\quad \gamma \approx \frac{h}{v}$

where $h$ is the perpendicular height of $B$ above the axis.

Snell's law in the paraxial limit: $n_1 i = n_2 r$, so

$n_1 (\alpha + \beta) = n_2 (\beta - \gamma)$

Substituting,

$n_1 \left(\frac{h}{-u} + \frac{h}{R}\right) = n_2 \left(\frac{h}{R} - \frac{h}{v}\right)$

Cancel $h$ and rearrange:

$\frac{n_2}{v} - \frac{n_1}{u} = \frac{n_2 - n_1}{R}$

This is the refraction-at-spherical-surface formula, valid with the Cartesian sign convention for any combination of object/image positions and surface orientation.

Worked Example

A point object lies 30 cm in front of a convex glass surface ($n = 1.5$) of radius 20 cm, in air. Find the image position.

Take $n_1 = 1$, $n_2 = 1.5$, $u = -30$ cm, $R = +20$ cm.

$\frac{1.5}{v} - \frac{1}{-30} = \frac{1.5 - 1}{20}$

$\frac{1.5}{v} = \frac{0.5}{20} - \frac{1}{30} = 0.025 - 0.0333 = -0.00833$

$v = \frac{1.5}{-0.00833} \approx -180\text{ cm}$

A virtual image, 180 cm on the same side as the object.

Common Confusions

• Sign of $R$: convex towards incident light means $R > 0$. A concave surface (centre on the incident side) gives $R < 0$.
• Don't substitute $u$ with a positive number for a real object. The formula's sign structure depends on $u$ being negative.
• The formula is for one surface only; a lens has two surfaces (combine them to obtain the lens-maker's formula).

Key Takeaways

• Refraction at a single spherical surface: $\frac{n_2}{v} - \frac{n_1}{u} = \frac{n_2 - n_1}{R}$.
• Paraxial approximation: small angles, near-axial rays.
• Sign convention is essential; use the standard Cartesian rules.
• This formula is the building block of the lens-maker's formula.