Spherical Surfaces and Thin Lenses — Part 1: Refraction at a spherical surface and the lens maker’s formula: visual lesson, ISC Class 12 Physics

Lesson

Refraction at Spherical Surfaces and Thin Lenses

Part 1 — one curved surface, then two: from a glass rod with a rounded end to the lens maker’s formula.

In this lesson you will

Derive n₂/v − n₁/u = (n₂ − n₁)/R for a convex surface facing the rarer medium, and use it with the New Cartesian signs
Derive the lens maker’s formula and the thin lens formula for a biconvex lens, and apply them to other lens shapes
Predict how the focal length of a lens changes when it is placed in a liquid

Concept

Signs for refracting surfaces

We keep the New Cartesian convention from mirrors, with the light drawn travelling from left to right. A spherical refracting surface has a pole P, a centre of curvature C and a radius of curvature R.

For a surface, R is positive when C lies on the side the light is going to (convex towards the incoming light) and negative otherwise.
Results hold for paraxial rays: close to the axis and making small angles with it.

Table of the New Cartesian signs for refracting surfaces and lenses: distances from the pole, real object negative, real image on the far side positive, radius positive if the centre lies on the outgoing side, converging lens positive focal length
Quantity	Sign (light from left to right)
Distances u, v, f, R	Measured from the pole P (or the optical centre O of a lens)
Real object	$u < 0$
Real image formed by refraction	$v > 0$ (on the far side)
Radius of curvature R	Positive if the centre C lies on the side the light goes to
Converging lens / diverging lens	$f > 0$ / $f < 0$

Diagram

Setting up the derivation

A convex surface separates a rarer medium ( $n_1$ ) from a denser one ( $n_2$ ). A ray from O meets the surface at A and bends towards the normal CA, crossing the axis at the real image I. Let it make angles $\alpha$ , $\beta$ , $\gamma$ with the axis at O, I and C.

Triangle OAC (exterior angle at A): $i = \alpha + \gamma$ .
Triangle AIC (exterior angle at C): $\gamma = r + \beta$ , so $r = \gamma - \beta$ .
D is the foot of the perpendicular from A; for paraxial rays D is at P.

Cookies