Q.Arrive at the lens equation from the lens maker's formula.

The Intuition: Why a Lens Bends Light

A lens works because light slows down when it enters glass. When a wavefront hits a curved surface at an angle, different parts of it slow down at different moments, and the wavefront bends. The stronger the curvature, the more it bends.

A lens has two surfaces. Each surface bends light by an amount that depends on its radius of curvature $R$ and the refractive index $n$ of the glass. The net bending — the focal length $f$ — is the combined effect of both surfaces.

If you had a single spherical surface separating air from glass, its contribution to bending power is $\frac{n-1}{R}$. A lens has two such surfaces: light goes from air into glass at the first surface, then from glass back into air at the second. Because the two surfaces face opposite directions relative to the travelling light, their radii typically carry opposite signs.

The Precise Statement

For a thin lens (thickness negligible compared to the radii), the Lens Maker's Formula is:

where:

• $f$ is the focal length of the lens (positive for converging, negative for diverging)
• $n$ is the refractive index of the lens material relative to the surrounding medium (usually air)
• $R_1$ is the radius of curvature of the first surface (the one light reaches first)
• $R_2$ is the radius of curvature of the second surface

How to Apply It: A Worked Example

Take a biconvex lens made of glass ($n = 1.5$) with both surfaces having the same radius of curvature magnitude, $20$ cm.

Light travels left to right. The first surface bulges toward the incoming light, so its centre of curvature lies to the right of the surface — on the side the light is travelling towards. By the rule above, $R_1 = +20$ cm.

The second surface also bulges outward (away from the lens), so its centre of curvature lies to the left of that surface — on the side the light is travelling from. So $R_2 = -20$ cm.

Plug in:

So $f = +20$ cm. Positive means converging — correct for a biconvex lens.

Why the Formula Works (Brief Derivation) …