Lensmaker's Equation and Combinations of Lenses

What Decides a Lens's Focal Length?

Why does a thick, strongly curved lens bend light more than a thin, gently curved one? Why does a glass lens lose most of its power under water? And how do microscopes and cameras combine several lenses to get the focal length they need? The answers come from the lensmaker's equation, which links a lens's focal length to its shape and material, and from the rules for combining lenses. This tutorial derives and applies the lensmaker's equation, explains the sign convention, shows how a lens behaves in different media, and covers lenses in contact, separated lenses and two-lens image problems. It builds on the tutorial on lenses and the eye.

Sign Convention

This tutorial uses the real-is-positive convention common in Nigerian textbooks and WAEC/JAMB questions:

distances to real objects and real images are positive; distances to virtual ones are negative;
a converging (convex) lens has a positive focal length; a diverging (concave) lens has a negative one;
in the lensmaker's equation, a surface that bulges outwards (convex to the outside air) has a positive radius of curvature; a surface that curves inwards (concave) has a negative radius; a flat surface has an infinite radius, so 1/R = 0.

The lens formula is then 1/f = 1/u + 1/v, and the power of a lens is P = 1/f (in dioptres, D, with f in metres).

The Lensmaker's Equation

For a thin lens of refractive index n in air, with surfaces of radii R₁ and R₂:

1/f = (n − 1)(1/R₁ + 1/R₂).

The formula shows two things:

Curvature: more strongly curved surfaces (smaller R) give a larger 1/f, so a shorter focal length and a more powerful lens.
Material: a higher refractive index n gives a larger (n − 1) and a stronger lens. This is why high-index plastic spectacle lenses can be made thinner for the same prescription.

The equation comes from applying the refraction formula at each surface in turn: the first surface forms an image, which acts as the object for the second surface. For a thin lens the thickness is ignored.

Worked Examples Using the Lensmaker's Equation

Example 1. A biconvex lens of glass (n = 1.5) has both surfaces of radius 20 cm. Find its focal length and power.

Both surfaces are convex, so R₁ = R₂ = +20 cm. 1/f = 0.5 × (1/20 + 1/20) = 0.5 × 0.1 = 0.05, so f = 20 cm and P = 1/0.20 m = +5 D.

A useful shortcut: for a symmetric biconvex lens with n = 1.5, f = R.

Example 2. A plano-convex lens (one flat side) has a curved surface of radius 15 cm and n = 1.5. Find f.

1/f = 0.5 × (1/15 + 0) = 1/30, so f = 30 cm. Flattening one side doubles the focal length compared with a biconvex lens of the same radii.

Example 3. A biconcave lens of n = 1.6 has surfaces of radius 25 cm and 40 cm. Find f.

Both surfaces are concave, so R₁ = −25 cm and R₂ = −40 cm. 1/f = 0.6 × (−1/25 − 1/40) = 0.6 × (−0.065) = −0.039, so f ≈ −25.6 cm, a diverging lens.

Example 4 (finding a radius). A symmetric biconvex lens of focal length 12 cm is to be made from glass of n = 1.6. What radius should both surfaces have?

1/12 = 0.6 × (2/R), so 2/R = 1/7.2 and R = 14.4 cm.

A Lens in a Different Medium

If the lens is surrounded by a medium of refractive index nm (such as water), only the relative refractive index matters:

1/f = (n/nm − 1)(1/R₁ + 1/R₂).

Example 5. The lens in Example 1 (f = 20 cm in air) is placed in water (nm = 1.33). Find its new focal length.

n/nm = 1.5/1.33 = 1.128, so (n/nm − 1) = 0.128. 1/f = 0.128 × 0.1 = 0.0128, so f ≈ 78 cm. The lens is now nearly four times weaker, because glass and water bend light by almost the same amount.

This explains why your vision is blurred under water: the cornea, which does most of the eye's focusing, has a refractive index close to that of water and loses almost all its power. Goggles restore a layer of air in front of the eyes. If a lens is placed in a liquid with a higher refractive index than its own, (n/nm − 1) becomes negative and a convex lens becomes diverging.

Thin Lenses in Contact

When thin lenses touch, their powers add:

P = P₁ + P₂, or 1/f = 1/f₁ + 1/f₂.

Example 6. A converging lens of focal length 20 cm is in contact with a diverging lens of focal length −30 cm. Find the focal length of the combination.

P = 5 D + (−3.33 D) = 1.67 D, so f = 1/1.67 = 0.60 m = 60 cm. The combination is converging, but weaker than the converging lens alone.

This idea is used in achromatic doublets: a converging crown-glass lens cemented to a weaker diverging flint-glass lens. Flint glass spreads colours more, so the diverging lens cancels the colour spreading (chromatic aberration) of the converging lens while the pair still converges.

Two Separated Lenses

For two thin lenses a distance d apart, the combined focal length is

1/f = 1/f₁ + 1/f₂ − d/(f₁f₂).

For example, lenses of 10 cm and 15 cm separated by 5 cm give 1/f = 1/10 + 1/15 − 5/150 = 0.1333, so f = 7.5 cm.

For image problems, however, it is usually clearer to treat the lenses one at a time: the image formed by the first lens becomes the object for the second.

Example 7. An object is 30 cm in front of lens A (f = 10 cm). Lens B (f = 15 cm) is 25 cm behind lens A. Find the final image.

Lens A: 1/v = 1/10 − 1/30 = 2/30, so v₁ = 15 cm. A real, inverted image forms 15 cm behind A, with magnification 15/30 = 0.5.
Lens B: this image is 25 − 15 = 10 cm in front of B, so u₂ = 10 cm (a real object). 1/v = 1/15 − 1/10 = −1/30, so v₂ = −30 cm: a virtual image 30 cm in front of B, on the same side as the light comes from. Magnification = 30/10 = 3.
Overall: total magnification = 0.5 × 3 = 1.5. The first image was inverted, and lens B acting as a magnifying glass keeps it the same way up, so the final image is virtual, inverted relative to the original object, and 1.5 times its size.

This is exactly how a compound microscope works: the objective forms a real, magnified image, and the eyepiece acts as a magnifying glass to view it, multiplying the magnification further.

Summary

Lensmaker's equation: 1/f = (n − 1)(1/R₁ + 1/R₂), with convex surfaces positive and flat surfaces 1/R = 0.
Stronger curvature or a higher refractive index gives a shorter focal length.
In another medium, use n/nm; a glass lens is much weaker in water.
Thin lenses in contact: powers add, P = P₁ + P₂.
For separated lenses, treat the first image as the object for the second lens and multiply the magnifications.