Lenses and Image Formation — Converging vs Diverging

Q: What is the difference between converging and diverging lenses?

A converging (convex) lens is thicker in the middle and brings parallel rays to a focal point. A diverging (concave) lens is thinner in the middle and spreads rays apart, as if from a virtual focal point.

Q: How do I use the thin lens equation?

Use 1/f = 1/do + 1/di with the convention that distances on the outgoing-light side are positive. Converging lenses have positive f; diverging lenses have negative f. Positive di means a real image on the far side.

Q: Why does a magnifying glass make things bigger?

When the object is inside the focal length of a converging lens, the refracted rays diverge on the far side — your eye traces them back to a magnified upright virtual image on the same side as the object. That is exactly how a magnifying glass works.

Q: Which lens corrects nearsightedness vs farsightedness?

Nearsightedness (myopia): the eye focuses too strongly, so diverging lenses spread light before it enters. Farsightedness (hyperopia): the eye focuses too weakly, so converging lenses add bending power.

High School⏱ 10 min readOptics

Converging and diverging lenses: the thin lens equation 1/f = 1/do + 1/di, ray diagrams, magnification, and how cameras and glasses use lenses.

Quick answer

Lenses shape images by refraction: converging lenses focus light to form real or magnified virtual images; diverging lenses spread light and always give small upright virtual images. The thin lens equation 1/f = 1/do + 1/di locates any image.

$\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}$

Unit: All distances in the same length unit

Example: A camera lens with f = 50 mm photographing a mountain (object at infinity) forms the image 50 mm behind the lens — that is why the sensor sits one focal length back.

What you will learn

Distinguish converging from diverging lenses by shape and behavior
Apply the thin lens equation to locate images
Draw principal-ray diagrams for lenses
Compute magnification and interpret its sign
Explain lens uses in glasses, cameras, and magnifiers

Two shapes, two behaviors

A mirror folds light back; a lens lets it through, bending it twice — once entering the glass, once leaving. Shape those two refractions correctly and you get cameras, glasses, microscopes, and telescopes.

Converging (convex) lenses are fat in the middle: they bend parallel rays toward a focal point.
Diverging (concave) lenses are thin in the middle: they spread rays apart as if from a virtual focus.

A convex lens converges parallel rays to its focal point. The same geometry, reversed, describes image formation.

The thin lens equation

The thin lens equation is the mirror equation's twin, with one sign change in convention:

\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}

f = focal length (+ converging, − diverging); do, di measured from the lens

Magnification

m = −di/do = hi/ho. Negative m means inverted (real image); |m| > 1 means enlarged. A diverging lens always gives |m| < 1, upright and virtual.

Solving the lens equation

Worked example. A converging lens has f = 10 cm. An object stands 25 cm in front. Where is the image?

$\frac{1}{d_i} = \frac{1}{10} - \frac{1}{25} = \frac{5 - 2}{50} = \frac{3}{50}$

$d_i \approx 16.7$ cm on the far side — a real, inverted image. Magnification $m = -16.7/25 \approx -0.67$ : smaller and upside down, like a camera's sensor image.

Ray diagrams for lenses

Three principal rays predict every lens image:

A ray parallel to the axis refracts through the far focal point (converging) or as if from the near one (diverging).
A ray through the near focal point emerges parallel to the axis.
A ray through the lens center continues straight, barely bent.

Trace any two; their meeting point (real) or apparent origin (virtual) is the image. This is the single most exam-useful drawing skill in optics.

Eyes, glasses, and cameras

Your eye is a lens system with a fixed image distance (lens to retina) — it focuses by changing the lens shape, not position. When the eye cannot manage:

Myopia (nearsighted): eyeball too long; diverging glasses push the focus back onto the retina.
Hyperopia (farsighted): eyeball too short; converging glasses add focusing power.

Cameras do the opposite: fixed-shape glass, moving position. Autofocus slides lens groups until the sensor sits exactly at di — the lens equation, solved mechanically, thousands of times a day.

Variables and units

Symbol	Name	Unit	Meaning
$f$	Focal length	m or cm	Distance from lens center to focal point
$do$	Object distance	m or cm	Distance from lens to object
$di$	Image distance	m or cm	Distance from lens to image
$m$	Magnification	unitless	Image height over object height; negative = inverted

Common mistakes

✗ Lenses work by reflection.

Lenses work by refraction — light bends twice, entering and leaving the glass, per Snell's law. The curved surfaces are shaped so both bends cooperate toward focusing or diverging.

✗ Covering half a lens gives half an image.

Every part of the lens contributes to the whole image. Covering half only dims the image — it stays complete, because rays from each object point pass through the entire lens.

✗ The focal point is where images form.

The focal point is where parallel rays converge. Real objects at finite distances form images elsewhere, located by the thin lens equation.

Real-world applications

Cameras: compound lens systems focus light onto sensors, with autofocus shifting elements to satisfy the lens equation.
Eyeglasses: converging lenses correct farsightedness; diverging lenses correct nearsightedness.
Microscopes: objective plus eyepiece lenses compound magnification to reveal cells and bacteria.
Projectors: a bright converging lens throws a magnified real image onto a distant screen.

Frequently asked questions

What is the difference between converging and diverging lenses?

A converging (convex) lens is thicker in the middle and brings parallel rays to a focal point. A diverging (concave) lens is thinner in the middle and spreads rays apart, as if from a virtual focal point.

How do I use the thin lens equation?

Use 1/f = 1/do + 1/di with the convention that distances on the outgoing-light side are positive. Converging lenses have positive f; diverging lenses have negative f. Positive di means a real image on the far side.

Why does a magnifying glass make things bigger?

When the object is inside the focal length of a converging lens, the refracted rays diverge on the far side — your eye traces them back to a magnified upright virtual image on the same side as the object. That is exactly how a magnifying glass works.

Which lens corrects nearsightedness vs farsightedness?

Nearsightedness (myopia): the eye focuses too strongly, so diverging lenses spread light before it enters. Farsightedness (hyperopia): the eye focuses too weakly, so converging lenses add bending power.