IMAT Physics · Optics

Lenses, Power and Vision Defects

A converging lens brings light to a focus and a diverging lens spreads it; the thin lens equation places the image, and lens power in dioptres sets the glasses that correct the eye.

All Optics notes

Why this matters

No past question has been set on lenses or the eye yet. They are core syllabus and an obvious place for a medicine-themed question, such as the power of the lens that corrects short sight.

Concept 1 of 3: Converging and diverging lenses and the thin lens equation

A converging (convex) lens is thicker in the middle; it bends parallel rays together to a real focus. A diverging (concave) lens is thinner in the middle; it spreads them out as if they came from a focus in front of it. One equation links the object distance, the image distance and the focal length, and the signs tell you whether the image is real or virtual.

Definition

Converging lens: focal length $f > 0$ . Object beyond $F$ : real, inverted image on the far side. Object inside $F$ : virtual, upright, magnified image on the same side (a magnifying glass).
Diverging lens: $f < 0$ . It always gives a virtual, upright, smaller image.
Sign rule used here: $d_o$ positive for a real object; $d_i$ positive for a real image (far side), negative for a virtual image (same side as the object).
Magnification $m = -d_i/d_o$ is the ratio of image height to object height. Negative $m$ means inverted; $|m| > 1$ means enlarged.
An object exactly at $F$ of a converging lens gives parallel rays: the image is at infinity.

Thin lens equation and magnification

\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \qquad m = -\frac{d_i}{d_o} = \frac{h_i}{h_o}

$f$ focal length: positive for converging, negative for diverging
$d_o$ distance of the object from the lens
$d_i$ distance of the image: positive real, negative virtual
$h_o,\ h_i$ heights of object and image

Worked example

An object stands 30 cm from a converging lens of focal length 20 cm. Find the position and nature of the image, and its magnification.

Practice this conceptself-check · 3 quick reps

A converging lens does not always give a real image

With the object closer to a converging lens than its focal point, the image is virtual, upright and magnified, on the same side as the object. That is how a magnifying glass works.

Concept 2 of 3: Lens power in dioptres

A strongly curved lens bends light sharply and has a short focal length. Power is just the reciprocal of the focal length, so a stronger lens has a bigger number. Opticians use it because the powers of thin lenses placed together simply add.

Definition

The power of a lens is $P = 1/f$ , with $f$ in metres. The unit is the dioptre (D), equal to $\text{m}^{-1}$ .
Converging lenses have positive power, diverging lenses negative power.
For thin lenses in contact, the powers add: $P = P_1 + P_2$ .

Power of a lens

P = \frac{1}{f} \qquad P_{\text{total}} = P_1 + P_2

$P$ power, in dioptres (D)
$f$ focal length, in metres

Worked example

Find the power of a converging lens with

f = 25\ \text{cm}

and of a diverging lens with

f = -50\ \text{cm}

. What is the focal length of the two placed together?

Practice this conceptself-check · 3 quick reps

Power needs the focal length in metres

A lens with

f = 20\ \text{cm}

has

P = 1/0.20 = 5\ \text{D}

. Using 20 instead of 0.20 gives 0.05 D, a hundred times too small, and that slip is a favourite wrong option.

Concept 3 of 3: The human eye, short sight and long sight

The eye is a converging lens system that throws a real, inverted image onto the retina. Most of the bending happens at the cornea; the lens fine-tunes it by changing shape. If the eyeball is too long or too short for its lenses, the image lands in front of or behind the retina, and glasses shift it back.

Definition

The cornea does most of the focusing; the lens adjusts it (accommodation): the ciliary muscles let the lens become fatter and more powerful for near objects.
A normal eye sees clearly from the near point (about 25 cm) to the far point (infinity).
Myopia (short sight): distant objects are blurred; the image forms in front of the retina. Corrected by a diverging lens, whose focal length equals minus the far point distance.
Hypermetropia (long sight): near objects are blurred; the image would form behind the retina. Corrected by a converging lens.

Part or defect	What it does or what goes wrong	Note or correction
Cornea	Does most of the focusing, at the curved air to cornea surface	Its power is fixed
Lens and ciliary muscles	Fine focusing: the muscles change the lens's shape (accommodation)	The lens becomes fatter for near objects
Iris and pupil	The iris changes the size of the pupil to control how much light enters	The pupil narrows in bright light
Retina	Light-sensitive layer where a real, inverted image forms	The brain interprets the image upright
Myopia (short sight)	Eyeball too long or lens too strong: image in front of the retina	Diverging lens, negative power
Hypermetropia (long sight)	Eyeball too short or lens too weak: image behind the retina	Converging lens, positive power
Presbyopia	With age the lens stiffens and cannot accommodate for near objects	Converging reading glasses

Practice this conceptself-check · 4 quick reps

Short sight needs a diverging lens

A myopic eye focuses too strongly, so the correction must weaken it: a diverging lens with negative power. Long sight is the opposite and needs a converging lens. Swapping the two is the usual wrong option.

The cornea, not the lens, does most of the focusing

The biggest change of refractive index is between air and the cornea, so most of the bending happens there. The lens only fine-tunes the focus by changing shape.

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (2)

Converging and diverging lenses and the thin lens equation
Thin lens equation and magnification
$\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \qquad m = -\frac{d_i}{d_o} = \frac{h_i}{h_o}$
Lens power in dioptres
Power of a lens
$P = \frac{1}{f} \qquad P_{\text{total}} = P_1 + P_2$

Reference tables (1)

The human eye, short sight and long sight7 rows

Part or defect	What it does or what goes wrong	Note or correction
Cornea	Does most of the focusing, at the curved air to cornea surface	Its power is fixed
Lens and ciliary muscles	Fine focusing: the muscles change the lens's shape (accommodation)	The lens becomes fatter for near objects
Iris and pupil	The iris changes the size of the pupil to control how much light enters	The pupil narrows in bright light
Retina	Light-sensitive layer where a real, inverted image forms	The brain interprets the image upright
Myopia (short sight)	Eyeball too long or lens too strong: image in front of the retina	Diverging lens, negative power
Hypermetropia (long sight)	Eyeball too short or lens too weak: image behind the retina	Converging lens, positive power
Presbyopia	With age the lens stiffens and cannot accommodate for near objects	Converging reading glasses

Watch out for (4)

A converging lens does not always give a real image
→ Converging and diverging lenses and the thin lens equation
Power needs the focal length in metres
→ Lens power in dioptres
Short sight needs a diverging lens
→ The human eye, short sight and long sight
The cornea, not the lens, does most of the focusing
→ The human eye, short sight and long sight

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