Lesson 8 of 10Light & Geometric Optics

Unit 4 · Lesson 8 of 10

Converging and Diverging Lenses with Ray Diagrams

How a curved piece of glass uses refraction to bring light together or spread it apart, the three principal rays for each kind of lens, and how the image changes as the object moves, from cameras to projectors to magnifying glasses.

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Key things to remember

1Converging (convex) lenses are thicker in the middle; diverging (concave) lenses are thinner in the middle.
2Converging rays: parallel → through F′; through O → straight; through F → parallel.
3Diverging rays: parallel → as if from F (dashed); through O → straight; toward F′ → parallel.
4Converging lens: object beyond 2F gives a smaller, inverted, real image (camera); between 2F and F gives a larger, inverted, real image (projector); inside F gives a larger, upright, virtual image (magnifier).
5Diverging lens: always smaller, upright, virtual, on the object's side between F and the lens.
6Real lens images form on the far side of the lens; virtual ones on the object's side.

Hold a magnifying glass in sunlight and you can focus the Sun into a tiny bright spot. Hold the same lens close to a page and the letters grow. Hold it at arm's length toward a window and you see the scene upside down. One lens, three very different images. A lens is a transparent object with curved surfaces that refracts light twice, once entering and once leaving, and the shape of the curves decides whether the rays come together or spread apart.

AConverging (convex) lens

Thicker in the middle than at the edges
Bends parallel rays together so they meet at a real focal point
Can make real or virtual images
Magnifying glass, camera, projector, eyeglasses for farsightedness

BDiverging (concave) lens

Thinner in the middle than at the edges
Spreads parallel rays apart as if from a virtual focal point
Always makes small, upright, virtual images
Peephole viewer, eyeglasses for nearsightedness

Parts of a lens

Optical centre (O): the centre of the lens. A ray through it passes straight through.
Principal axis: the line through O perpendicular to the lens.
Focal points: a lens has one on each side, the same distance $f$ from O because light can pass through either way. Here F is on the object's side and F′ (F-prime) is on the far side. (Some resources swap these two labels; always check which side is which.)
2F and 2F′: points at twice the focal length. An object at 2F makes an image at 2F′.
In ray diagrams the lens is drawn as a vertical line, and all the bending is shown happening at that line. This works well for thin lenses.

Principal rays for a converging lens

Ray 1 travels parallel to the principal axis and refracts through F′ on the far side. Ray 2 passes straight through the optical centre without bending. Ray 3 passes through F on the object's side and refracts parallel to the principal axis. Two rays locate the image; the third checks it.

Fig 1A converging lens with F and 2F marked on the object's side and F′ and 2F′ on the far side. The object stands beyond 2F; the parallel ray refracts through the far focal point, the centre ray goes straight through, and the ray through the near focal point leaves parallel. They meet between F′ and 2F′ at a real, inverted, smaller image.

Object position (converging lens)	Size	Attitude	Location	Type	Use
Beyond 2F	Smaller	Inverted	Between F′ and 2F′	Real	Camera, eye
At 2F	Same size	Inverted	At 2F′	Real	Photocopier
Between 2F and F	Larger	Inverted	Beyond 2F′	Real	Movie and slide projector
At F	No clear image	—	—	Rays leave parallel	Spotlight, lighthouse
Between F and the lens	Larger	Upright	Same side as the object	Virtual	Magnifying glass

The pattern matches the concave mirror, except that a lens forms real images on the far side, where the light has gone through.

Where do real images form?

A mirror sends light back, so its real images are in front of the mirror, on the object's side. A lens lets light through, so its real images are on the opposite side from the object, where a screen can catch them. Virtual images from a lens appear on the same side as the object.

Diverging lenses

For a diverging lens, Ray 1 travels parallel to the principal axis and refracts outward as if it came from F on the object's side; draw a dashed extension back to F. Ray 2 passes straight through the optical centre. Ray 3 is aimed at F′ on the far side and refracts parallel to the principal axis. The refracted rays spread apart, so only their dashed extensions meet, on the object's side of the lens.

Fig 2A diverging lens. The parallel ray refracts outward as if it came from the near focal point (dashed extension), the ray aimed at the far focal point leaves parallel, and the centre ray goes straight through. The image is virtual, upright and smaller, on the object's side between F and the lens.

Cover half the lens and you still see the whole image

Every point on the object sends rays through every part of the lens, and all of them meet at the same image point. Block half the lens and the remaining half still brings rays from every object point together, so the whole image forms, just dimmer. Ray diagrams use only three convenient rays, but countless others make the image.

Loading interactive…

Switch to a converging or diverging lens and drag the object past 2F, F and inside F to watch the principal rays and the image change.

Worked examples

Reveal one step at a time and try to beat it to the next line.

Example 1

A slide projector places the slide between 2F and F of a converging lens. Use the principal rays to describe the image and explain why slides are loaded upside down.

Example 2

You hold a converging lens (

f = 10 cm

)

6 cm

above a stamp. Describe the image and explain why this lens works as a magnifier.

Example 3

A peephole in a door uses a diverging lens. Describe the image of a visitor and explain the advantage.

Where marks get lost

Mixing up lens and mirror names. A convex lens converges light, but a convex mirror diverges it. Go by what the light does: converging or diverging.
Drawing the ray through the optical centre as bending. It passes straight through.
Putting a lens's real image on the same side as the object. Light passes through a lens, so real images are on the far side.
Using the far focal point F′ for the ray that should go through the near focal point F. The ray that leaves parallel is the one that passed through F on the object's side.
Thinking that covering half a lens cuts off half the image. The whole image still forms, only dimmer.

Flashcards 9

Converging (convex) lens

Diverging (concave) lens

Optical centre

Converging lens principal rays

+ 5 more

Practice

Which lens is thicker in the middle than at its edges?

Match each object position for a converging lens to the image it forms.

Beyond 2F

At 2F

Between 2F and F

Between F and the lens

A camera must make a small, real image of a distant scene on its sensor. Where is the scene relative to the converging lens?

A ray travelling parallel to the principal axis enters a diverging lens. Which describes the refracted ray?

An object is placed exactly at 2F of a converging lens with

f = 15 cm

. How far from the lens is the image, in cm? (Type the number.)

Half of a converging lens is covered with black tape. What happens to the image on a screen?

Which type of image can a diverging lens form?

Sort the steps for drawing a converging-lens ray diagram.

1Describe the image using SALT
2Mark the image tip where the refracted rays (or their dashed extensions) cross
3From the object's tip, draw two principal rays: one parallel to the axis that refracts through F′, and one straight through the optical centre
4Draw the principal axis, the lens line and mark F, 2F, F′ and 2F′
5Draw the object as an upright arrow at the given position

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