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E. Physics: Light and Geometric Optics · Light and Geometric Optics

Lenses and Ray Diagrams for Converging Lenses

E3.5 E3.4 E2.5 E2.1 A1.5About 75 minutes
Learning goals
  • I can identify converging and diverging lenses and describe what each does to parallel light.
  • I can explain that light refracts at both surfaces of a lens, with some partial reflection.
  • I can draw ray diagrams for converging lenses using the three rules.
  • I can describe the image for any object position using SALT and name a use.

Words to know

lens
A transparent object with at least one curved surface that refracts light to form an image.
converging lens
A lens that is thicker in the middle and brings parallel rays together (convex lens).
diverging lens
A lens that is thinner in the middle and spreads parallel rays apart (concave lens).
optical centre (O)
The centre of a lens; rays through it pass straight through.
principal focus (F)
For a converging lens, the point on the far side where parallel rays meet.
secondary focus (F′)
The focal point on the opposite side of the lens from F, the same distance from O.
thin lens model
Drawing all the refraction of a lens as happening at one line through its centre.

A curved piece of glass that can do almost anything

A lens is just a carefully shaped piece of glass or plastic, but it lets you read small print, see distant galaxies, photograph your friends, and see clearly if you need glasses. A lens is a transparent object with at least one curved surface that refracts light to form an image.

Two kinds of lenses

  • A converging lens (convex lens) is thicker in the middle than at the edges. It bends parallel rays towards each other so they meet at a focal point. Magnifying glasses and camera lenses are converging.
  • A diverging lens (concave lens) is thinner in the middle. It spreads parallel rays apart, as if they came from a focal point on the near side. Glasses for nearsighted people use diverging lenses.
Fconverging (convex) lensFdiverging (concave) lens
Parallel rays through a converging lens meet at the principal focus F. Through a diverging lens they spread apart as if they came from F on the near side (dashed lines).

How a lens bends light: two refractions

Light refracts twice: once entering the lens and once leaving it. At each surface the normal is a line from that surface's centre of curvature. Near the edges of a converging lens, the two surfaces are tilted like a prism, so rays there are bent the most. Near the centre, the surfaces are almost parallel, so a ray through the centre is hardly bent at all.

At each surface a little light is also partially reflected. That is why camera lenses and good eyeglasses have anti-reflective coatings.

In ray diagrams we use a thin lens model: we draw the lens as a thin shape and show all the bending happening at one vertical line through the middle of the lens. This gives the same image positions as a real thin lens.

crosses the axis near F1st bend2nd benddotted: weak partial reflections at each surface
One ray traced exactly through a real glass lens (n = 1.52) using Snell's law at each surface. The ray bends towards the normal entering the glass and away from the normal leaving it; both bends turn it towards the axis, which it crosses near F.

Lens vocabulary

  • Optical centre (O): the exact centre of the lens.
  • Principal axis: the line through O perpendicular to the lens.
  • Principal focus (F): where rays parallel to the principal axis meet after passing through a converging lens (on the far side). For a diverging lens, F is on the near side, where the refracted rays appear to come from.
  • Secondary focus (F′): the focal point on the other side of the lens, the same distance from O. (Light can pass through a lens either way, so there is a focus on each side.)
  • Focal length (f): the distance from O to F.
  • 2F and 2F′: points at twice the focal length; useful landmarks, like C for mirrors.

The three rays for a converging lens

  1. A ray parallel to the principal axis refracts through F (on the far side).
  2. A ray through F′ (on the near side) refracts parallel to the principal axis.
  3. A ray through the optical centre O goes straight through without bending.

Use at least two rays from the top of the object. Where they cross on the far side is a real image. If they spread apart, extend them backwards (dashed) to find a virtual image on the same side as the object.

Object beyond 2F′ (camera)FF′2F2F′objectimage
Object beyond 2F′. The rays cross between F and 2F on the other side: a smaller, inverted, real image. This is how a camera or your eye forms an image.
Worked example

Example 1: Let's draw one together

A converging lens has f = 5.0 cm. A 3.0 cm tall object is 15 cm from the lens. Draw a scale diagram and describe the image.

  1. Choose a scale: 1 cm on paper = 1 cm. Draw the principal axis and the lens. Mark F and F′ 5.0 cm from O on each side, and 2F and 2F′ at 10 cm.
  2. Draw the object at 15 cm on the left: it is beyond 2F′.
  3. Ray 1: from the tip, parallel to the axis to the lens, then through F.
  4. Ray 2: from the tip through F′ to the lens, then parallel to the axis.
  5. Ray 3: from the tip straight through O.
  6. The three rays cross 7.5 cm to the right of the lens, 1.5 cm below the axis.
  7. SALT: smaller (1.5 cm, half the height), inverted, 7.5 cm from the lens on the opposite side (between F and 2F), real.

The image is smaller, inverted, between F and 2F on the far side (7.5 cm from the lens), and real. (The diagram above is this example drawn at 1 cm = 16 px.)

Object between 2F′ and F′ (projector)FF′2F2F′objectimage
Object between 2F′ and F′. The rays cross beyond 2F: a larger, inverted, real image. Projectors use this arrangement to throw a big image on a screen.
Object inside F′ (magnifying glass)FF′2F2F′objectimage
Object inside F′. The refracted rays spread apart, so they are traced back (dashed) to a larger, upright, virtual image on the same side as the object. This is a magnifying glass.
Object positionImage locationSizeAttitudeTypeUse
beyond 2F′between F and 2Fsmallerinvertedrealcamera, eye
at 2F′at 2Fsame sizeinvertedrealphotocopier (same-size copies)
between 2F′ and F′beyond 2Flargerinvertedrealprojector
at F′no image (rays leave parallel)———spotlight, lighthouse lens
inside F′same side as object, farther from lenslargeruprightvirtualmagnifying glass
Images formed by a converging lens. Compare with the concave mirror table: the pattern is the same.

Diverging lenses

The three rays for a diverging lens:

  1. A ray parallel to the principal axis refracts as if it came from F (on the near side). Draw a dashed line back to F.
  2. A ray heading towards F′ (on the far side) refracts parallel to the axis.
  3. A ray through O goes straight through.

A diverging lens always forms an image that is smaller, upright, virtual and on the same side as the object, between F and the lens, just like a convex mirror.

Diverging lensFF′objectimage
A diverging lens. The refracted rays spread out; traced back (dashed), they meet on the object's side to form a smaller, upright, virtual image.
Just for fun

A single drop of water can act as a converging lens. Put a tiny drop of water on a phone screen and look closely: the drop magnifies the screen enough to show the separate red, green and blue dots that make up each pixel.

Hands-on activity

Magnifying-glass lab

You need: a magnifying glass, a sheet of white paper or a white wall, a room with a bright window, a ruler, a book with small print, a clear plastic sheet (such as a page protector) and a drop of water.

  1. Real image. Stand on the far side of the room. Hold the magnifying glass between the window and the white paper and move it until a sharp image of the window appears on the paper. Is it upright or inverted? Larger or smaller? Measure the lens-to-paper distance: that is approximately f.
  2. Virtual image. Hold the magnifying glass close to the book's small print (closer than f). Look through it. Describe the image with SALT. Slowly move the lens away from the page. What happens when the page is farther than f from the lens?
  3. Water-drop lens. Put a single drop of water on the plastic sheet over the small print. Look through the drop. What kind of lens is the drop, and why?
  4. Record your observations in a table and compare them with the summary table above.

What to look for: the window image is real, inverted and smaller (object beyond 2F′); the print looks upright and larger when the lens is closer than f, then becomes blurry and inverted when farther. The water drop is thicker in the middle, so it is a converging lens.

Safety (grown-up note): Never look at the Sun through a lens, and never leave a magnifying glass in sunlight: it can focus enough energy to start a fire.

Your turn

1

How can you tell a converging lens from a diverging lens by touch?

2

State the three rules for drawing rays through a converging lens.

3

An object is beyond 2F′ of a converging lens. Describe the image.

4

An object is inside F′ of a converging lens. Describe the image and name a device that uses this.

5

Why does light refract twice in a lens?

6

Describe the image formed by a diverging lens.

More practice: warm-up

1

What is another name for a converging lens?

2

Which ray passes through a lens without bending?

3

Where should an object be to make a same-size image with a converging lens?

4

Can a diverging lens make a real image of a real object?

5

What does a lens do to light: reflect or refract?

More practice: on your own

1

A projector must make a large, inverted, real image on a far screen. Where should the slide or display be placed relative to the lens?

2

Explain why a ray through the optical centre of a thin lens is not bent.

3

A student holds a magnifying glass at arm's length and sees an upside-down image of a distant house. Explain.

4

Why do camera lenses have anti-reflective coatings?

5

Compare a converging lens with a concave mirror.

Challenge

1

Using a scale diagram (or the pattern in the table), predict the image when a 2.0 cm tall object is 12 cm from a converging lens with f = 4.0 cm.

2

A converging lens is placed in water. Will its focal length be longer or shorter than in air? Explain.

3

A lighthouse needs a strong parallel beam. Where should the lamp be placed relative to a large converging lens, and why?

Check your understanding

Quiz

1.

A converging lens is:

2.

For a converging lens, a ray parallel to the principal axis refracts:

3.

Which ray passes through a thin lens without changing direction?

4.

A camera forms an image of a distant object on its sensor. The image is:

5.

A magnifying glass forms a larger, upright image when the object is:

6.

A diverging lens always forms an image that is:

For parents and teachers

Big idea: Expectations E3.5 and E2.5 ask students to describe and predict images formed by converging lenses with ray diagrams; E3.4 includes partial reflection and refraction in lenses. The three-ray method and the summary table parallel the concave-mirror lesson, which helps.

Common difficulties:

  • F vs F′. F is on the far side (where parallel rays meet); F′ is on the object's side. Ray 2 goes through F′.
  • Bending at the centre line. In the thin-lens model, rays are bent once at the lens's centre line.
  • Upright images. For a single converging lens, an upright image is always virtual and on the object's side.

How to help: The magnifying-glass lab is quick and uses common materials. Supervise to make sure the lens is never used with sunlight.