Converging lenses and ray diagrams

How a lens works

• A lens works entirely by refraction. Light refracts once entering the glass and again leaving it, and the curved surfaces mean different parts of the beam are bent by different amounts.
• A converging (convex) lens is thicker in the middle. It brings parallel rays together.
• A diverging (concave) lens is thinner in the middle. It spreads parallel rays apart.

The words you need

• Principal axis — the horizontal line through the centre of the lens, at right angles to it.
• Optical centre — the middle of the lens. A ray through it passes straight through, undeviated.
• Principal focus ($F$) — the point on the principal axis where rays parallel to the axis are brought to a focus.
  • A lens has a focus on each side, the same distance out.
• Focal length ($f$) — the distance from the optical centre to the principal focus.
  • A fatter lens has a shorter focal length and is more powerful.
• Object distance ($d_o$) — from the object to the lens.
• Image distance ($d_i$) — from the lens to the image.

The three construction rays

To locate an image, draw rays from the top of the object. Any two of these three are enough; the third is a check.

• Ray 1 — parallel then through the focus. A ray travelling parallel to the principal axis refracts through the lens and passes through the far principal focus.

• Ray 2 — through the optical centre. A ray aimed at the centre of the lens carries straight on, undeviated.

• Ray 3 — through the near focus then parallel. A ray passing through the near principal focus emerges parallel to the principal axis.

• Where the refracted rays cross is where the top of the image sits. Drop a perpendicular to the axis to complete the image.

Drag the object below and watch the image move, flip and change size as it crosses the focal point:

Real and virtual images

• A real image forms where refracted rays actually cross.
  • It can be projected onto a screen.
  • For a single converging lens it is always inverted.
• A virtual image forms where the rays only appear to come from, when traced backward.
  • It cannot be projected onto a screen.
  • It is always upright.
  • Draw the backward extensions as dashed lines.

The five cases for a converging lens

The image depends entirely on where the object sits relative to $F$ and $2F$:

Object position	Image position	Nature
Beyond $2F$	Between $F$ and $2F$	Real, inverted, smaller
At $2F$	At $2F$	Real, inverted, same size
Between $F$ and $2F$	Beyond $2F$	Real, inverted, larger
At $F$	No image (rays emerge parallel)	—
Inside $F$	Same side as the object	Virtual, upright, larger

• The one case that gives an upright image is the object inside the focal length — this is the magnifying glass.
• Everything else with a single converging lens is real and inverted.

Drawing an accurate diagram

• Draw the principal axis as a long horizontal line and the lens as a vertical line crossing it.
• Mark $F$ and $2F$ on both sides, to scale.
• Draw the object as a vertical arrow standing on the axis.
• Draw two construction rays from the top of the object.
• Mark the image where they cross, and describe it in words: real/virtual, upright/inverted, larger/smaller.
• Use a ruler — the specification requires one, and marks are given for accuracy.