Spherical lenses: images by convex and concave lenses
A convex lens converges light and gives real, inverted images unless the object is inside F₁. A concave lens diverges light and always gives virtual, erect, smaller images.
Lens words you must know
- A lens is a piece of transparent material with two surfaces, at least one of them spherical (part of a sphere).
- A convex lens bulges outwards and is thicker at the middle than at the edges. It converges (brings together) light, so it is a converging lens.
- A concave lens curves inwards and is thicker at the edges than at the middle. It diverges (spreads out) light, so it is a diverging lens.
- Each surface has a centre of curvature, C₁ and C₂. The principal axis is the line joining them.
- The optical centre (O) is the centre point of the lens. The aperture is the width of the lens.
- A lens has two principal foci, F₁ and F₂, one on each side. The focal length f is the distance from O to a focus. In diagrams, 2F₁ and 2F₂ mark points at twice the focal length.
Three rays for lens ray diagrams (any two are enough)
- Rule 1: a ray parallel to the principal axis passes through F₂ after a convex lens. After a concave lens it spreads out as if it came from F₁, on the same side as the object.
- Rule 2: a ray through F₁ (convex), or aimed towards F₂ (concave), comes out parallel to the principal axis.
- Rule 3: a ray through the optical centre O goes straight on without bending.
- Real rays are solid lines with arrowheads. Lines that only show where rays seem to come from are dotted.
- For every diagram: draw the axis and a thin lens standing on it at O. Mark F₁ and 2F₁ on the left, F₂ and 2F₂ on the right, evenly spaced. Put the object AB on the left.
Images formed by lenses
| Lens and object position | Image at | Size | Nature |
|---|---|---|---|
| Convex: at infinity | F₂ | Highly diminished, point-sized | Real and inverted |
| Convex: beyond 2F₁ | Between F₂ and 2F₂ | Diminished | Real and inverted |
| Convex: at 2F₁ | 2F₂ | Same size | Real and inverted |
| Convex: between F₁ and 2F₁ | Beyond 2F₂ | Enlarged | Real and inverted |
| Convex: at F₁ | Infinity | Highly enlarged | Real and inverted |
| Convex: between F₁ and O | Same side as the object | Enlarged | Virtual and erect |
| Concave: at infinity | F₁ | Highly diminished, point-sized | Virtual and erect |
| Concave: between infinity and O | Between F₁ and O | Diminished | Virtual and erect |
How to draw the six ray diagrams for a convex lens
Draw two rays from the top point A in every case: one parallel to the axis (it bends through F₂) and one through O (it goes straight). Where they cross is A′. Draw A′B′ to the axis.
- 1Object at infinity. Draw two rays coming from the left, parallel to the axis. After the lens, draw both passing through F₂.Image at F₂: point-sized, real, inverted.
- 2Object beyond 2F₁. Stand AB to the left of 2F₁. The ray parallel to the axis bends through F₂. The ray through O goes straight. They cross below the axis between F₂ and 2F₂.Image between F₂ and 2F₂: diminished, real, inverted.
- 3Object at 2F₁. Stand AB on 2F₁. The same two rays cross below the axis exactly at 2F₂. Draw A′B′ pointing down, as long as AB.Image at 2F₂: same size, real, inverted.
- 4Object between F₁ and 2F₁. Stand AB between F₁ and 2F₁. The two rays cross below the axis beyond 2F₂. Draw a long A′B′ pointing down.Image beyond 2F₂: enlarged, real, inverted.
- 5Object at F₁. Stand AB on F₁. The ray parallel to the axis bends through F₂. The ray through O goes straight. The two refracted rays are parallel, so they meet only at infinity.Image at infinity: highly enlarged, real, inverted.
- 6Object between F₁ and O. Stand AB between F₁ and the lens. The ray parallel to the axis bends through F₂. The ray through O goes straight. The rays spread apart on the right. Extend both backwards as dotted lines: they meet at A′ above the axis, on the object's side. Draw A′B′ upright, larger and dotted.Image on the object's side: enlarged, virtual, erect. This is how a magnifying glass works.
How to draw the two ray diagrams for a concave lens
Draw the concave lens as a thin lens with flared edges, standing on the axis at O. Mark F₁ on the left and F₂ on the right.
- 1Object at infinity. Draw two rays parallel to the axis. After the lens, draw them spreading outwards. Extend each backwards as a dotted line: they meet at F₁ on the left.Image at F₁: point-sized, virtual, erect.
- 2Object anywhere in front. From A, draw a ray parallel to the axis; after the lens, bend it outwards and dot it back to F₁. From A, draw a ray through O; it goes straight. The dotted line and this ray meet at A′ between F₁ and O. Draw A′B′ upright, small and dotted.Image between F₁ and O: diminished, virtual, erect.
A jeweller checking gold, or a grandparent reading small print, uses a magnifying glass: a convex lens with the object held inside F₁. Spectacles for short sight use concave lenses.
1 marks: “Where should an object be placed in front of a convex lens to get a real image of the same size?” At 2F₁. The image forms at 2F₂: real, inverted and the same size.
3 marks: “Draw a ray diagram to show the image formed by a concave lens for an object placed between infinity and its optical centre. State the position and nature of the image.” Ray parallel to the axis diverges as if from F₁ (dotted back); ray through O goes straight; image between F₁ and O, virtual, erect, diminished.
5 marks: “With ray diagrams, show how a convex lens forms (i) a real, inverted image of the same size and (ii) a virtual, erect, enlarged image. Name a device that uses case (ii).” (i) Object at 2F₁, image at 2F₂. (ii) Object between F₁ and O, image on the same side, enlarged. Device: magnifying glass.
A convex lens behaves like a concave mirror: real and inverted until the object crosses F. A concave lens behaves like a convex mirror: always small, upright and virtual.