Diverging rays make an image appear where no light arrives

1. Rays that spread apart after leaving a lens or a mirror are said to diverge, and rays like that never cross each other on the far side.
2. Your eye judges only the direction from which each ray arrives, never the path the ray took to get there.
3. Light travels in straight lines, so the eye and brain read every arriving ray as having come straight from wherever that line points back to.
4. Trace two diverging rays backwards and those two lines cross at a single point, and that crossing point is where the object seems to be.
5. What you see there is a virtual image, and it lies on the same side as the light that formed it.
6. No light energy ever reaches that place, because the rays are travelling away from it rather than towards it.
7. Hold a white card at that position and you get nothing sharp, only the general glow of light passing nearby, so a virtual image forms no visible projection on a screen.
   1. A screen shows a picture only where the rays from each point of the object are brought back together on the screen itself, and that is precisely what does not happen here.
8. You see your face roughly $30cm$ behind a bathroom mirror even though a solid wall stands there, and no film pressed against that wall would ever record it.

An object inside the principal focus gives an enlarged upright virtual image

1. A converging lens forms a virtual image when the object sits between the lens and the principal focus, so the object distance is smaller than the focal length.
2. Set the diagram up with the object standing on the principal axis, nearer the lens than the mark for $F$.
3. Draw the first construction ray from the top of the object parallel to the principal axis, and refract it at the lens so that it passes through $F$ on the far side.
4. Draw the second construction ray from the top of the object straight through the centre of the lens, where it carries on without changing direction.
5. Beyond the lens both rays slope downwards, but the ray through the centre falls more steeply, so the pair spread apart instead of crossing.
   1. The parallel ray is always bent by the same amount, set by the focal length, while the central ray gets steeper the closer the object comes, so inside $F$ the central ray is the steeper of the two.
6. Since the emerging rays never meet, extend both of them backwards as dashed lines into the space on the object's side of the lens.
7. The two dashed lines cross at one point, and that point is the top of the image, further from the lens than the object and higher above the axis.
8. Drop a vertical line from that crossing point down to the principal axis to draw the image, standing the same way up as the object.
9. The image is enlarged, upright and virtual, and it sits on the same side of the lens as the object.
10. The dashed lines are construction lines only, because no light ever travels along them.
11. An observer has to look through the lens from the far side to see this image, and no screen anywhere will catch it.

A magnifying glass keeps the object inside its own focal length

1. A hand lens is one converging lens, typically with a focal length somewhere between $5cm$ and $15cm$.
2. To use one, lay the lens almost flat on the print or the insect, then raise it slowly while looking down through it.
3. Stop while the object is still nearer the lens than the principal focus, which for a $10cm$ lens means keeping the gap under $10cm$.
4. Rays from one point of the object reach the lens strongly diverging, and the lens reduces that spread without quite reversing it, so they leave still diverging but much more gently.
   1. Gently spreading rays traced backwards meet a long way from the lens, well beyond the object, and the whole image is scaled up in the same proportion, which is why it looks enlarged.
5. The image is upright because both dashed extensions run back on the same side of the principal axis as the top of the object.
6. Everyday uses are reading the small print on a medicine label, checking the perforations round a stamp, and looking at the veins in a leaf.
7. With the object this close, the enlarged image is virtual, so it can only be looked at through the lens and never thrown onto a wall.
8. Lift the lens too far and the object ends up beyond the principal focus, the emerging rays then converge and cross, and the view flips upside down.

A plane mirror image is virtual for exactly the same reason

1. Rays from a point on an object strike a plane mirror and reflect, and the reflected rays that enter your eye are diverging.
2. Extending them straight back gives a crossing point as far behind the mirror as the object is in front of it, where the image is the same size, upright, laterally inverted and virtual.
3. No light gets behind the mirror at all, so a screen held there stays blank, and that is the test which separates an image of this kind from one the rays really pass through.

A diverging lens gives a virtual image wherever the object is

1. Send the first construction ray from the top of the object parallel to the principal axis, and a diverging lens spreads it outwards so that it leaves along a line tracing back to $F$ on the object's side.
2. Send the second ray through the centre of the lens, where again it carries straight on.
3. These two rays spread apart for every object distance, near or far, so the construction always ends with dashed backward extensions.
4. They cross between the lens and $F$, giving an image that is upright, diminished and virtual, on the same side as the object.
5. That is why the view through a diverging lens stays the right way up but shrinks, however far you hold it from what you are looking at.