Image formation by a lens
The hardest part of this chapter is drawing the ray diagram. So don't draw it — move the object and watch it happen. Every ray, focus and image here is computed from the real lens formula, so what you see is exactly what your exam expects.
Drag the object arrow left and right.
Read this off the diagram
The two rays you always draw
One ray parallel to the axis bends through the focus F. One ray through the optical centre goes straight. Where they meet is the image.
Real vs virtual
If the rays actually cross, the image is real (can be caught on a screen, always inverted). If only their backward extensions meet, it's virtual (erect) — shown dashed.
Sign convention
Distances are measured from the optical centre. Against the incoming light is negative, along it is positive — the same convention your board answers must use.
Convex vs concave
A convex lens can make real or virtual images depending on where the object sits. A concave lens always makes a small, erect, virtual image.
This is one of Priodemy for School — interactive Maths & Science for Class 9–12, free with every EduSuite school.
How a lens forms an image
The formula does the drawing for you
Every ray on screen is computed from the lens formula 1/v − 1/u = 1/f. Note the minus sign: this is not the mirror formula, which uses a plus. Mixing the two is the most common single error in this chapter, and it is worth writing both down side by side once, clearly, before an exam.
A convex lens is a converging lens with positive focal length; a concave lens diverges and has negative focal length. The construction rays are the same in every question: a ray parallel to the principal axis refracts through the focus on the far side, and a ray through the optical centre passes straight on undeviated. Where those two emergent rays meet is the image. If they diverge instead of meeting, extend them backwards — where the extensions meet is a virtual image.
Real and virtual, and how to tell them apart
A real image is formed where light actually converges. It can be caught on a screen, and it is always inverted for a single convex lens. A virtual image is formed where the light only appears to come from; no light reaches that point, so no screen will show it, and it is always erect. This is the difference between the image a projector throws onto a wall and the image you see through a magnifying glass.
Drag the object inwards from far away and watch the transition. Beyond 2F the image is real, inverted and diminished, sitting between F and 2F. At 2F it is real, inverted and the same size, at 2F on the other side. Between 2F and F it is real, inverted and enlarged, beyond 2F. At F the emergent rays are parallel and no image forms. Inside F the image flips to virtual, erect and enlarged, on the same side as the object — the magnifying-glass case.
Power, and why opticians use it
The power of a lens is P = 1/f with f measured in metres, and the unit is the dioptre. A converging lens has positive power, a diverging lens negative. Power is used in practice because lenses placed together simply add: two thin lenses in contact behave as one lens of power P₁ + P₂. Adding focal lengths would require a far messier calculation, which is why a prescription is written in dioptres rather than centimetres.
Mistakes that cost marks
Using the mirror formula for a lens. Lens is 1/v − 1/u = 1/f; mirror is 1/v + 1/u = 1/f. The sign convention also differs in effect: for a lens, a positive v means a real image on the far side, whereas for a mirror a negative v means a real image in front.
Forgetting that u is negative. A real object placed to the left of the lens has a negative object distance. Substituting a positive u produces a plausible-looking but wrong image position, and the error is invisible unless you check whether the answer makes physical sense.
Reporting magnification without its sign. For a lens m = v/u, and the sign carries the answer to "what is the nature of the image". Negative means inverted and real; positive means erect and virtual. Quoting only the size discards half the information the question is asking for.