Lens Magnification (m = −d_i/d_o)

This one falls out of similar triangles, and it is worth seeing why rather than accepting it. The ray that passes through the exact centre of a thin lens goes straight through undeviated, because at the centre the two surfaces are parallel. That single ray makes the same angle with the axis on both sides, so the triangle formed by the object and the triangle formed by the image are similar. Their proportions give $|h_i|/|h_o| = d_i/d_o$ — the image is as many times larger as it is farther away. The minus sign in $m = -d_i/d_o$ is a bookkeeping convention laid on top of that geometry, and it encodes orientation: under the standard convention a real image from a converging lens has both distances positive, so $m$ comes out negative, and negative means inverted.

A camera with a 50 mm lens focused on a subject 2 m away forms its image at $d_i = 51.3$ mm, so $m = -51.3/2000 = -0.0256$. A person 1.8 m tall images 46 mm tall, which will not fit on a full-frame sensor only 24 mm high — so you step back, or fit a wider lens. A projector runs the same equation the other way. Put a 24 mm slide 102 mm in front of a 100 mm lens and the thin-lens equation gives $d_i = 5.1$ m, so $m = -50$: an image 1.2 m across and upside down. That inversion is why slides go into a carousel the wrong way up.

The two magnification pages are meant to be used together. This one gets you from distances to a ratio; $m = h_i/h_o$ gets you from that ratio to a size. A case worth memorising sits between them: when $d_o = 2f$ the thin-lens equation gives $d_i = 2f$ as well, so $m = -1$ exactly — object and image the same size, symmetric about the lens, and separated by $4f$. That is the 1:1 setting macro photographers work at, and $4f$ is the smallest object-to-sensor distance at which a given lens can form a real image at all. Try to squeeze the two closer and no solution exists.

The sign is the trap, and sign conventions are the single largest source of wrong answers in geometric optics. A negative $m$ does not mean a negative size or a reduced image — it means inverted. Size lives in the magnitude, orientation lives in the sign, and $m = -0.03$ is telling you two separate things at once: strongly reduced, and upside down. This page uses the standard "real is positive" convention: $d_o$ positive for a real object, $d_i$ positive for a real image on the far side of the lens, negative for a virtual image on the near side. Hold a magnifying glass closer than its focal length and $d_i$ goes negative, so $m$ goes positive: upright and enlarged, exactly what your eye reports. Pick one convention and never borrow a formula from a textbook using another. One last distinction: this is the linear magnification of a real optical image. The "10×" stamped on a hand lens and the "40×" on a microscope objective are angular magnifications defined against a 25 cm reference viewing distance, a different quantity that cannot be compared with this one directly.