The thin lens equation and magnification

Descartes' conjugation relation (thin lens equation)

For a thin lens, with the origin at the optical center O and an axis oriented in the direction of light propagation, the position of the image A' of an object point A located on the axis satisfies:

1/OA' - 1/OA = 1/f' = C

All distances (OA, OA', f') are algebraic quantities (positive or negative depending on the chosen direction).

Magnification

It compares the size of the image A'B' to that of the object AB:

gamma = A'B'/AB = OA'/OA

• gamma > 0 -> upright image (same orientation as the object)
• gamma < 0 -> inverted image
• |gamma| > 1 -> enlarged image ; |gamma| < 1 -> reduced image

Worked example

Converging lens, f' = 10 cm. Object AB placed at OA = -30 cm (real object, placed before the lens).

1/OA' = 1/f' + 1/OA = 1/10 + (-1/30) = 3/30 - 1/30 = 2/30 = 1/15

so OA' = 15 cm: the image is real, located 15 cm after the lens.

gamma = OA'/OA = 15/(-30) = -0.5

The image is inverted and half the size of the object.

Classic pitfalls

• Follow the sign convention carefully: a real object placed before the lens has a negative OA coordinate.
• f' is negative for a diverging lens: remember to include it with its sign in calculations.
• Do not confuse magnification (a ratio of sizes) with focal length (a characteristic of the lens).