Thin lens, conjugation relation and magnification
1/OA’ − 1/OA = 1/f’
The Descartes conjugation relation, both ways: where the image of a given object forms, or which lens produced two measured positions. The tool returns the image position, its algebraic measure, the magnification, the optical power, the image height when the object height is known, and describes the image in three words: real or virtual, upright or inverted, enlarged or reduced. The sign convention, which is half the subject and where the exercise goes wrong, is set by the tool: you enter a measured, positive distance, and the negative algebraic measure is handed back to you for your working.
15
- Calculation
- 1/OA’ − 1/OA = 1/f’: 1/15 − 1/-30 = 1/10
- Magnification
- γ = OA’ / OA = 15 / -30 = -0.5
The image is real, inverted and reduced. It forms 15 from the lens, in the unit of your distances. The lens is converging.
The convention, to copy as is onto an exam paper: the origin is the optical centre O, the positive direction is that of the light. A real object therefore sits ON THE LEFT, and its algebraic measure OA is negative. That is where the exercise most often goes wrong, writing OA = +30 for an object measured at 30 cm. The tool asks for the measured, positive distance and sets the sign itself.
The image is real: it forms behind the lens, where the light comes out, and a screen placed there catches it. A real image of a real object is always inverted, not as a special case but as a consequence of the relation.
The model is the thin lens under Gauss conditions: rays little inclined and close to the axis. A real thick lens, or one lit outside those conditions, shows aberrations that shift and distort the image. The focal length also depends on wavelength, which produces chromatic aberration.
Scientific dossier
What the tool computes, what it assumes, where it stops being valid, and where its data comes from.
Method & formulas1/OA’ − 1/OA = 1/f’
1/OA’ − 1/OA = 1/f’
γ = OA’ / OA = A’B’ / AB
V = 1 / f’
real object: OA < 0
real image: OA’ > 0
One relation ties both positions to the lens, and one more draws the magnification from it. All the difficulty lies in the convention: the origin is the optical centre, the positive direction is that of the light, and positions are algebraic measures. A real object sitting in front of the lens, its measure is negative, which puzzles until the gesture has been made once.
- OA
- · algebraic measure of the object, negative for a real object.
- OA’
- · algebraic measure of the image. Positive behind the lens, where a screen catches it.
- f’
- · image focal length. Positive for a converging lens, negative for a diverging one.
- γ
- · magnification. Negative, the image is inverted; greater than 1 in absolute value, it is enlarged.
- Optical power
- · 1/f’, in dioptres when the focal length is in metres. It is what an eyeglass prescription carries.
Validity domainThe model is the thin lens under Gauss conditions, that is, rays little inclined and close to the optical axis.
The model is the thin lens under Gauss conditions, that is, rays little inclined and close to the optical axis. A thick lens, or one lit outside those conditions, shows geometric aberrations that shift and distort the image; the focal length also depends on wavelength, hence chromatic aberration. An object placed exactly at the object focus gives no image at finite distance: rays leave parallel, the image is at infinity, and the tool says so rather than returning a very large number. The tolerance for that case is relative to the focal length, so that it holds on five millimetres as on five metres.
Reading the resultThree words describe the image, and each is read off a sign.
Three words describe the image, and each is read off a sign. Real or virtual: the sign of OA’ says which side it sits on, hence whether a screen can catch it. Inverted or upright: the sign of the magnification. Enlarged or reduced: its absolute value. One consequence worth remembering rather than relearning: from a real object, a real image is always inverted, and an upright image is always virtual.