Lens Equation and Power
Every remaining lesson in this topic — the eye as a detector, short-sightedness, long-sightedness, astigmatism — rests on the same small set of quantitative tools. This lesson builds that toolkit properly, and spends real time on the one thing that causes almost every mistake here: getting the signs right.
Power: why dioptres, not focal length
Every converging or diverging lens has a focal length $f$, the distance from the lens at which parallel incoming rays are brought to a focus (or from which they appear to diverge, for a diverging lens). But when describing the eye and the lenses used to correct it, physicists almost always talk about power instead of focal length. Power is defined as:
with $f$ measured in metres, giving power the unit of dioptres, symbol D (equivalently $\text{m}^{-1}$). A strongly converging lens, with a short focal length, has a large positive power; a weakly converging lens, with a long focal length, has a small positive power; a diverging lens has negative power.
The reason power is the natural quantity here, rather than focal length, is that powers simply add when two thin lenses are placed close together (as happens, for instance, when a corrective lens sits in front of the eye's own lens system):
Focal lengths don't combine this simply — you can't just add two focal lengths together to get the combined focal length. Because the eye's total optical system is really several refracting surfaces (cornea, lens) acting one after another, and because correcting a vision defect always means adding an extra lens in front of an existing system, describing everything in dioptres makes the arithmetic of combining these effects genuinely straightforward. This becomes essential in the correction lessons later in this topic.
The lens equation and magnification
The relationship between where an object is placed and where its image forms is given by the lens equation:
where $u$ is the object distance (from the lens to the object), $v$ is the image distance (from the lens to the image), and $f$ is the focal length. The linear magnification produced is:
giving the ratio of image size to object size. A magnification with size greater than 1 means an enlarged image; less than 1 means a diminished image, as is the case for the eye itself, discussed in the previous lesson.
Sign convention: the single biggest source of error in this topic
The lens equation only gives correct answers if distances are assigned consistent signs depending on whether they describe something real or virtual. Getting this wrong — a sign flipped here or there — is by far the most common way marks are lost across every lesson in this topic that involves a calculation, so it's worth fixing one clear, consistent convention now and using it without exception from here on.
Object distance $u$: positive for a real object (the normal case — light actually diverging from it).
Image distance $v$: positive for a real image (formed by converging rays, on the opposite side of the lens from the object); negative for a virtual image (formed by diverging rays that only appear to come from a point, on the same side of the lens as the object).
Focal length $f$ (and power $P$): positive for a converging lens; negative for a diverging lens.
This single convention, applied consistently, is what every worked example in this lesson — and every calculation in the rest of this topic — is built on.
Worked examples
Two worked examples follow: a straightforward case with everything positive, then a case built specifically to drill the sign convention with a virtual image.
Key takeaways — quick-reference toolkit
- Power is defined as $P = \dfrac{1}{f}$, measured in dioptres (D), with $f$ in metres.
- Powers of lenses used together simply add: $P_{\text{total}} = P_1 + P_2$ — the reason power, not focal length, is used throughout this topic.
- The lens equation is $\dfrac{1}{u} + \dfrac{1}{v} = \dfrac{1}{f}$, and magnification is $m = \dfrac{v}{u}$.
- Sign convention (real-is-positive): real objects and real images are positive; virtual images are negative; converging lenses (and their power) are positive; diverging lenses (and their power) are negative.
- A positive $v$ means a real image on the far side of the lens from the object; a negative $v$ means a virtual image on the same side as the object.
- Working directly in dioptres (reciprocals of distance in metres) usually keeps the arithmetic cleaner than working in focal lengths and distances throughout.
- This sign convention is applied without exception in every later lesson in this topic, particularly the correction calculations for short-sightedness and long-sightedness.