Lens Equation
A lens is a carefully shaped piece of transparent material that refracts light to either converge it (a convex lens) or diverge it (a concave lens). Whether you are using a magnifying glass, a camera, spectacles, or a microscope, the same fundamental mathematics governs where an image forms. The lens equation — also called the thin-lens formula — is a compact algebraic relationship linking three distances: the object distance, the image distance, and the focal length. Mastering this single equation unlocks the ability to predict precisely where an image will appear and whether it will be real or virtual. Think of light rays from a distant lamp post arriving almost perfectly parallel at a convex lens. The lens bends them so that they all meet at one special point on the other side — the principal focus. The distance from the lens centre to that point is the focal length f. Now replace the distant lamp post with a nearby object at distance u. The refracted rays still converge, but they meet at a new distance v on the other side. The lens equation, 1/v − 1/u = 1/f (in the real-is-positive sign convention), captures exactly how u, v, and f are intertwined for every possible object position. The equation applies to any thin lens — one thin enough that we can treat refraction as happening at a single plane. It works for both converging and diverging lenses provided you apply the sign convention consistently. Once you know any two of the three quantities, the equation gives the third instantly, making it an indispensable tool in optical design and everyday problem-solving.
- Setting Up a Sign Convention
- Deriving the Formula from Geometry
- Interpreting the Three Variables
- Special Cases Worth Knowing
Setting Up a Sign Convention
Physics uses the New Cartesian (real-is-positive) sign convention. All distances are measured from the optical centre of the lens. Distances measured in the direction of the incoming light are positive; distances measured against it are negative. For a real object in front of the lens, u is negative. For a real image formed on the far side of a converging lens, v is positive. A converging lens has a positive focal length f; a diverging lens has a negative f.
Deriving the Formula from Geometry
Consider a converging lens with an object of height h placed at distance |u| beyond the focal point. Drawing two standard rays — one parallel to the principal axis (refracts through F') and one through the optical centre (passes straight) — their intersection locates the image. Using similar triangles on both sides of the lens, one obtains: image height / object height = v / u (magnification). Combining the two pairs of similar triangles yields 1/v − 1/u = 1/f. This is the thin-lens equation in its Cartesian form.
Interpreting the Three Variables
Object distance u: how far the object is from the lens — always negative for a real object in the Cartesian convention. Image distance v: how far the image forms from the lens — positive for a real image (same side as outgoing light), negative for a virtual image (same side as the object). Focal length f: a fixed property of the lens, determined by its curvature and refractive index. Positive for converging lenses, negative for diverging ones. Rearranging: v = uf / (u + f), which shows how v changes continuously as u varies.
Special Cases Worth Knowing
When the object is at infinity (u → −∞), 1/u → 0, so v = f: the image forms at the focal point. When the object is at 2f (u = −2f), you get v = 2f: the image is real, inverted, and the same size as the object — used in photocopiers set to 1:1. When the object is inside the focal length (|u| < f) of a converging lens, the formula gives a negative v, meaning the image is virtual, upright, and magnified — exactly what a magnifying glass does. For a diverging lens (f < 0), the image is always virtual, upright, and diminished regardless of object position.
The Key Equation
- v — Image distance (metres (m) or centimetres (cm)): Distance from the optical centre of the lens to where the image forms. Positive for a real image on the transmission side; negative for a virtual image on the same side as the object.
- u — Object distance (metres (m) or centimetres (cm)): Distance from the optical centre to the object. Negative for a real object placed in front of the lens (against the direction of incident light).
- f — Focal length (metres (m) or centimetres (cm)): Distance from the optical centre to the principal focus. Positive for a converging (convex) lens; negative for a diverging (concave) lens.
Worked Example
Problem. A converging lens has a focal length of 20 cm. An object is placed 30 cm in front of the lens. Find the image distance and state whether the image is real or virtual.
Answer. The image forms 60 cm beyond the lens. It is real, inverted, and magnified by a factor of 2.
Origins · 1 minJohannes Kepler and how this was discovered
Kepler published Dioptrice in 1611, the first systematic mathematical treatment of how lenses refract light and form images. He correctly explained how both convex and concave lenses work, described the astronomical telescope, and laid the geometric groundwork that later allowed Descartes and others to formalise the thin-lens equation. His analysis of conjugate points — object and image positions linked through the lens — is the conceptual heart of the modern lens formula. Kepler was so short-sighted that he used his own blurred vision as experimental data when working out the optics of the human eye — effectively running experiments on himself without a laboratory.
Related Concepts
- Power of a Lens — Power P = 1/f (in dioptres) is derived directly from the focal length in the lens equation; understanding f is essential for calculating lens power used in spectacle prescriptions.
- Magnification — Linear magnification m = v/u is calculated using the same image and object distances that appear in the lens equation, making the two concepts inseparable.
- Refraction at Curved Surfaces — The lens equation is derived from applying Snell's law at curved refracting surfaces; understanding surface refraction explains why the thin-lens formula takes the form it does.
- The Human Eye and Defects of Vision — Corrective lenses for myopia and hyperopia are prescribed using the lens equation to calculate the required focal length that shifts the image onto the retina.