--- title: "Thin lenses and the lens maker's equation" description: "Use thin-lens and lensmaker equations in the small-angle approximation. Apply explicit distance and curvature signs to find focal length, finite image position and magnification, and distinguish real " canonical: https://lightmysky.com/learn/science/thin-lenses-and-the-lens-makers-equation-mt_FXOcC5mpIM source: https://lightmysky.com/learn/science/thin-lenses-and-the-lens-makers-equation-mt_FXOcC5mpIM.md retrieved: 2026-11-12 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # Thin lenses and the lens maker's equation Use thin-lens and lensmaker equations in the small-angle approximation. Apply explicit distance and curvature signs to find focal length, finite image position and magnification, and distinguish real from virtual images. Subject: Science · Area: Waves, Light & Sound · Ages 19 to 20 Page: https://lightmysky.com/learn/science/thin-lenses-and-the-lens-makers-equation-mt_FXOcC5mpIM ## Ready when they can - Uses the lens equation and a consistent sign convention to locate an image and find its magnification - Relates focal length to surface curvature and refractive index - Distinguishes real from virtual images and says which can be caught on a screen ## Lesson: Placing images with thin lenses You have traced rays through a converging lens. Use an ideal thin lens in air, with rays close to the optical axis and making small angles with it. A central ray continues straight. For a converging lens, an axis-parallel ray leaves through the far focus. A ray whose incoming line passes through the near focus leaves parallel to the axis. For a real object before the lens, object distance do is positive. Focal length f is positive for a converging lens and negative for a diverging lens. Image distance di is positive for a real image on the outgoing side and negative for a virtual image on the object's side. Measure distances from the lens centre. Use 1/f = 1/do + 1/di, so 1/di = 1/f - 1/do. Keep length units consistent. **Example.** Calculate reciprocal image distance first, then invert it. For f = +8 cm and do = +24 cm, 1/di = 1/8 - 1/24 = 2/24 = 1/12 per cm. Thus di = +12 cm, a real image on the outgoing side. Keep any negative sign when taking a reciprocal. Magnification m = -di/do is image height divided by object height. Positive m means upright; negative m means inverted. Its magnitude gives the size ratio: |m| > 1 is enlarged, |m| = 1 is same-size, and |m| < 1 is reduced. For do = +25 cm and di = +10 cm, m = -10/25 = -0.4: inverted and 0.4 times the object's height. Trace rays from the same object point. Where the actual outgoing rays cross, they form a real image that a screen can receive at its image plane. If only backward extensions meet, they locate a virtual image. That lens alone cannot project it onto a screen at its apparent position. An empty screen does not prove an image is virtual: the screen might be misplaced. Another optical system can form a new image from the outgoing light. For a converging lens with 0 < do < f, the object is inside the focal length. Then 1/di = 1/f - 1/do is negative, so the image is virtual. Also m = -di/do = f/(f-do), which is positive and greater than one: upright and enlarged. At do = f, rays from each object point leave parallel and no finite image forms. For a thin spherical lens, 1/f = (nL/nM - 1)*(1/R1 - 1/R2). nL is lens index; nM is the same surrounding index on both sides. Let light travel left to right. R1 belongs to the first surface and R2 to the second. A radius is positive if its curvature centre is right of that surface and negative if left. A flat face contributes zero reciprocal curvature. **Example.** In air, take nM = 1 and write n = nL. Then 1/f = (n-1)*(1/R1 - 1/R2). For n = 1.6, R1 = +30 cm and R2 = -30 cm, 1/f = 0.6*(1/30 + 1/30) = 0.04 per cm. Thus f = +25 cm. Subtract the signed second reciprocal without losing its minus sign. For a fixed-f single-lens camera model with f > 0 and do > f, di = f*do/(do-f) = f + f*f/(do-f). Moving the object closer reduces do-f, so di increases. Refocus by increasing the lens-to-sensor separation. For a very distant object, di approaches f. Real cameras may use several lens elements and other focusing arrangements. *(drawing: This follows the distance equation for the single-lens camera model.)* **Recap.** Keep distance and curvature signs. The lens equation locates finite images; magnification gives orientation and size. Actual outgoing ray crossings form real images. *Lesson, questions and flashcards written for this dataset.* ## Practice 8 questions on this page, each with its working shown. ## Needs first - [Huygens's principle and the laws of reflection and refraction](https://lightmysky.com/learn/science/huygenss-principle-and-the-laws-of-reflection-and-refraction-mt_xxdniF2__w) ## Opens up - [Curved Mirrors, the Mirror Equation and Real Images](https://lightmysky.com/learn/science/curved-mirrors-the-mirror-equation-and-real-images-mt_7XqB2umthJ) - [Magnifiers, Microscopes and Telescopes](https://lightmysky.com/learn/science/magnifiers-microscopes-and-telescopes-mt_f30rUsqYkN)