Physics
Geometrical optics calculator: thin lenses and spherical mirrors
Pick a relation and clear the quantity you are after. The Gaussian convention applies throughout: p positive for a real object, q positive for a real image, f positive for anything that converges. Diverging lenses and virtual images fall out on their own, with a minus sign.
One equation, and the signs that make it work
Geometrical optics is one of the thriftiest chapters in physics: a single formula, 1/f = 1/p + 1/q, covers converging and diverging lenses, concave and convex mirrors, real and virtual images. All the difficulty has moved into the signs, and that is where nearly every mistake ends up. There are not two formulas, one for the converging lens and one for the diverging: there is one formula and a convention applied without exception.
The Gaussian convention says p is positive for a real object in front of the element, q is positive for a real image on the far side of a lens, and f is positive for anything that converges. The rest follows: if the calculation hands back a negative q, the image is virtual and sits on the object's side; no screen will catch it, but the eye sees it perfectly well — which is exactly the magnifying glass.
The magnification m = −q/p carries two pieces of information in one number. Its size says how much bigger or smaller the image is; its sign says whether the image stands upright or on its head. For a single lens the two are rigidly linked: real implies inverted, virtual implies upright. Only by combining several elements — as in a telescope — do the combinations come apart.
Common mistakes
- Switching formulas for a diverging lens: the formula is the same, only the sign of f changes. Writing 1/f = 1/p − 1/q to make the numbers come out is the fastest way to get every later exercise wrong.
- Forgetting that a mirror's focal length is half its radius: f = R/2. Using R in place of f doubles the image distance.
- Taking every radius in the lensmaker's equation as positive: in a biconvex lens the second radius is negative. With two equal positive radii the formula returns an infinite focal length — a flat plate.
- Adding the focal lengths of two lenses in contact: it is the optical powers that add, not the focal lengths. Two 50 cm lenses together give 4 dioptres, that is 25 cm, not 100 cm.
Frequently asked questions
What is the thin lens equation?
1/f = 1/p + 1/q, where f is the focal length, p the object distance and q the image distance. The same equation holds for spherical mirrors, with f = R/2.
How do I tell whether an image is real or virtual?
From the sign of q. A positive q means a real image: the rays genuinely converge there and a screen will catch it. A negative q means a virtual image, sitting on the object's side and visible only by looking through the lens.
What does a negative magnification mean?
That the image is upside down relative to the object. Its size tells you the scaling: m = −2 means twice as large and inverted, m = 0.5 means half as large and upright.
What are dioptres?
Optical power is the reciprocal of the focal length in metres: P = 1/f. A 50 cm lens has a power of 2 dioptres. The advantage is that for thin lenses in contact the dioptres add algebraically.
Why can't a convex mirror form a real image?
Because its focal length is negative: the reflected rays diverge and never meet. The image is always virtual, upright and reduced, which is why convex mirrors are used for surveillance and wide-angle rear views.
How this calculation works
Thin lens and spherical mirror equation: 1/f = 1/p + 1/q, so f = pq/(p+q), q = fp/(p−f) and p = fq/(q−f). Spherical mirror: f = R/2. Lensmaker's equation: 1/f = (n − 1)(1/R₁ − 1/R₂), with radii positive where the surface bulges towards the object. Transverse magnification: m = −q/p = h'/h. Optical power: P = 1/f with f in metres; for thin lenses in contact the powers add. Sign convention: p > 0 for a real object, q > 0 for a real image, f > 0 for a converging element; a negative result means a virtual object, a virtual image or a diverging element respectively.
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