Thin Lens Focal Length

Focal length from object and image distances

The Gauss thin-lens equation reads 1/f = 1/p + 1/p', where f is the focal length, p the object distance and p' the image distance. When you know p and p', rearranging gives f = (p·p')/(p + p'). Watch the signs: f > 0 for converging lenses (biconvex), and f < 0 for diverging lenses (biconcave). Linear magnification comes out as A = -p'/p. Vergence V = 1/f, with f expressed in meters, is reported in diopters (D), the same unit you'll see on an eyeglass prescription. A normal emmetropic eye carries roughly 60 D of refractive power. Take an object at 30 cm with the image landing at 60 cm: f = (30·60)/(30+60) = 20 cm, so V = 1/0.20 = 5 D and A = -2, meaning the image is real, inverted and twice the size.

Applications

You'll meet this calculation when working out the focal length of eyeglass and contact lenses. Myopia is corrected with diverging lenses and negative diopters, while hyperopia calls for converging lenses with positive diopters. It also turns up in LASIK refractive surgery, where the cornea is reshaped to change its effective focal length, in camera and microscope optics, in telescope design, and in any physics lab where students measure p and p' on an optical bench to back out f.

FAQ

What if I get a negative f? Then the lens is diverging, which is what you'd expect for myopia correction. A -2.00 D prescription works out to f = -50 cm.

How do I know the image is real? A positive p' means a real image, one you can project onto a screen sitting on the far side of the lens. A negative p' means it's virtual, on the same side as the object and only visible by looking through the lens.

Why does the formula assume a thin lens? The thin-lens approximation treats the lens thickness as negligible next to p and p'. Once that no longer holds, you need the lensmaker's equation, which brings in the two surface radii and the refractive index.