Thin Lenses, Mirrors, and Optical Instruments
The thin lens and mirror equation, 1/do+1/di=1/f, handles converging and diverging lenses and concave and convex mirrors all at once, provided you're careful with sign conventions: positive f for converging (lens) or concave (mirror), negative for diverging or convex, and a negative image distance signals a virtual image. Everything about where an image forms, whether it's upright or inverted, and how large it is falls out of this one equation plus the magnification relationship m=−di/do.
What you'll learn
- Apply the thin lens equation to find image distance given object distance and focal length.
- Apply the mirror equation, using the same sign conventions, to concave and convex mirrors.
- Calculate magnification and determine whether an image is upright or inverted, real or virtual.
- Distinguish converging (positive f) from diverging (negative f) lenses, and concave (positive f) from convex (negative f) mirrors.
- Trace principal rays to locate an image graphically for a simple lens or mirror.
- Analyze a simple two-lens optical system by treating the first lens's image as the second lens's object.
1. The Thin Lens and Mirror Equation
One equation, 1/do+1/di=1/f, governs converging lenses, diverging lenses, concave mirrors, and convex mirrors alike — the only thing that changes between them is the sign convention for f: positive for converging lenses and concave mirrors, negative for diverging lenses and convex mirrors.
Magnification follows as m=−di/do: a negative m means an inverted image, a positive m means upright; |m| greater than 1 means enlarged, less than 1 means reduced. A negative di signals a virtual image — one that can't be projected onto a screen, formed on the same side as the object.
2. Multi-Element Systems
When light passes through more than one optical element, the first element's image simply becomes the object for the next — with one geometric adjustment: the object distance for the second element is found from the physical separation between the elements and where the first image actually formed, which may require subtracting rather than reusing the first image distance directly.
Key equations
- 1/do + 1/di = 1/f — The thin lens equation (and, with the same sign convention, the mirror equation) — works for both lenses and mirrors, converging/concave (f>0) or diverging/convex (f<0).
- m = −di/do — Positive m means an upright image, negative means inverted; |m|>1 means enlarged, |m|<1 means reduced.
- di < 0 ⟹ virtual image — A negative image distance signals a virtual image — one that can't be projected onto a screen, formed on the same side as the object (for a lens) or behind the mirror's surface.