The Lens Equation
Simple Explanation
The lens equation relates an object's distance from a lens, the resulting image distance, and the lens's focal length. It lets you calculate exactly where an image will form, given the object's position and the lens being used.
Why Do We Need It?
The lens equation is the essential quantitative tool for designing and predicting how any optical system with lenses — cameras, telescopes, microscopes, eyeglasses — will actually form images.
Formula
The Lens Equation
1/v − 1/u = 1/f
Relates an object's distance from a lens, the resulting image distance, and the lens's focal length (using the convention that distances are measured from the lens, with light travelling in the positive direction).
- v
- — Image distance from the lens
- u
- — Object distance from the lens (negative, since the object is on the incoming side)
- f
- — Focal length of the lens (positive for a converging lens, negative for a diverging lens)
When to use it: Use to find the image distance, object distance, or focal length of a lens, given the other two.
Worked Example
Finding image distance
An object is placed 30 cm from a converging lens with a focal length of 10 cm. Find the image distance (u = -30 cm, f = +10 cm, using the sign convention).
Why Does This Work?
This formula follows from carefully applying refraction (Snell's law) at each of the lens's two curved surfaces and combining the results — the specific form (1/v − 1/u = 1/f) reflects a consistent sign convention where distances are measured from the lens, with the direction light travels taken as positive.
Real-Life Example
Focusing a camera lens
A camera automatically adjusts its lens position to keep a subject in sharp focus at different distances.
The camera's autofocus system essentially solves the lens equation in real time, adjusting the lens (and thus the effective object/image distance relationship) so the image forms precisely on the camera's sensor.
Practice
An object is placed 20 cm from a converging lens with a focal length of 5 cm (u = -20 cm, f = 5 cm). Find the image distance v.
MediumCommon mistake
Forgetting to apply the correct sign convention consistently — object distances, image distances, and focal lengths must all follow the same sign rules (e.g. object distance is conventionally negative), or the equation gives an incorrect result.
Quick Review
- 1/v − 1/u = 1/f relates object distance, image distance, and focal length.
- Consistent sign convention is essential (u is typically negative).
- Solving for v tells you exactly where the image forms.