Thin Lens Equation
Reference for thin lens equation 1/f = 1/do + 1/di and magnification m = -di/do.
Covers converging and diverging lenses with real and virtual image examples.
The Formula
The thin lens equation relates the focal length of a lens to the object distance and the image distance. It works for both converging (convex) and diverging (concave) lenses when you apply the correct sign conventions.
Converging lenses have positive focal lengths. Diverging lenses have negative focal lengths. A positive image distance means the image is real (on the opposite side from the object). A negative image distance means the image is virtual (on the same side as the object).
Variables
| Symbol | Meaning | Unit |
|---|---|---|
| f | Focal length of the lens | cm or m |
| dₒ | Object distance (from lens to object) | cm or m |
| dᵢ | Image distance (from lens to image) | cm or m |
Magnification
The magnification tells you how large the image is compared to the object. A positive M means the image is upright. A negative M means it is inverted. |M| greater than 1 means the image is enlarged. |M| less than 1 means it is reduced.
Example 1
An object is placed 30 cm from a converging lens with f = 20 cm. Find the image location.
1/f = 1/dₒ + 1/dᵢ
1/20 = 1/30 + 1/dᵢ
1/dᵢ = 1/20 − 1/30 = 3/60 − 2/60 = 1/60
dᵢ = 60 cm — a real, inverted image formed 60 cm on the other side
Example 2
An object is 10 cm from a diverging lens with f = −15 cm. Where is the image?
1/(−15) = 1/10 + 1/dᵢ
1/dᵢ = −1/15 − 1/10 = −2/30 − 3/30 = −5/30
dᵢ = −6 cm
M = −(−6)/10 = 0.6
dᵢ = −6 cm — a virtual, upright image at 60% of original size
When to Use It
- Designing camera lenses and optical instruments
- Calculating image position in microscopes and telescopes
- Prescribing corrective eyeglass or contact lenses
- Understanding projector and magnifying glass optics
- Solving physics optics problems in courses and exams