Thin Lens Equation
The thin lens equation relates focal length, object distance, and image distance.
Learn 1/f = 1/do + 1/di with examples.
The Formula
The thin lens equation describes how a thin lens forms an image. It applies to both converging (convex) and diverging (concave) lenses.
Variables
| Symbol | Meaning |
|---|---|
| f | Focal length of the lens |
| dₒ | Object distance (from lens to object) |
| dᵢ | Image distance (from lens to image) |
Sign Convention
| Quantity | Positive | Negative |
|---|---|---|
| f | Converging (convex) lens | Diverging (concave) lens |
| dₒ | Object on incoming side (real object) | Virtual object |
| dᵢ | Image on outgoing side (real image) | Image on incoming side (virtual image) |
Magnification
If |M| > 1, the image is enlarged. If |M| < 1, the image is reduced. If M is negative, the image is inverted.
Example 1 — Converging Lens (Real Image)
An object is 30 cm from a converging lens with focal length 10 cm. Where does the image form?
1/dᵢ = 1/f − 1/dₒ = 1/10 − 1/30
1/dᵢ = 3/30 − 1/30 = 2/30 = 1/15
dᵢ = 15 cm
M = −15/30 = −0.5
The image forms 15 cm behind the lens. It is real, inverted, and half the size of the object.
Example 2 — Diverging Lens (Virtual Image)
An object is 20 cm from a diverging lens with focal length −10 cm. Where does the image form?
1/dᵢ = 1/f − 1/dₒ = 1/(−10) − 1/20
1/dᵢ = −2/20 − 1/20 = −3/20
dᵢ = −20/3 ≈ −6.67 cm
M = −(−6.67)/20 = +0.33
The image forms 6.67 cm in front of the lens (virtual). It is upright and one-third the size.
When to Use It
- Designing camera and telescope optics
- Calculating magnification for microscopes and magnifying glasses
- Determining image placement in projectors
- Understanding how eyeglasses correct vision