Thin Lens Equation
Reference for the thin lens equation 1/f = 1/do + 1/di.
Calculate focal length, object distance, or image distance for converging and diverging lenses.
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
Key Notes
- Thin lens equation: 1/f = 1/do + 1/di: f is focal length, do is object distance, di is image distance — all measured from the lens center. For a converging (convex) lens, f > 0. For a diverging (concave) lens, f < 0.
- Sign convention for images: di > 0: real image (forms on the far side, can be projected on a screen). di < 0: virtual image (same side as the object, cannot be projected — e.g., magnifying glass used close to an object). Magnification m = −di/do; negative m means inverted image.
- Lensmaker's equation: 1/f = (n−1)(1/R₁ − 1/R₂): Relates focal length to the glass refractive index n and the radii of curvature R₁ and R₂ of the two lens surfaces. Sign convention: R is positive if the center of curvature is to the right of the surface.
- Power of a lens: P = 1/f (diopters): Measured in diopters (D = m⁻¹). A lens with f = 0.5 m has P = +2 D. Lenses in contact: P_total = P₁ + P₂ (powers add, focal lengths do not). Eyeglass prescriptions are written in diopters.
- Applications: The lens equation governs camera lens design and focusing, telescope and microscope objective selection, projector screen distance calculation, contact lens and eyeglass prescription calculation, and fiber optic coupling.