Library / Foundations: Physics / FPH-46
Foundations: Physics
Image distance for a thin lens
Distance from a thin converging lens to the image, from the focal length and the object distance.
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- Result
- v [cm]
For a thin converging lens, 1/f = 1/u + 1/v, where u is the object distance, v the image distance and f the focal length. Solving for v gives v = fu/(u − f). A positive v is a real image on the far side of the lens (it can be caught on a screen); a negative v is a virtual image on the same side as the object. An object at the focal point (u = f) gives an image at infinity.
Distance from a thin converging lens to the image, from the focal length and the object distance. A positive answer is a real image on the far side; a negative answer is a virtual image on the same side as the object.
Calculator
- Magnification, −v/u−0.5
- Total distance from object to image, u + v45 cm
- Power of the lens, 100/f (f in cm)10 dioptre
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- An object 30 cm from a 10 cm lens forms a real image 15 cm from the lens, magnified by −0.5 (inverted and smaller). Check: 1/10 = 1/30 + 1/15.
- Moving the object closer than 10 cm (inside the focal point) makes v negative: the image is virtual, upright and magnified, as in a magnifying glass.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (cm) | Local step | Change in result |
|---|---|---|---|---|---|---|
| f Focal length (converging lens, positive) | 1 to 50 cm | −2,280 | 699.7 | 2,980 | +10% | +15.8 % |
| u Object distance (real object, positive) | 1 to 200 cm | −32.25 | 33.44 | 65.69 | +10% | −4.35 % |
Sweep of Object distance (real object, positive)
| Object distance (real object, positive) (cm) | Image distance (cm) |
|---|---|
| 1 | −1.111 |
| 25.88 | 16.3 |
| 50.75 | 12.45 |
| 75.63 | 11.52 |
| 100.5 | 11.1 |
| 125.4 | 10.87 |
| 150.3 | 10.71 |
| 175.1 | 10.61 |
| 200 | 10.53 |
Sweep one variable
Variables and how each one contributes
Image distance
What the equation gives you. Everything else feeds it.
Focal length (converging lens, positive)
The focal length of the converging lens, in cm. It is positive for a converging lens. Raises the image distance for an object beyond the focal point, and the effect grows rapidly as the object approaches the focal point. Raising it by 1 cm (to 11 cm) moves the result by +15.8 %, from 15 to 17.37 cm.
Object distance (real object, positive)
The distance from the lens to the object, in cm, for a real object (positive). Lowers the image distance for an object beyond the focal point: moving the object away brings the image closer to the focal point. Raising it by 5 cm (to 35 cm) moves the result by −6.67 %, from 15 to 14 cm.
Worked example with the values above
- Numerator: f u = 10 × 30 = 300 cm².
- Denominator: u − f = 30 − 10 = 20 cm.
- v = 300 / 20 = 15 cm.
Assumptions and limits
- A thin lens in the paraxial approximation: rays close to the axis and small angles.
- A real object (u > 0) and a converging lens (f > 0); a virtual image gives a negative v. The object must not be at the focal point, where the image is at infinity.
Common mistakes
- Using a negative f for a converging lens.
- Mixing units, such as f in mm and u in cm.
- Reading a negative v as a mistake: it just means a virtual image.
Statistics and uncertainty
The image distance is very sensitive near the focal point: as u approaches f, a small error in u produces a large error in v. Measurements of f by this method are best made with the object well beyond 2f, where the result is least sensitive.
Where it is used
- Placing a detector, screen or eyepiece for a camera, sight or telescope.
- Finding a lens focal length from measured distances.
Maths you need
- Focal length: distance at which parallel rays meet
- Real and virtual images: on opposite sides of the lens