Thin Lens Equation
The thin lens equation 1/f = 1/do + 1/di relates focal length, object distance, and image distance.
Learn optics with worked examples.
The Formula
The thin lens equation relates the focal length of a lens to the distance of the object and the distance of the image it forms. It works for both converging (convex) and diverging (concave) lenses, using sign conventions.
This equation is the foundation of geometric optics and is used to design everything from eyeglasses to cameras to telescopes. The "thin lens" assumption means the lens thickness is small compared to the distances involved.
Magnification
Magnification tells you how large the image is compared to the object. A negative magnification means the image is inverted (upside down). |M| > 1 means the image is larger than the object.
Variables
| Symbol | Meaning |
|---|---|
| f | Focal length of the lens (positive for converging, negative for diverging) (meters) |
| do | Object distance from the lens (always positive for real objects) (meters) |
| di | Image distance from the lens (positive = real image, negative = virtual image) (meters) |
| M | Magnification (negative = inverted image) |
| ho, hi | Object height and image height (meters) |
Sign Convention
- Converging (convex) lens: f is positive
- Diverging (concave) lens: f is negative
- Real image (same side as outgoing light): di is positive
- Virtual image (same side as object): di is negative
Example 1
A converging lens has a focal length of 20 cm. An object is placed 50 cm from the lens. Where is the image formed, and what is the magnification?
Apply the thin lens equation: 1/f = 1/do + 1/di
1/20 = 1/50 + 1/di
1/di = 1/20 − 1/50 = 5/100 − 2/100 = 3/100
di = 100/3 ≈ 33.3 cm (positive, so it is a real image)
M = −di/do = −33.3/50 = −0.667
The image forms 33.3 cm from the lens. It is real, inverted, and 2/3 the size of the object.
Example 2
A diverging lens has a focal length of −15 cm. An object is placed 30 cm from the lens. Where is the image?
1/f = 1/do + 1/di
1/(−15) = 1/30 + 1/di
1/di = −1/15 − 1/30 = −2/30 − 1/30 = −3/30 = −1/10
di = −10 cm
M = −(−10)/30 = +0.333
The image forms 10 cm from the lens on the same side as the object (virtual). It is upright and 1/3 the size.
When to Use It
Use the thin lens equation for optics and lens design problems.
- Designing camera lenses and determining focus distances
- Prescribing corrective eyeglasses and contact lenses
- Building telescopes, microscopes, and projectors
- Calculating image size and position in physics problems
- Understanding how magnifying glasses work