The thin lens equation connects the position of an object, the position of its image, and the focal length of a lens. It is one of the most useful tools in geometric optics because it predicts where images form and whether they are real or virtual. Cameras, eyeglasses, microscopes, telescopes, and projectors all rely on this relationship.
A simple ray diagram helps turn the equation into a clear visual model of light bending through a lens.
For a thin lens, light rays are treated as if they refract at one central plane through the lens. A convex lens can bring parallel rays together at a focal point, while a concave lens spreads rays as if they came from a focal point. The equation 1/f = 1/do + 1/di works with sign conventions to describe real and virtual images.
Magnification then connects image size to object size using m = hi/ho = -di/do.
Understanding Physics: The Thin Lens Equation
The word thin is an approximation, not a description of every real lens. A real lens has two curved surfaces and some thickness. Light bends at both surfaces.
In the thin lens model, those bends are treated as one bend at a central plane. This works well when the lens thickness is small compared with the object and image distances.
It becomes less accurate for thick camera lenses, very wide angle systems, or lenses used far from their central region. Lens makers handle those cases with more detailed models that track each surface separately.
Ray diagrams show why image position can be calculated. A ray that enters parallel to the main axis leaves in a direction linked to the focal point. A ray sent through the center of a thin lens changes direction very little.
Where these rays meet gives the image location. For a diverging lens, the outgoing rays do not actually meet. Their backward extensions meet on the object side.
This is why a virtual image can be seen by an eye but cannot be caught on a screen. The eye lens redirects the rays again and forms a real image on the retina.
The equation uses reciprocals of distance, so careful arithmetic matters. First convert every distance into the same unit, such as centimetres or metres. Then isolate the reciprocal of the unknown distance before taking the final reciprocal.
A negative result is useful information rather than a mistake. It tells you that the rays only appear to come from that location. There are important boundary cases.
When an object is at a converging lens focal point, the outgoing rays are parallel and the image is effectively extremely far away. When the object is very far away, its image forms close to the focal point. This explains why a camera focused on distant scenery places its sensor near a fixed position.
Focusing devices change a distance inside the system. In a phone camera, small lens elements move by tiny amounts so that light from objects at different distances lands sharply on the sensor. In a projector, moving the lens changes where the real image forms on the wall.
Reading glasses work because they create a virtual image at a distance the eye can focus on comfortably. When solving problems, draw the principal axis, mark the focal points, and decide which side of the lens each point belongs on before using numbers. Keep the object height above the axis positive by convention.
The sign of the image height then shows whether the image is upright or upside down. A quick sketch often catches an answer that is mathematically calculated but physically impossible.
Key Facts
- Thin lens equation: 1/f = 1/do + 1/di.
- Magnification equation: m = hi/ho = -di/do.
- For a converging lens, f is positive; for a diverging lens, f is negative.
- A real image has positive di and forms on the opposite side of the lens from the object.
- A virtual image has negative di and forms on the same side of the lens as the object.
- If |m| > 1 the image is enlarged, if |m| < 1 the image is reduced, and a negative m means the image is inverted.
Vocabulary
- Thin lens
- A lens whose thickness is small compared with the object distance, image distance, and radii of curvature, so refraction can be modeled at one plane.
- Focal length
- The distance from the center of a lens to the focal point where parallel rays converge or appear to diverge.
- Object distance
- The distance do from the object to the center of the lens, measured along the optical axis.
- Image distance
- The distance di from the center of the lens to the image location, measured along the optical axis.
- Magnification
- The ratio of image height to object height, which also equals negative image distance divided by object distance.
Common Mistakes to Avoid
- Using the wrong sign for focal length, because converging lenses have positive f while diverging lenses have negative f under the standard convention.
- Forgetting that a negative image distance means a virtual image, because the image is on the same side of the lens as the object and cannot be projected on a screen.
- Dropping the negative sign in m = -di/do, because that sign tells whether the image is upright or inverted.
- Mixing units in the thin lens equation, because f, do, and di must all use the same distance unit before solving.
Practice Questions
- 1 A convex lens has focal length f = 10.0 cm. An object is placed do = 30.0 cm from the lens. Find the image distance di and the magnification m.
- 2 A lens forms a real image 24.0 cm from the lens when the object is 12.0 cm away. Find the focal length and state whether the lens is converging or diverging.
- 3 An object is placed inside the focal length of a convex lens. Explain whether the image is real or virtual, upright or inverted, and larger or smaller than the object.