Light and Lenses

Lenses form images by refracting light rays. When parallel rays enter a converging lens, they meet at the focal point a distance $f$ from the lens. For real objects at finite distances, the image forms at a distance $\mathrm{d\_i}$ determined by the thin lens equation $\frac{1}{f}=\frac{1}{\mathrm{d\_o}}+\frac{1}{\mathrm{d\_i}}$.

Deriving the Thin Lens Equation

By tracing rays through the lens and assuming its thickness is negligible compared to the radii of curvature, one can derive the relationship above. The object distance $\mathrm{d\_o}$ is measured from the lens to the object. Positive values indicate objects on the side from which light approaches. The image distance $\mathrm{d\_i}$ is positive when the image forms on the opposite side of the lens. Solving for $\mathrm{d\_i}$ yields $\mathrm{d\_i}=\frac{\mathrm{f\; d\_o}}{\mathrm{d\_o}-f}$.

Magnification

The magnification $M$ of the image is the ratio of image height to object height. It is given by $M=\frac{\mathrm{h\_i}}{\mathrm{h\_o}}=-\frac{\mathrm{d\_i}}{\mathrm{d\_o}}$. A negative value indicates the image is inverted relative to the object. When $\left|M\right|$ is greater than one, the image is enlarged; when it is less than one, the image is reduced.

Image Orientation

An image formed on the side opposite the object is real, meaning light actually converges to that point and the image can be projected on a screen. If the object is closer to the lens than the focal length, $\mathrm{d\_i}$ becomes negative, indicating a virtual image. Virtual images cannot be projected but are seen by looking through the lens.

Example Calculation

Imagine a camera lens with $f=50$ cm focusing on an object $\mathrm{d\_o}=200$ cm away. The image distance is $\mathrm{d\_i}=\frac{\mathrm{50\times 200}}{200-50}=66.7$ cm. If the object is 10 cm tall, the image height is $\mathrm{h\_i}=M\mathrm{h\_o}$ with $M=-\frac{66.7}{200}$, giving $\mathrm{h\_i}=-3.3$ cm. The negative sign shows the image is inverted.

Impact of Focal Length

Shorter focal lengths allow lenses to form images of nearby objects but often distort perspective. Longer focal lengths yield larger magnifications at greater distances, which is useful for telescopes and zoom lenses. Understanding how $f$ interacts with $\mathrm{d\_o}$ helps photographers choose the right lens for a desired composition.

Real and Virtual Images

Placing the object at a distance equal to the focal length causes the rays to exit the lens parallel. The image distance tends toward infinity, meaning the lens cannot bring the rays to a focus on a screen. This configuration is useful for magnifying glasses, where the eye interprets the parallel rays as coming from a large, distant object.

Using the Calculator

Input the focal length, object distance, and object height. The calculator computes the image distance and magnification, reporting whether the image is real or virtual based on the sign of $\mathrm{d\_i}$. It also provides the estimated image height, assisting with optical bench experiments or photography projects.

Limitations

The thin lens approximation assumes the lens thickness is negligible and that the optical axis is centered. For complex or thick lenses, additional corrections may be required. Nevertheless, the formula offers a remarkably accurate prediction for many practical situations, including simple cameras, eyeglasses, and lab setups.

Insights for Students

Working through lens calculations builds intuition about how images form and how optical devices operate. From microscopes to projectors, the same basic principles apply. Experimenting with different object distances and focal lengths reveals why certain lenses are chosen for specific tasks.

Real-World Applications

Photographers, astronomers, and even smartphone designers rely on lens formulas every day. Understanding how a change in focal length alters the field of view or how magnification affects depth of field can mean the difference between a sharp image and a blurry one. This calculator provides quick feedback when planning optical setups—whether you are configuring a macro lens for detailed close-ups or designing a telephoto system to capture distant planets. By trying various parameters, you can immediately see how lens choices influence the final image.