How far apart are the object and image produced by a converging lens with 35 -cm focal length when the object is (a) \(40 \mathrm{cm}\) and (b) \(30 \mathrm{cm}\) from the lens?

A converging lens, also known as a convex lens, is designed to bring light rays together at a point, known as the focal point. Structurally, the lens is thicker at the center than at the edges, which allows it to bend the light rays inward towards a line that passes through the center of the lens, referred to as the optical axis.

When an object is placed in front of a converging lens, the light rays emanating from the object are refracted by the lens, potentially forming an image that can be real or virtual, inverted or upright, and magnified or reduced in size, depending on the object's distance from the lens. Understanding the behavior of converging lenses is crucial in many applications, including eyeglasses, cameras, and microscopes, to enhance visual capability and to capture or magnify images.

The image distance, denoted as 'v' in lens equations, represents the distance from the lens to the image formed by the lens. This distance can be positive or negative, signifying the nature of the image; a positive value usually indicates a real image on the opposite side of the lens from the object, while a negative value signifies a virtual image on the same side as the object.

In our example problem, the calculated negative image distances signify that the images are virtual and on the same side of the lens as the object. It's helpful to visualize this by imagining the light rays diverging after they pass through the lens, projecting a virtual image that could be seen by looking back through the lens from the opposite side.

The object distance 'u' is measured from the lens to the object and is key in determining the nature and proportions of the image produced. By convention, the object distance is given a negative value when the object is positioned on the real/positive side of the lens, which is the case for all conventional, everyday use of lenses.

Calculating the object distance, which was provided as part of our problem, is straightforward. However, interpreting its relationship to the image distance and the focal length using the lens-maker's formula provides comprehensive insights into the lens's imaging properties. A change in the object's distance from the lens drastically affects the characteristics of the image formed, as demonstrated in the variation of results between parts (a) and (b) of our example problem.

The focal length, denoted by 'f', is perhaps the most critical parameter of a lens, defining its converging or diverging capability. It is the distance between the lens and the focal point, where parallel rays of light converge after passing through the lens. The focal length is determined by the lens's shape and the material from which it's made.

In practical terms, the shorter the focal length, the more powerful the lens and the more it bends the light. In the context of our problem, the focal length remained constant at 35 cm, yet its relationship with changing object distances revealed different image positions, underlining the significance of the focal length in optical calculations and lens design.