An object is placed in front of a converging lens in such a...

An object is placed in front of a converging lens in such a position that the lens $(f=12.0 \mathrm{cm})$ creates a real image located $21.0 \mathrm{cm}$ from the lens. Then, with the object remaining in place, the lens is replaced with another converging lens $(f=16.0 \mathrm{cm}) .$ A new, real image is formed. What is the image distance of this new image?

Key Concepts

Thin Lens Equation

The thin lens equation, given by 1/f = 1/d_o + 1/d_i, is a fundamental relation in geometric optics that connects the focal length of a lens (f) with the object distance (d_o) and the image distance (d_i). This equation is used to determine the position of the image formed by a lens, whether it is real or virtual, by appropriately substituting the known values and solving for the unknown quantity.

Focal Length

Focal length is a measure of a lens's ability to converge or diverge light, and it is defined as the distance from the lens at which parallel rays of light converge to a single point (for converging lenses) or appear to diverge from (for diverging lenses). In problems involving lens replacement, the focal length is a critical property of the lens that directly influences the image distance for a given object distance.

Object Distance Consistency

When an optical system is modified, such as replacing one lens with another, the object distance remains unchanged if the object is kept in the same position. This consistency allows one to first determine the object distance using the initial lens parameters and then apply the same object distance in the lens equation for the new lens to find the new image distance.

Real Image Formation

A real image is formed when light rays actually converge at a point after passing through a lens. In the context of lens problems, recognizing that a real image is produced indicates that the resultant image distance will be positive (according to sign conventions), and it can be directly calculated using the lens equation with the appropriate positive values for object distance and focal length.

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