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Diverging lenses, also known as concave lenses, are optical lenses that spread out light rays that are initially traveling parallel to its principal axis. This is characterized by a negative focal length, meaning that the focal point where the light rays appear to diverge from is on the same side as the incident light.

• Diverging lenses are typically thinner at the center and thicker at the edges.
• They are used in various applications such as in eyeglasses for people with myopia (nearsightedness).

The nature of a diverging lens is such that it always forms a virtual, erect, and diminished image compared to the object. When engaging with problems involving diverging lenses, consider the orientation of the image (erect vs. inverted) and its nature (virtual vs. real). These clues help in understanding the lens equation outcomes and the image formed by the lens, such as seen with the concept of a virtual image used in our example problem.

Converging lenses, often referred to as convex lenses, have the opposite effect of diverging lenses. They focus incoming parallel light rays to a single point known as the focal point, which results in a positive focal length.

• These lenses are usually thicker at the center and thinner at the edges.
• They are frequently used in applications such as magnifying glasses, cameras, and optical instruments.

In our exercise, a converging lens is used in conjunction with a diverging lens to form a real, inverted image. A key aspect of converging lenses is their ability to form real images, which can be projected onto a screen. However, under certain conditions (like when the object is within the focal length), they can also form virtual images. The focal length of the converging lens in our exercise was given as 12.0 cm, and we used the lens formula to determine the object's effective distance in the two-lens system.

The calculation of focal length for lenses is an essential concept in lens optics, as it determines how the lens will converge or diverge light. The lens formula is pivotal in these calculations: \[\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}\]Here, \( f \) represents the focal length, \( d_o \) the object distance from the lens, and \( d_i \) the image distance from the lens. The sign conventions (positive or negative) for \( d_o \) and \( d_i \) depend on the lens type and whether the image is real or virtual.

• Positive focal length indicates a converging lens, while a negative focal length indicates a diverging lens.
• Solve for \( f \) by rearranging the lens formula according to given \( d_o \) and \( d_i \).

In our given problem, we used the formula twice: first, to calculate the object distance for the given real image formed by the converging lens, and then to calculate the focal length of the diverging lens using its respective known distances. This two-step lens optical approach demonstrates the practical application of these calculations in systems where multiple lenses interact.