Show that if two thin lenses are close together (s, the...

Show that if two thin lenses are close together $(s$, the distance between the lenses, is negligibly small), the two lenses can be replaced by a single equivalent lens with focal length $f_{\mathrm{eq}} .$ Find the value of $f_{\mathrm{eq}}$ in terms of $f_{1}$ and $f_{2}$.

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Key Concepts

Optical Power and Its Additivity

Optical power, defined as the reciprocal of the focal length (P = 1/f), is a measure of a lens’s ability to converge or diverge light. When two thin lenses are placed in close contact, their individual powers can be added to obtain the equivalent power of the combined system. This leads to the relation 1/f_eq = 1/f1 + 1/f2, which is pivotal in determining the effective focal length of the lens combination.

Lens Formula

The thin lens formula (1/f = 1/s_o + 1/s_i, where f is the focal length, s_o is the object distance, and s_i is the image distance) is a fundamental relation in geometrical optics. It helps in relating object and image positions to the lens’s focal length and plays a crucial role in analyzing lens systems by determining how lenses converge or diverge light.

Thin Lens Approximation

This concept involves the assumption that the thickness of a lens is negligible compared to the object and image distances, which simplifies the optical analysis by allowing the use of simplified formulas. It enables us to treat the lens as a single plane where all refraction occurs, making it easier to analyze combinations of lenses when they are in close proximity.

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