Two-lens systems. In Fig. 34-45, stick figure O (the object)...

Two-lens systems. In Fig. $34-45$, stick figure $O$ (the object) stands on the common central axis of two thin, symmetric lenses, which are mounted in the boxed regions. Lens 1 is mounted within the boxed region closer to $O$, which is at object distance $p_{1}$. Lens 2 is mounted within the farther boxed region, at distance $d .$ Each problem in Table $34-9$ refers to a different combination of lenses and different values for distances, which are given in centimeters. The type of lens is indicated by $\mathrm{C}$ for converging and D for diverging; the number after C or $\mathrm{D}$ is the distance between a lens and either of its focal points (the proper sign of the focal distance is not indicated). Find (a) the image distance $i_{2}$ for the image produced by lens 2 (the final image produced by the system) and (b) the overall lateral magnification $M$ for the system, including signs. Also, determine whether the final image is (c) real (R) or virtual (V), (d) inverted (I) from object $O$ or noninverted (NI), and (e) on the same side of lens 2 as object $O$ or on the opposite side. $\begin{array}{lllll}\mathbf{8 1} & +12 & \mathrm{C} .8 .0 & 32 & \mathrm{C}, 6.0\end{array}$

Two-lens systems. In Fig. $34-45$, stick figure $O$ (the object) stands on the common central axis of two thin, symmetric lenses, which are mounted in the boxed regions. Lens 1 is mounted within the boxed region closer to $O$, which is at object distance $p_{1}$. Lens 2 is mounted within the farther boxed region, at distance $d .$ Each problem in Table $34-9$ refers to a different combination of lenses and different values for distances, which are given in centimeters. The type of lens is indicated by $\mathrm{C}$ for converging and D for diverging; the number after C or $\mathrm{D}$ is the distance between a lens and either of its focal points (the proper sign of the focal distance is not indicated).
Find (a) the image distance $i_{2}$ for the image produced by lens 2 (the final image produced by the system) and (b) the overall lateral magnification $M$ for the system, including signs. Also, determine whether the final image is (c) real (R) or virtual (V),
(d) inverted
(I) from object $O$ or noninverted (NI), and
(e) on the same side of lens 2 as object $O$ or on the opposite side.
$\begin{array}{lllll}\mathbf{8 1} & +12 & \mathrm{C} .8 .0 & 32 & \mathrm{C}, 6.0\end{array}$

Key Concepts

Converging and Diverging Lenses

Converging lenses (convex) focus parallel light rays to a point and have positive focal lengths, whereas diverging lenses (concave) cause parallel rays to spread out and are characterized by negative focal lengths. Knowing the behavior of these lenses is critical when analyzing and designing optical systems, as their different effects on light propagation determine the nature and location of the resulting images.

Real vs Virtual Images

Understanding the difference between real and virtual images is key in optical systems. A real image is formed when light rays converge and can be projected onto a screen, while a virtual image occurs when rays appear to diverge from a location; such images cannot be captured on a screen. This distinction affects both the sign of the image distance and the interpretation of image properties.

Multistep Imaging System

In a two-lens system, the image produced by the first lens becomes the object for the second lens. Analysis requires sequential application of the thin lens equation and magnification formula for each lens, ensuring that the object distances and image distances are correctly interpreted between the successive lenses.

Magnification

Magnification is defined as the ratio of the image height to the object height, which in lens systems is given by m = -i/p. This value not only indicates the size of the image relative to the object, but the sign also indicates image orientation (inversion or non-inversion). In multi-lens systems, the overall magnification is the product of the magnifications of the individual lenses.

Thin Lens Equation

The thin lens equation, given by 1/p + 1/i = 1/f, relates the object distance (p), the image distance (i), and the focal length (f) of a lens. It is fundamental in determining where an image is formed relative to the lens and is used for both single and multi-lens systems.

Sign Conventions

Accurate use of sign conventions is crucial when applying the thin lens formula and calculating magnification. Object and image distances take signs depending on their positions relative to the lens, and focal lengths are positive for converging lenses and negative for diverging lenses. Proper application of these conventions ensures correct determination of image characteristics.

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