Visitors at a science museum are invited to sit in a chair...

Visitors at a science museum are invited to sit in a chair to the right of a full-length diverging lens $\left(f_{1}=-3.00 \mathrm{m}\right)$ and observe a friend sitting in a second chair, 2.00 $\mathrm{m}$ to the left of the lens. The visitor then presses a button and a converging lens $\left(f_{2}=+4.00 \mathrm{m}\right)$ rises from the floor to a position 1.60 m to the right of the diverging lens, allowing the visitor to view the friend through both lenses at once. Find (a) the magnification of the friend when viewed through the diverging lens only and (b) the overall magnification of the friend when viewed through both lenses. Be sure to include the algebraic signs $(+\text { or }-)$ with your answers.

Visitors at a science museum are invited to sit in a chair to the right of a full-length diverging lens $\left(f_{1}=-3.00 \mathrm{m}\right)$ and observe a friend sitting in a second chair, 2.00 $\mathrm{m}$ to the left of the lens. The visitor then presses a button and a converging lens $\left(f_{2}=+4.00 \mathrm{m}\right)$ rises from the floor to a position 1.60 m to the right of the diverging lens, allowing
the visitor to view the friend through both lenses at once. Find (a) the magnification of the friend when viewed through the diverging lens only and (b) the overall magnification of the friend when viewed through both lenses. Be sure to include the algebraic signs $(+\text { or }-)$ with your answers.

Key Concepts

Combined Systems of Lenses

When dealing with a system of multiple lenses, the overall effect is determined by considering each lens sequentially. The image produced by the first lens serves as the object for the next lens, and the combined magnification is the product of the individual magnifications. This concept is crucial for understanding complex optical instruments and setups involving multiple lens elements.

Converging Lenses

Converging lenses, or convex lenses, have a positive focal length and can produce either real or virtual images depending on the position of the object relative to the focal point. These lenses are commonly used in optical instruments such as cameras and telescopes to focus light and form clear images.

Diverging Lenses

Diverging lenses, also known as concave lenses, have a negative focal length and always produce images that are reduced in size and virtual when the object is placed at a finite distance from the lens. Understanding the behavior of diverging lenses is important in applications where image correction or specific optical effects are required.

Magnification

Magnification in optics quantifies how much larger or smaller the image is compared to the object. It is given by the formula m = -d_i/d_o, where d_i is the image distance and d_o is the object distance. The negative sign accounts for image inversion; a positive magnification indicates an upright image, while a negative magnification indicates an inverted image. This concept is essential when comparing the sizes of images produced by lens systems.

Thin Lens Equation

The thin lens equation, 1/f = 1/d_o + 1/d_i, relates the focal length of a lens to the object distance and image distance. It is fundamental in geometric optics for determining where an image will form relative to a lens. The equation applies to both converging and diverging lenses, with sign conventions playing a critical role in identifying whether the image is real or virtual.

Sign Conventions in Lens Systems

Sign conventions in lens systems are essential for correctly applying the thin lens equation and determining image characteristics. They involve assigning positive or negative values to distances based on the direction of light travel relative to the optical axis. These conventions help decide whether an image is real or virtual, and they affect the calculation of magnification and image location.

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