A certain microscope is provided with objectives that have...

A certain microscope is provided with objectives that have focal lengths of $16 \mathrm{~mm}, 4 \mathrm{~mm},$ and $1.9 \mathrm{~mm}$ and with eyepieces that have angular magnifications of $5 \times$ and $10 \times .$ Each objective forms an image $120 \mathrm{~mm}$ beyond its second focal point. Determine (a) the largest overall angular magnification obtainable and (b) the smallest overall angular magnification obtainable.

Key Concepts

Compound Microscope

A compound microscope uses two sets of lenses—an objective and an eyepiece—to achieve a high level of magnification. The objective creates an enlarged real image of the specimen at an intermediate position, which is then further magnified by the eyepiece for the viewer. This multi-stage process allows the microscope to produce a total magnification that is the product of the individual magnifications.

Objective Lens

The objective lens is the first optical element in a microscope system, positioned closest to the specimen. Its function is to gather light from the specimen and to form an enlarged, real intermediate image. The magnification provided by the objective is mainly determined by its focal length and the fixed distance (tube length) from the specimen to the intermediate image plane.

Eyepiece Lens

The eyepiece (or ocular) acts as a magnifier for the intermediate image formed by the objective lens. It provides an angular magnification, making the final image appear larger to the observer. The eyepiece’s angular magnification, often given as a multiple (e.g., 5× or 10×), directly contributes to the overall magnification of the microscope.

Angular Magnification

Angular magnification refers to the factor by which an optical instrument increases the visual angle under which an object is seen, compared to the naked eye. In microscopes, the overall angular magnification is the product of the angular magnification provided by the eyepiece and the magnification from the objective lens, enhancing the apparent size of the specimen.

Magnification Calculation in Compound Microscopy

The overall magnification of a compound microscope is obtained by multiplying the magnification contributions from the objective lens and the eyepiece. The objective magnification is typically calculated using the ratio of the tube length (distance from specimen to intermediate image) to the focal length of the objective, while the eyepiece magnification is provided directly. By selecting different combinations of objectives and eyepieces, one can achieve different total magnifications, ranging from a maximum to a minimum value.

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