Question

The near point of a naked eye is $25 \mathrm{cm} .$ When placed at the near point and viewed by the naked eye, a tiny object would have an angular size of $5.2 \times 10^{-5}$ rad. When viewed through a compound microscope, however, it has an angular size of $-8.8 \times 10^{-3}$ rad. (The minus sign indicates that the image produced by the microscope is inverted.) The objective of the microscope has a focal length of $2.6 \mathrm{cm},$ and the distance between the objective and the eyepiece is $16 \mathrm{cm} .$ Find the focal length of the eyepiece.

   The near point of a naked eye is $25 \mathrm{cm} .$ When placed at the near point and viewed by the naked eye, a tiny object would have an angular size of $5.2 \times 10^{-5}$ rad. When viewed through a compound microscope, however, it has an angular size of $-8.8 \times 10^{-3}$ rad. (The minus sign indicates that the image produced by the microscope is inverted.) The objective of the microscope has a focal length of $2.6 \mathrm{cm},$ and the distance between the objective and the eyepiece is $16 \mathrm{cm} .$ Find the focal length of the eyepiece.

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Physics

John D. Cutnell,… 11th Edition

Chapter 26, Problem 93

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Solved by Expert Adnan Gill

05/04/2020

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We have the following data: - Near point of the naked eye, \( D = 25 \, \text{cm} \) - Angular size of the object viewed by the naked eye, \( \theta_{\text{naked}} = 5.2 \times 10^{-5} \, \text{rad} \) - Angular size of the object viewed through the microscope, \( Show more…

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The near point of a naked eye is $25 \mathrm{cm} .$ When placed at the near point and viewed by the naked eye, a tiny object would have an angular size of $5.2 \times 10^{-5}$ rad. When viewed through a compound microscope, however, it has an angular size of $-8.8 \times 10^{-3}$ rad. (The minus sign indicates that the image produced by the microscope is inverted.) The objective of the microscope has a focal length of $2.6 \mathrm{cm},$ and the distance between the objective and the eyepiece is $16 \mathrm{cm} .$ Find the focal length of the eyepiece.

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Key Concepts

Microscope Tube Length

The microscope tube length is the distance between the objective and the eyepiece lenses in a compound microscope. This distance is crucial because it determines the position where the intermediate image (formed by the objective lens) is located relative to the eyepiece, thereby affecting the overall magnification and optical configuration of the microscope.

Lens Focal Length

The focal length of a lens is the distance from the lens at which parallel incoming light rays are brought to a focus. In optical systems, the focal lengths of the objective and eyepiece lenses play a critical role in determining the magnification and resolution of the image. Shorter focal lengths generally offer higher magnification but may also affect the depth of field and overall image quality.

Near Point of the Eye

The near point of the eye is the closest distance at which the human eye can focus on an object clearly, typically around 25 centimeters for a normal eye. This concept is important in optics because it establishes a reference for the angular size of objects as perceived by the eye and serves as a basis for designing optical instruments like microscopes and magnifying glasses.

Angular Magnification

Angular magnification is a measure of how much larger an object appears when viewed through an optical instrument compared to the naked eye. It is defined as the ratio of the angular size of the image as seen through the instrument to the angular size of the object when viewed directly, taking into account the optics of the instrument and the standard viewing distance of the human eye.

Compound Microscope

A compound microscope is an optical instrument that uses two or more lenses (typically the objective and the eyepiece) to create a magnified image of a small object. The objective lens forms a real, enlarged image of the object, which is then further magnified by the eyepiece, resulting in a large and detailed virtual image that can be examined with the human eye.

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