Problem 88 - Physics

Question

In a compound microscope, the objective has a focal length of \(0.60 \mathrm{~cm},\) while the eyepiece has a focal length of \(2.0 \mathrm{~cm} .\) The separation between the objective and the eyepiece is \(L=12.0 \mathrm{~cm}\). An other microscope that has the same angular magnification can be constructed by interchanging the two lenses, provided that the distance between the lenses is adjusted to a value \(L^{\prime}\). Find \(L^{\prime}\).

Step-by-Step Solution

Answer

The new length \(L'\) must be \(13.4\,\mathrm{cm}\).

Step 1: Understanding Angular Magnification

Angular magnification of a compound microscope is given by the formula:\[M = \left(\frac{L - f_o}{f_o}\right) \left(\frac{25}{f_e}\right) \]where \(f_o\) is the focal length of the objective lens, \(f_e\) is the focal length of the eyepiece lens, and \(L\) is the tube length (distance between the lenses). Initially, \(f_o = 0.60\,\mathrm{cm}\) and \(f_e = 2.0\,\mathrm{cm}\). The tube length \(L = 12.0\,\mathrm{cm}\).

Step 2: Calculate Initial Angular Magnification

Substitute the known values into the angular magnification formula:\[M = \left(\frac{12.0 - 0.60}{0.60}\right) \left(\frac{25}{2.0}\right) \]Calculate the expression inside each parenthesis. First:\(\frac{12.0 - 0.60}{0.60} = \frac{11.4}{0.60} = 19\).Second: \(\frac{25}{2.0} = 12.5\).Multiply them: \(M = 19 \times 12.5 = 237.5\).

Step 3: Interchange Lenses and Use Same Magnification

Now, switch the lenses so the new objective focal length \(f'_o = 2.0\,\mathrm{cm}\) and the eyepiece focal length \(f'_e = 0.60\,\mathrm{cm}\). We need the new length \(L'\) such that the angular magnification \(M'\) is the same, 237.5.Thus:\[M' = \left(\frac{L' - 2.0}{2.0}\right) \left(\frac{25}{0.60}\right) = 237.5\]

Step 4: Calculate New Tube Length \(L'\)

First, solve the magnification equation for \(L'\):\[\left(\frac{L' - 2.0}{2.0}\right) \times 41.67 = 237.5\] where \(\frac{25}{0.60} \approx 41.67\).Divide both sides by 41.67: \[\frac{L' - 2.0}{2.0} = \frac{237.5}{41.67}\approx 5.7\]Multiply by 2.0: \[L' - 2.0 = 11.4\]Add 2.0 to both sides: \[L' = 13.4\]\ cm.

Key Concepts

Compound Microscope

A compound microscope is a powerful optical instrument used for magnifying small objects. It consists of two main lenses: an objective lens and an eyepiece lens. The structure and function of the compound microscope enhance its magnification capabilities, making it an essential tool for scientific research and education.

The compound microscope works by focusing light through the objective lens to form a real, inverted image inside the microscope. This image is further magnified by the eyepiece lens, providing much greater magnification than a simple magnifying lens.

• Objective Lens: Close to the specimen, forms an enlarged image.
• Eyepiece Lens: Magnifies the image created by the objective.

Combining these two lenses allows the compound microscope to reach high levels of magnification, making it invaluable for viewing tiny structures, such as cells and microorganisms.

Focal Length

The focal length is a critical aspect of lens functionality, determining how strongly the lens bends light to converge at a point. In the context of a microscope, the focal length of each lens—objective and eyepiece—plays a crucial role in determining the microscope’s overall magnifying power.

The focal length tells us how far light rays must converge to a point after passing through the lens. Shorter focal lengths mean greater bending of light, which generally yields higher magnification. In our problem:

• Objective lens has a focal length ( $f_o$) of 0.60 cm.
• Eyepiece lens has a focal length ( $f_e$) of 2.0 cm.

These values affect how each lens magnifies the specimen and the constellation of their focal lengths determines the microscope's angular magnification ability.

Objective Lens

The objective lens in a microscope is the lens closest to the specimen. It is responsible for creating the initial magnified image of the sample. The characteristics of the objective lens, such as its focal length, directly influence the microscope's ability to enlarge the specimen adequately.

In the exercise, the initial setup uses an objective lens with a focal length of 0.60 cm. This short focal length indicates that the lens can significantly bend light rays to create a magnified image. The objective lens's position, close to the specimen, allows for precise focusing and sharpness, crucial for detailed observations.

When the lenses are interchanged, the objective lens's role switches, which directly influences the overall tube length required to maintain the desired magnification.

Eyepiece Lens

The eyepiece lens, also known as the ocular lens, is the part of the microscope that magnifies the image produced by the objective lens. The viewer looks through the eyepiece lens to see the final enlarged image.

In our scenario, the eyepiece lens has an original focal length of 2.0 cm and is interchanged to act as an objective lens. The eyepiece's function is vital, as it amplifies the already-magnified image, enabling the detailed observation needed for minute objects.

• Shorter focal lengths allow the eyepiece to provide higher magnification.
• The eyepiece's positioning further away from the specimen than the objective allows it to properly enlarge the pre-magnified image.

This double amplification setup by the eyepiece lens enables the compound microscope to reach significant angular magnification.