Problem 58 - College Physics

Question

\bullet A microscope with an objective of focal length 8.00 \(\mathrm{mm}\) and an eyepiece of focal length 7.50 \(\mathrm{cm}\) is used to project an image on a screen 2.00 \(\mathrm{m}\) from the eyepiece. Let the image distance of the objective be 18.0 \(\mathrm{cm} .\) (a) What is the lateral magnification of the image? (b) What is the distance between the objective and the eyepiece?

Step-by-Step Solution

Answer

(a) Calculate total magnification using lens equations. (b) Sum objective image distance and modified eyepiece screen distance to find distances.

Step 1: Understanding the Problem

We are given a microscope with an objective lens and an eyepiece lens. The focal lengths of the lenses, the image distance from the objective, and the final image distance from the eyepiece are given. We need to determine the lateral magnification and the distance between the objective and the eyepiece.

Step 2: Convert All Units to Meters

Convert the given focal lengths from millimeters and centimeters to meters: For the objective: 8.00 mm = 0.008 m For the eyepiece: 7.50 cm = 0.075 m The image distance of the objective is given as 18.0 cm, so convert it to meters: 18.0 cm = 0.18 m.

Step 3: Calculate Objective Lens Magnification

The lateral magnification ( \(M_o\) ) of the objective lens can be calculated using the formula: \[M_o = \frac{-d_i}{d_o}\]Where \(d_i\) is the image distance (0.18 m), and \(d_o\) is the object distance. Calculate \(d_o\) using the lens formula: \[\frac{1}{f_o} = \frac{1}{d_o} + \frac{1}{d_i}\]Rearrange to find \(d_o\):\[d_o = \frac{1}{\frac{1}{f_o} - \frac{1}{d_i}} = \frac{1}{\frac{1}{0.008} - \frac{1}{0.18}}\]Calculate \(d_o\) and then substitute to find \(M_o\).

Step 4: Determine Total Magnification

The total magnification of the microscope system ( \(M_{total}\) ) is the product of the magnifications by the objective and the eyepiece. The magnification by the eyepiece ( \(M_e\) ) when projecting an image is given by: \[M_e = \frac{s}{f_e}\]Where \(s\) is the distance from the eyepiece to the screen (2.00 m) and \(f_e\) is the focal length of the eyepiece (0.075 m). Calculate \(M_e\) and multiply by \(M_o\) to get \(M_{total}\).

Step 5: Calculate Distance Between Lenses

The distance ( \(L\) ) between the objective and the eyepiece is:\[L = d_o + (s - f_e)\]This accounts for the distance where each lens forms its part of the image. Substitute the values for \(d_o\) and (s - f_e) to find \(L\).

Step 6: Final Answers

(a) Using the calculations from the previous steps, determine \(M_{total}\) as the lateral magnification of the image.(b) Substitute appropriate values into the equation for \(L\), providing the final answer.

Key Concepts

Lateral Magnification

Lateral magnification is an important concept when using microscopes. It helps us understand how much larger the image is compared to the object. This is crucial for anyone working in fields involving microscopes, such as biology and material sciences. In simple terms, lateral magnification (\(M\) ) is calculated using the ratio of the image distance (\(d_i\) ) to the object distance (\(d_o\) ).

• Formula: \(M_o = \frac{-d_i}{d_o}\)
• The negative sign indicates image inversion, a common outcome in optical systems.

Before calculating, ensure you have values for both \(d_i\) (image distance) and \(d_o\) (object distance). The higher the magnification, the larger the image appears compared to the object.

Lens Formula

The lens formula is fundamental to understanding optics, especially when working with microscopes. It relates the focal length (\(f\) ), image distance (\(d_i\) ), and object distance (\(d_o\) ) in a straightforward equation:\[ \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}\]The lens formula allows you to calculate any one of these variables if the other two are known. This makes it a versatile tool for determining how lenses will behave in specific setups. To find \(d_o\) or \(d_i\) , rearrange the equation accordingly:

• For \(d_o\): \[d_o = \frac{1}{\frac{1}{f} - \frac{1}{d_i}}\]
• For \(d_i\): \[d_i = \frac{1}{\frac{1}{f} - \frac{1}{d_o}}\]

Calculating these values precisely is important for optimizing microscope setup and achieving the desired magnification and clarity.

Focal Length Conversion

Converting focal lengths into consistent units is key to solving optical equations accurately. In optics, the focal length represents how strongly a lens converges or diverges light. When beginning any calculations, ensure that all measurements are in the same unit, typically meters or centimeters.

Here’s how conversions work:

• Millimeters to meters: divide by 1000.
• Centimeters to meters: divide by 100.

For example, converting a focal length of 8.00 mm to meters involves:\[8.00\, \text{mm} = 0.008\, \text{m}\]Converting correctly avoids errors and makes equations like the lens formula usable without conversion-related mistakes.

Distance Between Lenses

The distance between lenses in a microscope affects how images are formed and magnified. This measurement is crucial as it involves both the objective lens and the eyepiece lens. Understanding this distance helps in setting up the microscope for best performance. To find the distance \(L\) between the objective and eyepiece, you consider both object distance (\(d_o\)) and the projection distance (\(s - f_e\)).

The formula to calculate the distance is:\[L = d_o + (s - f_e)\]Here,

• \(d_o\) is the distance from the objective lens to the object.
• \(s\) is the distance from the eyepiece to where the image is projected.
• \(f_e\) is the focal length of the eyepiece.

Properly calculating \(L\) ensures the compounded magnification from both lenses functions as intended, providing clear and enlarged images. This knowledge is fundamental for both amateur and professional microscope users.