Problem 63 - University Physics with Modern Physics

Question

The focal length of the eyepiece of a certain microscope is 18.0 \(\mathrm{mm}\) . The focal length of the objective is 8.00 \(\mathrm{mm}\) . The distance between objective and eyepiece is 19.7 \(\mathrm{cm} .\) The final image formed by the eyepiece is at infinity. Treat all lenses as thin. (a) What is the distance from the objective to the object being viewed? (b) What is the magnitude of the linear magnification produced by the objective? (c) What is the overall angular magnification of the microscope?

Step-by-Step Solution

Answer

(a) 7.73 mm, (b) 23.38, (c) 324.46

Step 1: Understanding the Setup

The microscope consists of two lenses: the objective lens and the eyepiece lens. The total tube length (distance between eyepiece and objective) is given as 19.7 cm, which is equivalent to 197 mm to match the units of the focal lengths.

Step 2: Calculate the Image Distance for Objective

For the eyepiece to focus the image at infinity, the intermediate image formed by the objective lens must be at the focal point of the eyepiece. Given the total tube length, we can calculate the image distance from the objective by subtracting the eyepiece focal length from the total length. Thus, the image distance (\( v_o \)) is \( 197 \text{ mm} - 18 \text{ mm} = 179 \text{ mm} \).

Step 3: Using Lens Formula for Objective

For the objective lens, use the lens formula \( \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \) where \( f \) is the focal length, \( v \) is the image distance, and \( u \) is the object distance. For the objective, \( f_o = 8 \text{ mm} \) and \( v_o = 179 \text{ mm} \). Substitute into the equation to find \( u \): \[ \frac{1}{u_o} = \frac{1}{f_o} - \frac{1}{v_o} = \frac{1}{8} - \frac{1}{179}. \]Calculate \( u_o \) to find the object distance from the objective.

Step 4: Calculate Magnification by Objective

Linear magnification by the objective is given by \( m_o = \frac{v_o}{u_o} \). Use the values of \( v_o \) and \( u_o \) calculated in the previous steps to find the magnification produced by the objective lens.

Step 5: Calculate Overall Angular Magnification

Overall angular magnification \( M \) of the microscope is \( M = m_o \times m_e \). For the eyepiece, when the final image is formed at infinity, angular magnification is \( m_e = \frac{25}{f_e} \), where \( f_e \) is the focal length of the eyepiece in centimeters. Substitute \( m_o \) and \( m_e \) into this formula to find \( M \).

Step 6: Calculate and Convert Eyepiece Magnification

Given that \( f_e = 18 \text{ mm} = 1.8 \text{ cm} \), substituting into the angular magnification formula for the eyepiece gives \( m_e = \frac{25}{1.8} \). Calculate this value to determine the eyepiece magnification.

Step 7: Calculate the Final Distance and Magnifications

Compute the distances and magnifications using the step-wise calculations:(a) The object distance \( u_o \) from the objective based on the lens formula.(b) The linear magnification \( m_o \) from the objective.(c) The overall angular magnification \( M \) of the microscope using the derived expressions.

Key Concepts

Focal Length

In optics, the focal length of a lens is the distance between the lens and the point where all incoming parallel light rays converge to a single point called the focus. It is a crucial measurement that determines how much a lens converges or diverges light. In our microscope example, we have two lenses: the objective with a focal length of 8 mm and the eyepiece with a focal length of 18 mm.

Focal length provides insight into the optical power of a lens. A shorter focal length means greater lens power, allowing for stronger magnification. In a microscope, the objective lens, with its short focal length, plays a critical role in enlarging small objects for detailed viewing.

Understanding focal length is vital as it affects how light behaves through lenses and how images are formed. The focal length of each lens in a microscope influences how it magnifies and resolves detail.

Linear Magnification

Linear magnification refers to how much larger (or smaller) an image is compared to the actual object. This concept is especially important when using an objective lens in a microscope. Linear magnification is calculated as the ratio of the image distance to the object distance, expressed in the formula:

\( m_o = \frac{v_o}{u_o} \).

In this formula, \( v_o \) represents the image distance (the distance from the lens to the image), and \( u_o \) is the object distance (the distance from the lens to the object). For instance, if the image distance is 179 mm, and the object distance is calculated using the lens formula, this ratio tells us how much the object is magnified by the objective lens.

Linear magnification is an intrinsic property of lenses and is useful in determining how powerful the lens is in making small objects visible in significant detail through a microscope. It helps in examining the fine details of specimens at an elevated size.

Angular Magnification

Angular magnification describes how an optical system like a telescope or microscope increases the angular size of the image compared to the object, as seen from the eye's perspective. It is crucial in microscope optics to find out how much bigger an object appears when using both the objective and eyepiece lenses.

Angular magnification is calculated by the formula:

\( M = m_o \times m_e \),where \( m_o \) is the linear magnification by the objective, and \( m_e \) is the magnification by the eyepiece.

For the eyepiece, specifically when the image is at infinity, its magnification is given by:

\( m_e = \frac{25}{f_e} \),where \( f_e \) is the focal length of the eyepiece. This highlights its power to enlarge the angular size of the image.

Understanding angular magnification is essential in designing microscopes that deliver a sufficiently large, clear image of tiny objects, making it easier to study their details at higher resolutions.

Lens Formula

The lens formula is a simple yet powerful equation connecting three key elements in optics: focal length \( f \), image distance \( v \), and object distance \( u \). It's expressed as:

\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \].

This relationship is fundamental in predicting how lenses form images and their characteristics. By knowing any two of these quantities, the third can be effortlessly found using this formula.

In our microscope example, the lens formula helps determine the object distance \( u_o \) from the objective lens when \( f \) (8 mm) and \( v_o \) (179 mm) are given. Solving this equation offers insights into where the object must be for a sharp image formation at a given distance.

The lens formula serves as a bridge to understanding how to position an object and lens to achieve desired optical outcomes, making it an indispensable tool in optics and lens design.