Problem 79 - Physics for Scientists and Engineers with Modern Physics

Question

(II) An inexpensive instructional lab microscope allows the user to select its objective lens to have a focal length of \(32 \mathrm{~mm}, 15 \mathrm{~mm},\) or \(3.9 \mathrm{~mm} .\) It also has two possible eyepieces with magnifications \(5 \times\) and \(10 \times\). Each objective forms a real image \(160 \mathrm{~mm}\) beyond its focal point. What are the largest and smallest overall magnifications obtainable with this instrument?

Step-by-Step Solution

Answer

Largest magnification: 410.3, smallest magnification: 30.

Step 1: Understanding Objective Lens Magnification

The magnification of the objective lens is given by the formula: \( M_{obj} = \frac{d_i}{f_{obj}} \), where \( d_i \) is the image distance and \( f_{obj} \) is the focal length of the objective lens. Here, \( d_i = 160 \, \text{mm} + f_{obj} \) since the image is formed 160 mm beyond the focal point.

Step 2: Calculate Objective Lens Magnifications

For each objective lens:- **32 mm focal lens**: \( M_{obj1} = \frac{160 \, \text{mm} + 32 \, \text{mm}}{32 \, \text{mm}} = 6 \).- **15 mm focal lens**: \( M_{obj2} = \frac{160 \, \text{mm} + 15 \, \text{mm}}{15 \, \text{mm}} \approx 11.67 \).- **3.9 mm focal lens**: \( M_{obj3} = \frac{160 \, \text{mm} + 3.9 \, \text{mm}}{3.9 \, \text{mm}} \approx 41.03 \).

Step 3: Combine with Eyepiece Magnifications

The overall magnification \( M \) is the product of the objective and eyepiece magnifications: \( M = M_{obj} \times M_{eye} \). Use both eyepieces with each objective to find the combinations.

Step 4: Compute Maximum Overall Magnification

- Objective 3.9 mm with 10x eyepiece: \( M = 41.03 \times 10 \approx 410.3 \). This is the largest magnification possible.

Step 5: Compute Minimum Overall Magnification

- Objective 32 mm with 5x eyepiece: \( M = 6 \times 5 = 30 \). This is the smallest magnification possible.

Key Concepts

Objective Lens Focal Length

Understanding the objective lens focal length is key when using a microscope for magnification purposes. The focal length is the distance from the center of the lens to the focus point. It determines how much the microscope can magnify an object. A shorter focal length results in greater magnification potential.
The lab microscope in the problem provides objective lenses with focal lengths of 32 mm, 15 mm, and 3.9 mm. Each focal length affects the lens’s ability to enlarge an image, with the shortest at 3.9 mm offering the highest level of magnification.

• This is because a shorter focal length bends the light more sharply, forming a larger image.
• Changing the focal length alters the image distance, ensuring the image is formed 160 mm from the focal point.

Understanding this relationship helps you select the right focal length lens for your desired magnification level.

Eyepiece Magnification

The eyepiece magnification is an additional magnifying component that works together with the objective lens. It is the second stage in the two-step magnification process of microscopes.
In the original exercise, the microscope offers eyepieces with magnifications of 5x and 10x. These numbers represent how many times the eyepiece enlarges the image produced by the objective lens.

• For instance, a 5x eyepiece makes the image appear five times larger than it is when viewed directly.
• A 10x eyepiece further increases the enlargement, making objects ten times larger.

This element is crucial for achieving the desired overall magnification, as it multiplies the image size formed by the objective lens. By choosing between different eyepiece magnifications, users can achieve precise levels of detail in their observations.

Image Formation

Image formation in a microscope involves creating a visible and magnified depiction of a small object. It requires two lenses to work: the objective lens and the eyepiece. The image distance is the total distance from the tipped edge of the lens to where the image comes into focus, described as 160 mm plus the focal length of the objective.
The process can be broken down into:

• The objective lens creates a real, inverted image beyond its focal point. This image is not yet seen by the user.
• The eyepiece then takes this real image and magnifies it further, allowing users to see a larger, virtual, and upright final image.

The right combination of the objective focal length and eyepiece magnification will determine the clarity and size of this final image.

Optical Instruments

Optical instruments like microscopes are designed to manipulate and enhance light rays to create magnified views of small details. These tools combine multiple lenses, each with a specific role in image formation and enhancement.
A microscope operates by effective coordination between:

• The objective lens, which captures and focuses light to form an initial image.
• The eyepiece, boosting that image for the viewer's eye.

This symbiotic relationship allows microscopes to achieve high overall magnifications, limited only by the quality and arrangement of the lenses. Thus, selecting and adjusting optical components, such as the ones available in the described lab microscope, enable precise and varied manipulations, perfect for exploring microscopic worlds.