Problem 77 - Physics for Scientists and Engineers with Modern Physics

Question

(II) A microscope has a 1.8 -cm-focal-length eyepiece and a \(0.80-\mathrm{cm}\) objective. Assuming a relaxed normal eye, calculate (a) the position of the object if the distance between the lenses is \(16.8 \mathrm{~cm},\) and \((b)\) the total magnification.

Step-by-Step Solution

Answer

The position of the object is 1 cm from the objective; total magnification is 31.25.

Step 1: Understand the Lens Distances

The total distance between the lenses is given as 16.8 cm. This distance is the sum of the image distance of the objective (\(d_{i_obj} \)) and the object distance for the eyepiece (\(d_{o_eye}\)). Mathematically this can be expressed as \[ d_{i_{obj}} + d_{o_{eye}} = 16.8 \text{ cm} \].

Step 2: Use Lens Formula for Objective

The lens formula is \[\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \]. For the objective lens with focal length \(f_{obj} = 0.80 \text{ cm}\), the position of the object is \(d_{o_{obj}}\), and the image distance is \(d_{i_{obj}}\). The formula becomes: \[ \frac{1}{0.80} = \frac{1}{d_{o_{obj}}} + \frac{1}{d_{i_{obj}}} \].

Step 3: Combine lens separation and lens formula equations

Replace \(d_{i_{obj}}\) from Step 1 in the formula \[ \frac{1}{0.80} = \frac{1}{d_{o_{obj}}} + \frac{1}{16.8 - d_{o_{org}}} \] and solve for \(d_{o_{obj}}\).

Step 4: Calculate Magnification of Objective

The magnification of the objective lens is \(M_{obj} = \frac{d_{i_{obj}}}{d_{o_{obj}}}\).

Step 5: Use Lens Formula for Eyepiece

Using the lens formula for the eyepiece (focal length \(f_{eye} = 1.8 \text{ cm}\)), and because image at eyepiece is at infinity (relaxed viewer), setup the equation \[\frac{1}{d_{o_{eye}}} = \frac{1}{1.8}\]. Solve for \(d_{o_{eye}}\).

Step 6: Calculate Total Magnification

The total magnification \(M\) of the microscope is the product of the magnifications of the objective and the eyepiece \( M = M_{obj} \times M_{eye} \), where \(M_{eye} = \frac{25}{f_{eye}}\) assuming near point distance of 25 cm. Calculate this product.

Key Concepts

Lens Formulas

Understanding lens formulas is fundamental for calculating distances and magnifications in optical systems like microscopes. A microscope uses lenses to magnify small objects, and this operation is grounded on lens formulas. The key lens formula relates the focal length (abla), object distance (abla ext{o}) and image distance (abla ext{i}) by:

\[\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}.\]

This formula allows you to calculate the unknown distances if you know the focal length and one of the distances. In the context of a microscope, two key lens formulas are used: one for the objective lens and one for the eyepiece lens.

• The objective lens formula helps determine where the object should be placed relative to the objective to get a focused image.
• The eyepiece lens formula is used to adjust the focus for a relaxed eye, usually directing the final image to infinity.

Understanding these formulas is vital because it forms the backbone of how microscopes work and how they achieve magnification.

Focal Length

The focal length is the distance between the lens and the focus point, where parallel rays of light converge after passing through the lens. Focal length is crucial in the design and use of microscopes because it directly influences the imaging capability.

In a typical microscope setup, there are two lenses, each with its own focal length:

• The objective lens, which has a short focal length and provides the primary magnification. In the exercise, the objective lens has a focal length of 0.80 cm.
• The eyepiece lens, which usually has a longer focal length, helps to magnify the image formed by the objective lens. For the eyepiece in this exercise, the focal length is 1.8 cm.

The combination of these two lenses and their respective focal lengths allows for the significant magnification capability of a microscope. The shorter the focal length, the higher the magnification potential of a lens, making the choice of focal length critical for achieving desired magnification levels.

Objective Lens

The objective lens is a crucial component of microscopes, located near the object being observed. Its role is to gather light from the object and create a magnified real image. The quality and capability of the objective lens are essential for determining the microscope's overall performance.

Key considerations for the objective lens include:

• Focal length: A shorter focal length, such as 0.80 cm in the exercise, allows for higher magnification of the object.
• Magnification power: This is calculated by the ratio of the image distance to the object distance, using \[ M_{obj} = \frac{d_{i_{obj}}}{d_{o_{obj}}} \]. The goal is to achieve the first level of enlargement before further magnification by the eyepiece.

In microscopes, the objective lens is critical for setting the foundation of total magnification. It captures the initial image, which the eyepiece lens further amplifies. Choosing the right objective lens based on its focal length and magnification power can significantly impact viewing quality.

Eyepiece Lens

The eyepiece lens, also known as the ocular lens, is the second lens in a compound microscope and plays an important role in amplifying the image produced by the objective lens. It provides the final stage of magnification.

The eyepiece lens has some distinct features to be aware of:

• Focal Length: The focal length of the eyepiece lens in the exercise is 1.8 cm, longer than the focal length of the objective lens. This feature helps to comfortably view the final enlarged image.
• Magnification Function: The magnification through the eyepiece is often calculated by \[ M_{eye} = \frac{25}{f_{eye}} \] assuming the near point distance of a relaxed eye is 25 cm.
• Focusing: The eyepiece helps focus the image at infinity for a relaxed eye, which is crucial for viewing comfort and clarity.

Together, the eyepiece and objective lenses in a microscope enable significant magnification and detailed observation of small objects. Selecting an appropriate eyepiece can further enhance clarity and ease of use.