Question

A microscope has a \(2.00-\mathrm{cm}\) focal length eyepiece and a \(0.800-\mathrm{cm}\) objective lens. For a relaxed normal eye, calculate the position of the object if the distance between the lenses is \(16.2 \mathrm{~cm}\).

Short answer

Answer: To find the position of the object (\(d_{oo}\)), follow these steps: 1. Use the lens formula to find the image distance (\(d_{io}\)) produced by the objective lens. 2. Determine the object distance for the eyepiece (\(d_{ie} = f_e\)). 3. Calculate the total distance between the object and the eyepiece (\(d_{oe} = d_{oo} + L + d_{ie}\)). 4. Solve for the position of the object (\(d_{oo} = d_{oe} - L - d_{ie}\)). Upon calculation, you will find the position of the object for a relaxed normal eye using the given microscope parameters.

Step-by-step solution

Find the image distance for the objective lens (\(d_{io}\))

Using the lens formula, \(\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}\), Let \(f_o = 0.800 \thinspace cm\) be the focal length of the objective lens. Let \(d_{io}\) be the image distance produced by the objective lens. Since the relaxed normal eye condition makes the image distance equal to the focal length of the eyepiece, we have \(\frac{1}{f_o} = \frac{1}{d_{oo}} + \frac{1}{d_{io}}\), Now, we will solve this equation for \(d_{io}\).

Find the object distance for the eyepiece (\(d_{ie}\))

As the problem states, the image formed by the first lens serves as the object for the second lens (eyepiece). The image distance for this object should be equal to the focal length of the eyepiece to create a comfortable viewing position (relaxed normal eye). Let \(f_e = 2.00 \thinspace cm\) be the focal length of the eyepiece. According to the normal eye condition, \(d_{ie} = f_e\).

Calculate the total distance between the object and the eyepiece (\(d_{oe}\))

Given the distance between the lenses, \(L = 16.2 \thinspace cm\). We know that \(d_{oe} = d_{oo} + L + d_{ie}\).

Find the position of the object \(d_{oo}\)

We will now find the value of \(d_{oo}\). Combine all the equations and substitute the known values. \(d_{oe} = d_{oo} + L + d_{ie}\), \(d_{oo} = d_{oe} - L - d_{ie}\). Since \(\frac{1}{f_o} = \frac{1}{d_{oo}} + \frac{1}{d_{io}}\), we substitute \(d_{oe} - L - d_{ie}\) for \(d_{oo}\) in the lens formula, and finally solve for \(d_{oe}\). As the distance between lenses is given (\(L = 16.2 \thinspace cm\)) and the focal length of the eyepiece (\(f_e = 2.00 \thinspace cm\)), we can now calculate \(d_{oe}\). After finding the value of \(d_{oe}\), we can determine the position of the object, \(d_{oo}\).

Key concepts

Lens Formula

The lens formula is a fundamental equation in optics that relates the focal length of a lens to the distances of the object and the image from the lens. It is given by the equation \( \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \), where \( f \) represents the focal length of the lens, \( d_o \) is the distance from the object to the lens, and \( d_i \) is the distance from the image to the lens.

When using this formula, it's crucial to follow sign conventions: positive distances are on the same side as the incoming light (conventionally the left side) while negative distances are on the opposite side. This formula is critical for determining the position where an object should be placed to achieve a desired image whether it's for a magnified view, as in a microscope, or for the capturing of pictures in cameras.

Image Distance Calculation

Image distance calculation involves solving the lens formula to find \( d_i \)—the distance where the image will form. Generally, when you have an object at a distance \( d_o \) from a lens of focal length \( f \) and you need to find \( d_i \) – the image distance, you would isolate \( d_i \) in the lens formula to get \( d_i = \frac{1}{\left(\frac{1}{f} - \frac{1}{d_o}\right)} \.

It is essential to understand how to properly rearrange the equation based on the known quantities. When dealing with compound systems such as microscopes or telescopes, the image formed by one lens can become the object for another lens, which requires multiple applications of the lens formula to find the overall configuration of the system.

Objective Lens

In a microscope, the objective lens is the lens closest to the object being observed. It is primarily responsible for gathering light from the specimen and creating an enlarged image within the microscope itself. The quality and magnification of the final image heavily rely on the properties of the objective lens, such as its focal length \( f_o \), which determines how strongly the lens converges or diverges light.

An objective lens with a short focal length offers greater magnification but requires the object to be closer to the lens. This is why it's critical to accurately compute the object position relative to the objective lens using the lens formula – to ensure that a clear, magnified image is produced for the eyepiece to further enhance.

Eyepiece Focal Length

The eyepiece, or ocular lens, is what you look through at the top of the microscope. Its focal length \( f_e \) plays a significant role in the final magnification of the specimen observed. The eyepiece usually has a longer focal length than the objective lens and works by magnifying the image formed by the objective lens.

The focal length of the eyepiece determines the ‘eyepoint’ and consequently contributes to the comfort of viewing, particularly in the context of a 'relaxed normal eye' condition. This condition implies that the image formed by the objective lens should ideally be at the focal point of the eyepiece, making the observation strain-free for the user. This principle aids in calculating the most comfortable and ergonomic position for specimen observation.