Question

A physics student epoxies two converging lenses to the opposite ends of a \(2.0 \cdot 10^{1}-\mathrm{cm}\) -long tube. One lens has a focal length of \(f_{1}=6.0 \mathrm{~cm}\) and the other has a focal length of \(f_{2}=3.0 \mathrm{~cm}\). She wants to use this device as a microscope. Which end should she look through to obtain the highest magnification of an object?

Short answer

Answer: The student should look through the end with the lens of focal length 3.0 cm to obtain the highest magnification.

Step-by-step solution

Write down the magnification formula for converging lenses

The magnification formula for a converging lens is given by: $$M = 1 + \frac{D}{f}$$

Calculate the magnification power for the first lens

Given the first lens has a focal length of \(f_1=6.0\,\mathrm{cm}\), we can now calculate the magnification power of the first lens: $$M_1 = 1 + \frac{D}{f_1}$$

Calculate the magnification power for the second lens

Given the second lens has a focal length of \(f_2=3.0\,\mathrm{cm}\), we can now calculate the magnification power of the second lens: $$M_2 = 1 + \frac{D}{f_2}$$

Compare the magnification powers of both lenses

Let's compare the magnification powers of both lenses as a ratio of \(M_1\) to \(M_2\). This will help us determine which lens provides the highest magnification. $$\frac{M_1}{M_2} = \frac{1 + \frac{D}{f_1}}{1 + \frac{D}{f_2}}$$ Substitute the values of \(f_1\) and \(f_2\): $$\frac{M_1}{M_2} = \frac{1 + \frac{D}{6.0}}{1 + \frac{D}{3.0}}$$ Since \(D>0\) and \(f_2<f_1\), this ratio will be greater than 1, which means \(M_1<M_2\). Therefore, the student should look through the end with the lens of focal length \(f_2=3.0\,\mathrm{cm}\) to obtain the highest magnification of an object.

Key concepts

Converging Lenses

Converging lenses, also known as convex lenses, are a type of lens that bends incoming light rays towards a common focal point. These lenses are thicker at the center compared to the edges, allowing them to focus light effectively.

The primary characteristic of a converging lens is its ability to bring parallel rays of light to a focus. This makes them essential for many optical devices like cameras, glasses, and microscopes.

To understand how they work, imagine light rays entering the lens from one side. As these rays pass through the lens, they refract due to the change in medium—from air to the lens material, and back to air again. Because of the shape of the lens, all these refracted rays converge at a single point, known as the focal point.

The distance from the center of the lens to this focal point is the focal length, a vital element that determines how strong or powerful the lens is.

Focal Length

The focal length of a lens is a key concept in optics and dictates much about a lens's functionality. It is defined as the distance from the lens to the point where parallel rays of light meet after passing through the lens, known as the focal point.

A short focal length signifies that a lens has strong converging power. In simpler terms, such lenses can bend light sharply, focusing it over a shorter distance. Conversely, a lens with a longer focal length will bring light to focus more gently and over a longer distance.

• A lens with a short focal length is more compact and creates a larger magnification as it brings objects closer than they appear.
• A lens with a long focal length produces less magnification and is often used where a larger field of view is required.

For our microscope problem, examining the lens with a focal length of 3.0 cm reveals higher magnification, allowing greater detail visibility in an object.

Microscope

A microscope is a complex optical device composed of multiple lenses that allow us to view tiny details of small objects. Its main purpose is to magnify objects, making them easier to study in detail. The microscopes commonly used in labs consist of two main lenses: the objective lens and the eyepiece lens, both of which are converging lenses.

The objective lens, placed close to the object, has a crucial role since it provides the initial level of magnification. The eyepiece, which is closer to your eye, further magnifies the enlarged image produced by the objective lens. By selecting lenses with shorter focal lengths, such as the lens with a focus of 3.0 cm used in our example, the device can achieve a significant magnification.

• The composition and alignment of these lenses inside the barrel or tube greatly impact the performance and effectiveness of the microscope.
• Using lenses with different focal lengths strategically can optimize the setup for various magnification needs.

Understanding these components and how they enhance the ability to see fine details is vital for scholars and students when conducting observational studies.