Question

You are experimenting with a magnifying glass (consisting of a single converging lens) at a table. You discover that by holding the magnifying glass \(92.0 \mathrm{~mm}\) above your desk, you can form a real image of a light that is directly overhead. If the distance between the light and the table is \(2.35 \mathrm{~m},\) what is the focal length of the lens?

Short answer

Answer: The focal length of the lens is 0.088 m or 88.0 mm.

Step-by-step solution

Convert the given distances to meters

First, we need to convert the given distances into the same unit, in this case, meters. Object distance, \(d_o = 2.35 \mathrm{~m}\) (already in meters) Image distance, \(d_i = 92.0 \mathrm{~mm} = 0.092 \mathrm{~m}\) (converted from millimeters to meters)

Apply the lens equation

Now, we can apply the lens equation to find the focal length \(f\) of the lens: \(\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}\) Plug in the values of \(d_o\) and \(d_i\) we found in step 1: \(\frac{1}{f} = \frac{1}{2.35} + \frac{1}{0.092}\)

Solve the equation for the focal length

Solve the equation for \(f\): \(\frac{1}{f} = \frac{2.35 + 0.092}{2.35 \times 0.092}\) \(f = \frac{2.35 \times 0.092}{2.35 + 0.092}\)

Compute the focal length

Now, calculate the value of \(f\): \(f = \frac{2.35 \times 0.092}{2.35 + 0.092} = 0.088 \mathrm{~m}\)

Present the focal length

The focal length of the lens is \(0.088 \mathrm{~m}\) or \(88.0 \mathrm{~mm}\).

Key concepts

Focal Length Calculation

To calculate the focal length of a lens, we use the lens formula. This formula helps us determine how strongly a lens converges (or diverges) light. The lens formula is given by: \[ \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \] Where:

• \(f\) is the focal length of the lens.
• \(d_o\) is the distance between the lens and the object (here the light source).
• \(d_i\) is the distance between the lens and the image (the real image formed).

In the exercise, we had an object distance, \(d_o = 2.35\) meters, and an image distance \(d_i = 0.092\) meters. After substituting these values, we compute the focal length \(f\) using the formula: \[ \frac{1}{f} = \frac{1}{2.35} + \frac{1}{0.092} \] By solving this, we first find the common denominator and calculate the inverse to find \(f\). This careful calculation helps reveal that the focal length \(f\) is 0.088 meters. This result translates to an 88.0 mm focal length when converted back into millimeters.

Converging Lens

A converging lens, often called a convex lens, bends light rays toward a common point. This converging property allows such lenses to form real images on the opposite side of the object. Applications of a converging lens can be found in everything from cameras to correctional lenses for farsightedness.
When held at a suitable distance, a converging lens will take parallel light rays, such as those from a distant light source, and focus them at its focal point. This ability is precisely what makes a magnifying glass effective at focusing light and enlarging images.
In the exercise example, we used a converging lens as a magnifying glass. By holding it at a specific height (92.0 mm above the desk), a real image of the overhead light is formed. This real image is situated on the opposite side of the lens, aligning with the core characteristic of converging lenses.

Magnifying Glass

A magnifying glass is a simple optical device made from a single converging lens. It magnifies objects by focusing light in such a way that the image seen through the glass appears larger than the object itself. The effectiveness of a magnifying glass is determined by the focal length: the shorter it is, the stronger the magnification.
To use a magnifying glass effectively, one needs to adjust the distance between the lens and the object. Holding the lens closer to the object allows for greater magnification, while moving it farther away reduces the magnification up to the point where a real image is formed.
This exercise demonstrated the fundamental principles of how a magnifying glass functions. By positioning it at 92 mm above the table, we formed a real image from the original light source 2.35 m away. This setup clearly highlights the versatility and functionality of a magnifying glass in manipulating light to create visible magnified images. Understanding this basic property empowers us to use magnifying glasses more effectively for tasks like reading small text or inspecting tiny objects.