Question

When an object is placed at the proper distance to the left of a converging lens, the image is focused on a screen 30.0 cm to the right of the lens. A diverging lens is now placed 15.0 cm to the right of the converging lens, and it is found that the screen must be moved 19.2 cm farther to the right to obtain a sharp image. What is the focal length of the diverging lens?

Short answer

The focal length of the diverging lens is -10.4 cm.

Step-by-step solution

Understanding the Lens Setup

First, identify the positions and roles of both lenses. Initially, a converging lens forms a focused image 30.0 cm to the right of it. A diverging lens is then placed 15.0 cm to the right of the converging lens.

Determine New Image Distance for Diverging Lens

After placing the diverging lens, the screen must be moved 19.2 cm further to the right, making the total image distance from the converging lens \( 30.0 \, \text{cm} + 19.2 \, \text{cm} = 49.2 \, \text{cm} \). The distance of the image formed by the diverging lens is from its own position, so it is \\( 49.2 \, \text{cm} - 15.0 \, \text{cm} = 34.2 \, \text{cm} \). Thus, the new image distance for the diverging lens is \(34.2 \, \text{cm}\) to the right.

Apply Lens Formula for Diverging Lens

Use the lens formula \( \frac{1}{f} = \frac{1}{v} - \frac{1}{u} \) for the diverging lens, where \( v = 34.2 \, \text{cm} \) (image distance) and \( u = -15.0 \, \text{cm} \) (since the object for diverging lens is virtual, to the left). Solve for \( f \), the focal length of the diverging lens:\[\frac{1}{f} = \frac{1}{34.2} - \frac{1}{-15.0}.\]Calculate this to find \( f \).

Calculation

Calculate the reciprocal values:\[\frac{1}{f} = \frac{1}{34.2} + \frac{1}{15.0} \approx 0.0292 + 0.0667 = 0.0959.\]Thus, solve for \( f \):\[f = \frac{1}{0.0959} \approx 10.4 \, \text{cm}.\] The focal length of the diverging lens is \(-10.4 \, \text{cm}\) since it is a diverging lens and should be negative.

Key concepts

Converging Lens

A converging lens, often known as a convex lens, is a type of lens that brings light rays together to focus at a single point. When an object is placed in front of a converging lens, light rays from the object converge to form an image. This image can be projected onto a screen if it is real.
A hallmark of converging lenses is that they can create both real and virtual images. These lenses are commonly used in devices like cameras, glasses, and telescopes.
To analyze how they form images, we often use the lens formula, which assists in predicting the position and size of the image formed.

Diverging Lens

In contrast to converging lenses, a diverging lens, or concave lens, spreads out light rays. This lens causes parallel rays of light to diverge away from each other upon passing through the lens.
Diverging lenses always create virtual, upright, and reduced images. They are often used in optical devices like peepholes and eyeglasses for correcting nearsightedness.
Understanding how a diverging lens affects light can be crucial in solving optics problems. In a combined lens system exercise, like the one given, it is essential to analyze the systemic impact both lenses have on the image accurately.

Lens Formula

The lens formula is a fundamental equation in optics that helps us find the relationship between the object distance (\( u \)), the image distance (\( v \)), and the focal length (\( f \)) of a lens. Mathematically, it is represented as:

• \( \frac{1}{f} = \frac{1}{v} - \frac{1}{u} \)

This formula is indispensable when determining properties of the image formed by lenses.
For converging lenses, the focal length is positive, while for diverging lenses, it is negative. In the context of the exercise, the lens formula was used to determine the focal length of a diverging lens when combined with a converging lens.
By substituting the known values into the lens formula, we can calculate the unknown values, which allows us to understand how the image changes with different lens setups.

Optics Problem Solving

Optics problem solving often involves mathematical and conceptual understanding of how lenses and light interact. Let's breakdown the steps that are usually involved:

• Understand the problem: Clearly identify the type of lenses involved and their arrangements.
• Use known values: Measure or use given distances for object and images, and apply to relevant formulas.
• Apply the lens formula: This assists in finding unknown distances or focal lengths when some variables are known.
• Solve for unknowns: Perform calculations carefully to solve for unknown values like focal length.

In exercises like these, patience and practice are necessary. By making step-by-step analyses and calculations, one can determine how different lenses work together. It is crucial to recognize how each step contributes to finding the correct solution, ensuring a comprehensive understanding of the problem.