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5.8 - Numerical Examples

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Understanding the Lens Formula

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Teacher
Teacher

Welcome class! Today we’ll discuss the lens formula. The formula is \( \frac{1}{f} = \frac{1}{v} - \frac{1}{u} \). Can anyone tell me what each symbol represents?

Student 1
Student 1

Is \(f\) the focal length?

Teacher
Teacher

Exactly! And what about \(v\) and \(u\)?

Student 2
Student 2

\(v\) is the image distance, and \(u\) is the object distance.

Teacher
Teacher

Great! Just remember, the distances are measured from the optical center of the lens. Let’s move to our first example!

Example: Convex Lens

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Teacher
Teacher

In our first example, we have a convex lens forming an image at 20 cm when the object is placed at 30 cm. Can we identify the values of \(u\) and \(v\)?

Student 3
Student 3

Yes, \(u = -30 \, \text{cm}\) and \(v = +20 \, \text{cm}\).

Teacher
Teacher

Perfect! Now, substituting these values into the lens formula, what do we get?

Student 4
Student 4

We calculate \( \frac{1}{f} = \frac{1}{20} + \frac{1}{30} = \frac{5}{60} = \frac{1}{12}\), hence \(f = 12 \, \text{cm}\)!

Teacher
Teacher

Well done! Remember, this shows how the convex lens converges light to a point. Let’s go to the next example.

Example: Concave Lens

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Teacher
Teacher

Now, we’ll examine a concave lens. Here, we have an object placed 10 cm from the lens with a focal length of -15 cm. Can we set up our values for \(u\) and \(f\)?

Student 1
Student 1

\(u = -10 \, \text{cm}\) and \(f = -15 \, \text{cm}\).

Teacher
Teacher

Excellent! Applying the lens formula \( \frac{1}{v} = \frac{1}{f} + \frac{1}{u}\), what do you calculate?

Student 2
Student 2

We find \( \frac{1}{v} = -\frac{1}{15} - \frac{1}{10} = -\frac{8}{30}\), leading to \(v = -3.75 \, \text{cm}\)!

Teacher
Teacher

Exactly! The negative value indicates a virtual image. Remember the characteristics of images formed by concave lenses!

Characteristics of Lens Images

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Teacher
Teacher

Now let’s summarize the image characteristics for different types of lenses. Who remembers what happens with a convex lens?

Student 3
Student 3

It can produce real, inverted images that can be enlarged or diminished depending on the object's position!

Teacher
Teacher

Correct! And what about concave lenses?

Student 4
Student 4

Concave lenses always form virtual, erect, and diminished images.

Teacher
Teacher

Well done everyone! You’ve grasped important concepts of lenses. Remember, both types of lenses serve different applications.

Introduction & Overview

Read a summary of the section's main ideas. Choose from Basic, Medium, or Detailed.

Quick Overview

This section provides practical numerical examples related to lens formulas, focusing on convex and concave lenses.

Standard

This section examines two numerical examples involving a convex lens and a concave lens, demonstrating how to calculate focal lengths and image positions using the lens formula. These examples serve to enhance comprehension of lens behavior in real-world scenarios.

Detailed

In this section, we explore numerical examples that apply the lens formula to real situations involving convex and concave lenses. The lens formula is established as \( \frac{1}{f} = \frac{1}{v} - \frac{1}{u} \), where \( f \) represents the focal length, \( v \) the image distance, and \( u \) the object distance. Two examples are provided:

  1. Convex Lens: An image is formed 20 cm away from the lens when an object is placed 30 cm from it. Using the formula, we calculate the focal length to find it is 12 cm.
  2. Concave Lens: In a second example, an object 10 cm from a concave lens with a focal length of 15 cm results in a virtual image positioned at -3.75 cm. This reinforces understanding of how different lens types affect image characteristics (virtual, erect, diminished) and emphasizes practical applications of the lens formula for both spherical configurations.

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Audio Book

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Example 1: Finding Focal Length of a Convex Lens

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Example 1
A convex lens forms an image 20 cm away from the lens when the object is placed 30 cm from it. Find the focal length.

Solution:
Given: u = −30 cm, v = +20 cm
1/f = 1/v - 1/u = 1/20 + 1/30 = 5/60 = 1/12 ⇒ f = 12 cm

Detailed Explanation

In this example, we are tasked with finding the focal length of a convex lens. The object distance (u) is -30 cm (negative because the object is in front of the lens), and the image distance (v) is +20 cm (positive because the image is formed on the opposite side of the lens). We use the lens formula, which relates these distances to the focal length (f). The formula can be rearranged to find f. By substituting the values of u and v, we calculate the focal length to be 12 cm.

Examples & Analogies

Think of a convex lens as a magnifying glass. If you hold it 30 cm away from a small object, the lens will create a clear image of that object at 20 cm on the other side. The focal length is like its effectiveness in focusing light—12 cm means it can focus rays coming parallel to the axis at that distance.

Example 2: Image Position and Characteristics for a Concave Lens

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Example 2
Find the image position when an object is placed 10 cm from a concave lens of focal length 15 cm.

Solution:
Given: u = −10 cm, f = −15 cm
1/v = 1/f + 1/u = −1/15 + (−1/10) = −8/30 ⇒ v = −3.75 cm
Image is virtual, erect, and diminished.

Detailed Explanation

In this second example, we are tasked with finding where an image is formed by a concave lens. The object distance (u) is -10 cm, as objects are conventionally treated as negative in front of the lens, and the focal length (f) is -15 cm because concave lenses have a negative focal length. Using the lens formula again, we combine the two fractions and solve for v. The negative image distance indicates the image is virtual, formed on the same side as the object, and it is described as being erect and diminished in size.

Examples & Analogies

Imagine you're looking at yourself in a spoon that is shaped like a concave lens. When you place your face close to it (10 cm away), the image you see is virtual, meaning you can't actually touch it; it's just a reflection. It's smaller and upright, just like the image produced by a concave lens.

Definitions & Key Concepts

Learn essential terms and foundational ideas that form the basis of the topic.

Key Concepts

  • Lens Formula: \( \frac{1}{f} = \frac{1}{v} - \frac{1}{u} \) describes the relationship between object distance, image distance, and focal length.

  • Convex Lens: Converging lens that can create real or virtual images depending on object's position.

  • Concave Lens: Diverging lens that only forms virtual and diminished images.

Examples & Real-Life Applications

See how the concepts apply in real-world scenarios to understand their practical implications.

Examples

  • Example 1: A convex lens forms an image at 20 cm from the lens when an object is placed at 30 cm. The focal length is determined to be 12 cm.

  • Example 2: For a concave lens with a focal length of -15 cm, an object placed 10 cm from the lens results in a virtual image at -3.75 cm.

Memory Aids

Use mnemonics, acronyms, or visual cues to help remember key information more easily.

🎵 Rhymes Time

  • Convex brings bright, Inverted from light; Concave makes it stray, Virtual all day.

📖 Fascinating Stories

  • Imagine a lens shop. The convex lenses help in focusing the sun, while the concave lenses scatter light like a playful kid scattering toys!

🧠 Other Memory Gems

  • For lens images: 'Cinderella's Images Are Real' - Convex lenses give real images, Concave gives virtual.

🎯 Super Acronyms

F = focal length, v = image distance, u = object distance (FVU).

Flash Cards

Review key concepts with flashcards.

Glossary of Terms

Review the Definitions for terms.

  • Term: Focal Length

    Definition:

    The distance from the optical center of the lens to the principal focus.

  • Term: Image Distance (v)

    Definition:

    The distance from the optical center of the lens to the image formed.

  • Term: Object Distance (u)

    Definition:

    The distance from the optical center of the lens to the object being viewed.

  • Term: Convex Lens

    Definition:

    A lens that is thicker in the middle than at the edges and converges light rays.

  • Term: Concave Lens

    Definition:

    A lens that is thinner in the middle than at the edges and diverges light rays.