Thin Lenses | Physics for EmSAT Achieve PDF Download

Learning Objectives

By the end of this section, you will be able to:

• Use ray diagrams to locate and describe the image formed by a lens
• Employ the thin-lens equation to describe and locate the image formed by a lens

Lenses and Their Properties

• Lenses are vital components in various optical instruments, such as magnifying glasses, cameras, and even the human eye.
• The term "lens" originates from the Latin word for a lentil bean, resembling the shape of a convex lens.
• There are different lens shapes, each with unique properties for manipulating light.

Lens Shapes and Functionality

• A convex lens converges all incoming parallel light rays to a focal point on the opposite side, while a concave lens diverges these rays.
• When light passes through a lens, it bends towards or away from the optical axis based on the lens's refractive index.
• Converging lenses have a focal point where light converges, while diverging lenses have a virtual focal point.
• The distance between the lens center and the focal point is known as the focal length.

Ray Manipulation by Lenses

• Converging lenses bend light rays towards the optical axis, while diverging lenses bend them away.
• Understanding Snell's law helps explain how light behaves as it passes through lenses.
• The focal point of a converging lens is where light rays converge, while for a diverging lens, it's where they appear to originate.
• The focal length is crucial in determining the optical properties of a lens.

In conclusion, lenses play a crucial role in optics by manipulating light rays to form images. Understanding the behavior of light through different lens shapes is fundamental in various optical applications.

Understanding Thin Lenses

• Definition of Thin Lenses: A lens is categorized as thin when its thickness is significantly smaller than the radii of curvature of its surfaces. In simpler terms, if a lens is thin, light rays can be assumed to bend once at the center of the lens.

• Behavior of Light Rays: When dealing with thin lenses, light rays that are parallel to the optical axis will bend at the center of the lens and pass through the focal point. Additionally, light rays passing through the center of a thin lens remain undeviated.

Reversibility of Light Rays

• Snell's Law Reversibility: According to Snell's law, the paths of light rays are entirely reversible. This indicates that the direction of light rays can be reversed for all rays within a specific optical system.

• Example with Convex Lens: For instance, if a point-light source is positioned at the focal point of a convex lens, parallel light rays will emerge from the opposite side of the lens. This demonstrates the reversible nature of light rays in optical systems.

• Understanding Ray Tracing and Thin Lenses

  Ray tracing is a method used to track the paths that light rays take. When it comes to thin lenses, the process of ray tracing is akin to what we do with spherical mirrors. Just like mirrors, ray tracing is a useful tool for explaining how lenses work. The principles of ray tracing for thin lenses closely resemble those for spherical mirrors:

  • Rule 1:

    When a ray enters a converging lens parallel to the optical axis, it passes through the focal point on the opposite side of the lens. Conversely, for a diverging lens, a ray entering parallel to the optical axis exits through the focal point on the same side of the lens.

  • Rule 2:

    A ray that goes through the center of either a converging or diverging lens does not bend.

  • Rule 3:

    In a converging lens, a ray passing through the focal point exits parallel to the optical axis. In contrast, for a diverging lens, a ray approaching along the line passing through the focal point on the opposite side exits parallel to the axis.

• Chromatic Aberrations in Thin Lenses

  Thin lenses perform well with monochromatic light but face challenges with polychromatic light like white light. This limitation arises from the fact that the refractive index of a material varies with the light's wavelength. This phenomenon leads to colorful effects such as rainbows but also results in aberrations in images produced by lenses.

  • Chromatic Aberrations:

    Due to the dependency of focal distance on the wavelength of light, different wavelengths focus at diverse points, causing chromatic aberrations where images appear colored and blurred.

  • Correction with Doublets:

    Special lenses known as doublets can rectify chromatic aberrations. A doublet consists of a converging lens and a diverging lens combined, resulting in significantly reduced chromatic aberrations.

Chromatic Aberrations and Doublets- Chromatic aberrations in lenses can be corrected using doublets, which are lens combinations designed to minimize this distortion of colors.

Image Formation by Thin Lenses

- When we talk about how lenses create images, we often use a technique called ray tracing. This method helps us understand the different types of images lenses can produce. - A real image is one that can be projected onto a screen, like what we see in a movie theater. On the other hand, a virtual image is one that cannot be displayed on a screen.- For instance, when we wear eyeglasses, the image formed is a virtual one. We can't project it onto a screen.- To analyze how thin lenses create images, we rely on ray tracing techniques. By tracing the paths of specific light rays coming from an object, we can determine the location and size of the resulting image.- Let's consider an object placed at a distance from a converging lens. By tracing the paths of light rays originating from a single point on the object, we can determine the characteristics of the image formed.

• Ray 1: Enters the lens parallel to the optical axis and goes through the focal point on the other side.
• Ray 2: Passes through the center of the lens without being deviated.
• Ray 3: Goes through the focal point before reaching the lens and exits parallel to the optical axis.

- These rays intersect at a specific point on the opposite side of the lens, indicating the location of the image. All light rays coming from the same point on the object are refracted by the lens and converge at this intersection point.- This process demonstrates how thin lenses form images and how we can quantitatively analyze their properties using equations.- Understanding these principles helps us comprehend how lenses work in various optical devices and applications, providing insights into the behavior of light when it interacts with these optical elements.

Understanding Optical Axis and Image Formation

• When dealing with images, it's crucial to identify reference points. In the case of an arrow, locating the tip's image is just the beginning.

• Choosing the base of the arrow, usually positioned along the optical axis, helps orient the entire arrow image. This base lies above the tip's image due to the symmetry of lenses.

• For a comprehensive image, rays from various points on the arrow converge at specific points. This process ensures that the complete image is formed.

• While multiple rays can be traced, only two are necessary to pinpoint an image location. It's advisable to follow simple ray-tracing principles for accuracy.

• Key distances to consider include the object distance (d), which is the object's distance from the lens center, and the image distance (di), representing the image's distance from the lens center.

• Additionally, object height (ho) and image height (hi) play vital roles in image formation. Positive heights indicate upright images, while negative heights signify inverted images.

• By applying ray-tracing rules and creating scale drawings, such as those in Figure 2.5.6, we can accurately describe image locations and sizes.

• Ray tracing not only aids in visualization but also offers insights into how images are created in various scenarios.

Understanding Oblique Parallel Rays and Focal Plane

When we explore the behavior of light rays that are not parallel to the optical axis in the context of lenses, interesting phenomena occur. Let's delve into this concept further.

Convergence of Non-Parallel Rays

• Parallel rays that are not aligned with the optical axis behave uniquely.
• In a converging lens, these rays do not converge at the focal point as expected.
• Instead, they intersect at a specific point within a plane known as the focal plane.

Focal Plane Characteristics

• The focal plane is perpendicular to the optical axis of the lens.
• It includes the focal point, which is crucial in understanding the behavior of light rays.

Focusing of Parallel Rays

• When non-parallel rays pass through a converging lens, they focus at a particular point on the focal plane.
• This focal point is where the ray passing through the center of the lens intersects the focal plane.

Understanding how light behaves when not following the optical axis is fundamental in optics and lens studies. By comprehending the concept of oblique parallel rays and the focal plane, we gain insights into the intricate world of light manipulation.

Thin-Lens Equation

Ray tracing provides a way to understand how images are formed. For precise calculations, we use the thin-lens equation and the lens maker's equation to analyze thin lenses quantitatively.

When dealing with a bi-convex lens surrounded by mediums with different refractive indices, we aim to establish a relationship between the object distance (do), image distance (di), and lens parameters.

To derive the thin-lens equation, we first consider the image produced by the initial refracting surface and then treat this image as the object for the subsequent refracting surface.

The image formed by the first surface is virtual, represented by Q'. To find the corresponding image distance d'i, we utilize the object distance (do), image distance (di), and radius of curvature (R1) in the refraction equations.

Optics and Refraction Summary

• When light passes through a refracting surface, the behavior can be understood using specific equations.

• For a virtual image formed on the same side as the object:

  • The image is virtual when di' < 0="" and="" do="" /> 0.
  • The first surface is convex towards the object, so R1 > 0.

• To determine the object distance for the refracted object from the second interface:

  • Interchange the roles of the refractive indices n1 and n2 in the equation.
  • Consider the object distance as do' and image distance as di.
  • Utilize R2 for the radius of curvature in the equation.

• For a real image formed on the opposite side from the object:

  • The image is real if di > 0 and do' > 0.
  • The second surface is convex away from the object, so R2 < />

• Simplify Equation 2.5.2 by considering:

  • Expressing do' = |di| * t, where t is a positive value.
  • Using absolute values to ensure positive values for do' and t.

• Introduction to Thin Lens Equations
  • When light passes through a lens, it undergoes refraction, bending either towards or away from the normal, based on the lens's shape and refractive index.
• Derivation of Thin Lens Equations
  • Equations 2.5.2 and 2.5.3 are combined to yield a formula that involves the object distance, image distance, and lens radii.
  • Through the thin-lens approximation, where the lens thickness is negligible compared to distances involved, simplifications are made for calculations.
  • The resultant formula resembles the mirror equation for spherical mirrors, demonstrating a similar relationship between object and image distances.
  • A key equation, 1/do + 1/di = 1/f, known as the thin-lens equation, defines the focal length of a thin lens.
• Significance of Thin-Lens Equation
  • The thin-lens equation asserts that the focal length of a thin lens remains consistent from both sides of the lens.
  • It emphasizes the dependence of focal length on the lens's curvature and refractive indices of the lens and surrounding medium.
• Application of Lens Maker's Equation
  • The lens maker's equation, 1/f = (n2/n1 - 1)(1/R1 - 1/R2), elucidates the relationship between focal length, radii of curvature, and refractive indices.
  • This equation underscores that the focal length is determined by the lens and surrounding medium's optical properties.
  • For a lens in air (n1 = 1.0) with n2 representing the refractive index, the lens maker's equation simplifies to 1/f = (n - 1)(1/R1 - 1/R2).

Optical Physics Concepts

Thin Lens Equation

• Distance of the image (di) is considered positive if the image is on the side opposite the object, indicating a real image. If the image is on the same side as the object, di is negative, representing a virtual image.
• The focal length (f) is positive for a converging lens and negative for a diverging lens.
• The radius of curvature (R) is positive for a surface that is convex towards the object and negative for a surface that is concave towards the object.

Magnification

When dealing with finite-size objects on the optical axis and performing ray tracing, the magnification factor (m) of an image can be determined. The formula for magnification is m = -hi/ho = -di/do, where hi and ho represent the heights of the image and object, respectively.

If the magnification (m) is greater than 0, the image maintains the same vertical orientation as the object, referred to as an "upright" image. Conversely, if m is less than 0, the image displays the opposite vertical orientation compared to the object, known as an "inverted" image.

• Understanding the Thin-Lens Equation

  The thin-lens equation and the lens maker's equation are fundamental tools used in scenarios involving thin lenses. These equations help us analyze various aspects of image formation, which we will delve into through examples.

• Image Formation by a Converging Lens

  Let's consider a scenario with a thin converging lens. When an object moves towards the lens from infinity, where does the image form, and what characteristics does it possess?

  • By employing the thin-lens equation and a specific focal length, we can determine where the image forms concerning the object distance. The thin-lens equation can be represented as di=(1/f - 1/do)^-1.
  • For instance, if the focal length is 1 cm, the outcome is depicted in Figure 2.5.9a.

• Object at a Distance from the Lens

  When the object is significantly further away from the lens than its focal length, the image is expected to form close to the focal plane. This occurs because the second term in the equation becomes negligible compared to the first term, resulting in di ≈ f.

  • As shown in part (a) of the figure, the image distance gradually approaches the focal length of 1 cm for larger object distances.
  • As the object nears the focal plane, the image distance extends towards positive infinity. This behavior is anticipated since an object at the focal plane emits parallel rays that converge to form an image at infinity, far from the lens.
  • When the object is beyond the focal length, the image distance is positive, indicating a real image that is inverted and positioned on the opposite side of the lens compared to the object.
  • If the object is closer to the lens than its focal length, the image distance becomes negative, suggesting a virtual image that is upright and located on the same side of the lens as the object.

Summary: Thin Lenses

Thin Diverging Lens Characteristics:

• For a thin diverging lens with a focal length of f = -1.0 cm, the image distance is negative for all positive object distances.
• This indicates that the image formed is virtual, on the same side of the lens as the object, and upright.
• These characteristics can also be understood through ray-tracing diagrams.

Examples of Upright and Inverted Images with Converging Lenses:

• Refer to Figure 2.5.11 for images formed by converging lenses at different object distances.
• In part (a), when the object is placed beyond one focal length from the lens, the image is inverted.
• In part (b), when the object is placed within one focal length from the lens, the image is upright.

Understanding Thin Lenses: Worked Examples

• Work through the provided examples to gain a better understanding of how thin lenses operate.

Problem-Solving Strategy: Lenses

• Paraphrase and explain the information provided, elaborating with examples where necessary.
• Present all key points in bullet format under appropriate headings.
• Ensure a clear explanation without missing any topics.
• Keep the language simple and easy to comprehend for better understanding.

• Problem-Solving Approach for Optics:
  • Determine the applicability of ray tracing or the thin-lens equation, or both, in a given problem. Even if ray tracing is not utilized, a well-drawn sketch with symbols and values is highly beneficial.
  • Identify the unknown quantities that need to be determined within the problem scenario.
  • Create a list of the known values or information that can be inferred from the problem at hand.
  • If ray tracing is essential, refer to the ray-tracing principles provided at the beginning of the relevant section.
  • For most quantitative problems, employ the thin-lens equation and/or the lens maker's equation. Solve these equations to find the unknowns and incorporate the given values, or use both equations simultaneously to solve for two unknowns.
  • Verify the obtained solution for reasonableness. Ensure that the signs are correct and that the sketch or ray tracing aligns with the calculated values.
• Example: Using the Lens Maker's Equation
  • Scenario: Determine the radius of curvature of a biconcave lens symmetrically crafted from a glass with a refractive index of 1.55, resulting in a focal length of 20 cm in air. (In a biconcave lens, both surfaces possess the same radius of curvature.)
  • Strategy: Utilize the thin-lens variant of the lens maker's equation, where R1 < 0="" and="" r2="" /> 0. Given the symmetric nature of the lens, |R1| equals |R2|.
  • Solution: The radius of curvature, R, can be determined using the formula 1/f = (n2/n1 - 1)(2/R). By considering f = 20 cm, n2 = 1.55, and n1 = 1.00, the calculation yields the value of R.

Understanding Converging Lenses and Object Distances

• Introduction to Converging Lenses

  • A converging lens is a curved piece of glass that refracts light rays to converge them to a focal point.

• Thin-Lens Equation

  • The thin-lens equation, represented as 1/di + 1/do = 1/f, is used to calculate image distances.

• Example Scenario

  • Consider a convex lens with a focal length of 10.0 cm and a 3.0 cm high object placed at different distances.

• Calculating Image Distances and Magnification

  • For instance, when the object distance is 50.0 cm and the focal length is 10.0 cm:

    • The image distance is found to be 12.5 cm, indicating a real image located 12.5 cm from the lens.

    • The magnification, calculated as -0.250, suggests the image is inverted and smaller than the object.

Paraphrased Physics Concepts on Image Formation by Lenses

• Magnification and Image Distance Relationship

  When an object is placed at a distance from a lens, the image distance is determined by the formula di=(1/f - 1/do)^-1.

  The magnification (m) of an image formed by a lens is calculated using the formula m=-di/do.

  An example scenario: If an object is placed 5.00 cm away from a lens with a focal length of 10.0 cm, the image distance is 10.0 cm from the lens.

• Characteristics of Images Formed by Lenses

  When the image distance is negative, the image is virtual, located on the same side as the object, and magnified.

  In the case where |m| > 0, the image is upright and larger than the object.

  For instance, if |m| = 2.00, and the object is 3.0 cm tall, the image will be 6.0 cm tall.

• Real Image Formation

  When the image distance is positive, the image is real, positioned on the opposite side of the lens from the object.

  If, for example, an object is located 20.0 cm away from a lens with a focal length of 10.0 cm, the image distance is 20.0 cm from the lens.

Understanding Geometric Optics

• Definition of Magnification: Magnification (m) in optics refers to the ratio of the height of an image to the height of an object. A negative magnification indicates an inverted image, while a positive magnification means an upright image.
• Image Formation: When dealing with geometric optics problems, we often combine ray tracing with lens equations to determine the characteristics of an image formed by lenses.
• Example of Lens Selection: Let's consider an example where we need to project an image of a light bulb onto a screen located at a specific distance. We must decide whether to use a converging or diverging lens based on the desired image characteristics.
• Finding Focal Length: To choose the right lens, we calculate the focal length using the thin-lens equation, which involves the object distance (do) and the image distance (di).
• Thin-Lens Equation: The thin-lens equation 1/f = 1/do + 1/di helps us determine the focal length of the lens required to form a clear image on the screen.
• Magnification Calculation: Magnification can be calculated using the formula m = -di/do, where di is the image distance and do is the object distance.

Summary and Explanation

Magnification in Optics

• When the magnification (m) is negative in optics, it indicates that the image is inverted. For example, if m = -2.0, it means the image is twice the size of the object and inverted.

Significance of Magnification

• The negative sign in magnification signifies an inverted image. Additionally, a positive focal length, typical for a converging lens, is crucial. This can be confirmed through ray tracing, as demonstrated in Figure 2.5.12.
• Upon calculation, the resulting image is found to be inverted, real, and larger than the object.

Topics Related to Optics

• Images Formed by Refraction (2.4): This topic delves into how images are formed through the process of refraction.
• The Eye (2.6): Exploring the mechanisms and functions of the human eye.