Images Formed by | AP Physics 2 Unit 5 Study Guide

Getting Started

This chapter explores how lenses, common optical components found in everything from eyeglasses to telescopes, form images by refracting light. We will investigate the physical system of an object, a thin lens, and the path of light rays that create a predictable image. Our core goal is to answer the question: How can we determine the location, size, orientation, and type of image formed by a given lens for any object position?

What You Should Be Able to Do

After completing this section, you will be able to:

Distinguish between the behavior of converging (convex) and diverging (concave) lenses.
Construct principal-ray diagrams to qualitatively locate the image formed by a thin lens.
Apply the thin-lens and magnification equations to quantitatively determine an image's position, size, and orientation.
Differentiate between real and virtual images based on ray diagrams and calculated values.
Predict how an image's properties change as the object's position relative to the lens is altered.

Key Concepts & Mechanisms

To analyze how a lens forms an image, we use two complementary representations: a visual, geometric model (the ray diagram) and a quantitative, algebraic model (the lens equations). Each represents the same physical process—the refraction of light by a lens—but provides different insights and serves a different purpose. The "thin-lens approximation," which assumes the lens's thickness is negligible compared to its focal length, is used in both models.

Representation	What It Encodes	How to Read/Use It	Typical Pitfalls
Principal-Ray Diagram	The geometric path of specific, easy-to-draw light rays as they pass through the lens. It visually encodes the principles of refraction for a given lens shape.	To find the image, draw at least two of the three principal rays from the top of the object. The point where these rays (or their backward extensions) intersect marks the top of the image. This method gives a qualitative sense of the image's location, size, and orientation.	Forgetting that rays from a diverging lens must be extended backward to find the virtual image. Drawing rays that bend at the lens surfaces instead of at the central midline. Incorrectly drawing a ray through the focal point.
Mathematical Equations	The precise algebraic relationship between the object's position, the lens's intrinsic properties, and the resulting image's position and size.	Use the thin-lens equation to solve for an unknown distance ( $s_{o}$ , $s_{i}$ , or $f$ ). Use the magnification equation to find the image's size and orientation. The signs of the results are critical for interpretation (+ for real, - for virtual, etc.).	Mixing up signs for focal length (positive for converging, negative for diverging) or image distance (positive for real, negative for virtual). Forgetting to take the reciprocal when solving the thin-lens equation for a distance.

Key Models & Diagrams

The characteristics of an image depend critically on the type of lens and the object's position relative to the focal point. The following table summarizes the predictable outcomes, linking object placement to the observable image.

Object Position ( $s_{o}$ )	Image Characteristics (Converging Lens)	Image Characteristics (Diverging Lens)
Far from lens ( $s_{o} > 2 f$ )	Real, Inverted, Smaller ( $∣ M ∣ < 1$ )	Virtual, Upright, Smaller ( $∣ M ∣ < 1$ )
At twice the focal length ( $s_{o} = 2 f$ )	Real, Inverted, Same Size ( $∣ M ∣ = 1$ )	Virtual, Upright, Smaller ( $∣ M ∣ < 1$ )
Between f and 2f ( $f < s_{o} < 2 f$ )	Real, Inverted, Larger ( $∣ M ∣ > 1$ )	Virtual, Upright, Smaller ( $∣ M ∣ < 1$ )
At the focal point ( $s_{o} = f$ )	No image formed (rays are parallel)	Virtual, Upright, Smaller ( $∣ M ∣ < 1$ )
Inside the focal point ( $s_{o} < f$ )	Virtual, Upright, Larger ( $∣ M ∣ > 1$ )	Virtual, Upright, Smaller ( $∣ M ∣ < 1$ )

Key Components & Evidence

Converging (Convex) Lens: A lens that is thicker in the middle. It causes incident light rays parallel to the principal axis to refract and converge at a single point.
Diverging (Concave) Lens: A lens that is thinner in the middle. It causes incident light rays parallel to the principal axis to refract and spread out as if they originated from a single point.
Principal Axis: The imaginary line that passes horizontally through the center of the lens and is perpendicular to its surface.
Focal Length ( $f$ ): The distance from the center of the lens to its focal point, measured in meters (m). By convention, $f$ is positive for a converging lens and negative for a diverging lens.
Object Distance ( $s_{o}$ ): The distance from the object to the midline of the lens, measured in meters (m). By convention, $s_{o}$ is always positive for a single-lens system.
Image Distance ( $s_{i}$ ): The distance from the image to the midline of the lens, measured in meters (m). A positive $s_{i}$ indicates a real image (formed on the opposite side of the lens from the object), while a negative $s_{i}$ indicates a virtual image (formed on the same side as the object).
Thin-Lens Equation: The fundamental equation relating the three distances: $\frac{1}{s _{o}} + \frac{1}{s _{i}} = \frac{1}{f}$ . This equation models the geometric relationship established by refraction.
Magnification ( $M$ ): A dimensionless ratio that describes the image's size and orientation relative to the object. It is defined by two equivalent relations: $M = \frac{h _{i}}{h _{o}} = - \frac{s _{i}}{s _{o}}$ , where $h_{i}$ and $h_{o}$ are the image and object heights, respectively. A negative $M$ signifies an inverted image; a positive $M$ signifies an upright image.

Skill Snapshots

Causation

Placing an object inside the focal length of a converging lens causes the refracted rays to diverge, forcing our eyes to trace them back to a larger, upright, virtual image.
Using a diverging lens causes parallel light rays to spread out, which always results in the formation of a smaller, upright, virtual image, regardless of the object's position.
Moving an object from a position far away toward a converging lens causes its real image to form farther from the lens and increase in size.

Comparison

A converging lens has a positive focal length and can form both real and virtual images, whereas a diverging lens has a negative focal length and can only form virtual images.
A real image is formed by the actual intersection of light rays and has a positive image distance ( $s_{i}$ ), whereas a virtual image is formed where rays only appear to intersect and has a negative image distance ( $s_{i}$ ).
Ray diagrams provide a qualitative, visual prediction of the image's properties, whereas the thin-lens and magnification equations provide precise, quantitative predictions.

Change Over Time

Baseline State: An object is placed at a distance of $3 f$ from a converging lens, forming a small, inverted, real image between $f$ and $2 f$ on the other side.
Change 1: As the object moves from $3 f$ to $2 f$ , the real image moves away from the lens (from $1.5 f$ to $2 f$ ) and grows in size until it is the same size as the object.
Change 2: As the object moves from $2 f$ to just outside $f$ , the real image continues to move away from the lens and becomes larger than the object.
Continuity: Throughout this entire process, the focal length ( $f$ ) of the lens, an intrinsic property of its curvature and material, remains constant.

Common Misconceptions & Clarifications

Misconception: A negative magnification ( $M < 0$ ) means the image is smaller.
- Clarification: The sign of the magnification indicates orientation, not size. A negative sign means the image is inverted relative to the object. The size is determined by the absolute value, $∣ M ∣$ . If $∣ M ∣ < 1$ , the image is smaller; if $∣ M ∣ > 1$ , it is larger.
Misconception: Virtual images are imaginary and cannot be seen.
- Clarification: Virtual images are perfectly visible. When you look through a magnifying glass or wear corrective eyeglasses for nearsightedness, you are viewing a virtual image. The term "virtual" means that the light rays only appear to originate from the image location; they do not physically converge there. Your eye's lens takes these diverging rays and focuses them onto your retina.
Misconception: The thin-lens equation is an exact law of physics.
- Clarification: This equation is an approximation that works extremely well for lenses whose thickness is much smaller than their focal length and the object/image distances. For thick lenses or high-precision applications, more complex formulas are required to account for the refraction at both surfaces of the lens separately.

One-Paragraph Summary

The formation of images by thin lenses is a predictable phenomenon governed by the principles of refraction. We can analyze this system using two key tools: principal-ray diagrams for a qualitative understanding and the thin-lens and magnification equations for quantitative precision. A converging lens, with a positive focal length, can produce both real, inverted images and virtual, upright images depending on the object's location. In contrast, a diverging lens, with a negative focal length, always produces a virtual, upright, and smaller image. The sign conventions used in the equations are crucial, as they encode the physical characteristics of the image, such as whether it is real or virtual, upright or inverted. By mastering these models, we can predict the properties of an image formed by any thin lens.

Images Formed by Lenses - AP Physics 2: Algebra-Based Study Guide