Images Formed by | AP Physics 2 Unit 5 Study Guide

Getting Started

We will investigate how curved and plane mirrors manipulate light to form images. This chapter focuses on the geometric optics of reflection, where we treat light as rays traveling in straight lines. Our central question is: Given an object and a mirror, can we predict the location, size, orientation, and type of the image that will be formed?

What You Should Be Able to Do

Draw a ray diagram to locate the image formed by a concave or convex mirror for a given object position.
Use the mirror and magnification equations to calculate the image position, height, and magnification.
Distinguish between real and virtual images based on the behavior of light rays and mathematical sign conventions.
Describe the characteristics of an image (real/virtual, upright/inverted, larger/smaller) based on the object's position relative to the mirror's focal point.

Key Concepts & Mechanisms

The primary tool for understanding and predicting image formation is the ray diagram. It is a graphical representation that models the physical behavior of light rays as they reflect from a mirror's surface. By translating a physical setup into this diagrammatic form, we can determine the properties of the resulting image, which can then be confirmed with mathematical equations.

Representation	What It Encodes	How to Read/Use It	Typical Pitfalls
Ray Diagram	The geometric relationship between the object, mirror, principal axis, focal point, and the resulting image. It visually encodes the image's location ( $s_{i}$ ), size ( $h_{i}$ ), orientation (upright/inverted), and type (real/virtual).	To construct a diagram, place the object and mirror on a principal axis (a line perpendicular to the mirror's center). Then, draw at least two of the three principal rays from the top of the object:1. A ray parallel to the principal axis reflects through the focal point, F (for a concave mirror) or appears to come from F (for a convex mirror).2. A ray passing through F (or heading toward it) reflects parallel to the principal axis.3. A ray striking the center of the mirror reflects at an equal angle below the principal axis.The image is formed where these reflected rays (or their extensions) intersect.	- Mixing up rules: Applying the ray rules for a concave mirror to a convex one (e.g., having a ray pass through a convex mirror's focal point).- Misinterpreting intersections: A real image forms where reflected rays physically cross. A virtual image forms where the extensions of diverging reflected rays cross behind the mirror.- Sign convention errors: Forgetting that the focal length ( $f$ ) is negative for a convex mirror, which can lead to incorrect calculations.

Key Models & Diagrams

The characteristics of an image are entirely determined by the type of mirror and the object's position relative to the focal point ( $f$ ) and center of curvature ( $C$ , where $C = 2 f$ ). This relationship can be visualized with ray diagrams and calculated with two key equations.

The Mirror Equation: Relates the object distance ( $s_{o}$ ), image distance ( $s_{i}$ ), and focal length ( $f$ ).

$\frac{1}{s _{o}} + \frac{1}{s _{i}} = \frac{1}{f}$

The Magnification Equation: Relates the image height ( $h_{i}$ ) and object height ( $h_{o}$ ) to the image and object distances.

$M = \frac{h _{i}}{h _{o}} = - \frac{s _{i}}{s _{o}}$

This matrix summarizes the connection between the physical setup and the predicted image characteristics.

Mirror Type	Object Position ( $s_{o}$ )	Image Location ( $s_{i}$ ) & Type	Image Orientation & Size ( $M$ )
Concave	$s_{o} > 2 f$	$f < s_{i} < 2 f$ Real	Inverted, Smaller ( $- 1 < M < 0$ )
Concave	$s_{o} = 2 f$	$s_{i} = 2 f$ Real	Inverted, Same Size ( $M = - 1$ )
Concave	$f < s_{o} < 2 f$	$s_{i} > 2 f$ Real	Inverted, Larger ( $M < - 1$ )
Concave	$s_{o} < f$	$s_{i} < 0$ (behind mirror) Virtual	Upright, Larger ( $M > 1$ )
Convex	Any $s_{o} > 0$	$s_{i} < 0$ (behind mirror) Virtual	Upright, Smaller ( $0 < M < 1$ )

Key Components & Evidence

Object distance ( $s_{o}$ ): The distance from the object to the center of the mirror's surface, measured along the principal axis. By convention, $s_{o}$ is always positive. (SI unit: meters, m).
Image distance ( $s_{i}$ ): The distance from the image to the mirror's surface. A positive $s_{i}$ indicates a real image on the same side as the object; a negative $s_{i}$ indicates a virtual image behind the mirror. (SI unit: m).
Focal length ( $f$ ): The distance from the mirror to the focal point. It is positive for a concave (converging) mirror and negative for a convex (diverging) mirror. For a spherical mirror, $f = R /2$ , where $R$ is the radius of curvature. (SI unit: m).
Real Image: An image formed by the actual convergence of reflected light rays. It can be projected onto a screen.
Virtual Image: An image formed at a location from which reflected light rays appear to diverge. It cannot be projected onto a screen because no light is actually present at the image location.
Magnification ( $M$ ): A dimensionless ratio describing how large the image is relative to the object. A negative value ( $M < 0$ ) signifies an inverted image, while a positive value ( $M > 0$ ) signifies an upright image. An absolute value $∣ M ∣ > 1$ means the image is larger than the object.
Object height ( $h_{o}$ ) and Image height ( $h_{i}$ ): The physical heights of the object and image. An inverted image has a negative $h_{i}$ . (SI unit: m).
The Law of Reflection: The fundamental principle stating that the angle of incidence equals the angle of reflection. This law governs the path of every ray in a diagram.

Skill Snapshots

Causation:
1. Placing an object closer to a concave mirror (but still outside the focal point) causes the real image to form farther away and become larger.
2. Moving an object from far away toward a convex mirror causes the virtual image to move from the focal point toward the mirror's surface, growing in size but always remaining smaller than the object.
3. The curvature of a mirror causes parallel light rays to either converge at a focal point (concave) or appear to diverge from a focal point (convex), which is the basis for image formation.
Comparison:
1. A real image is formed by converging rays and has a positive image distance ( $s_{i}$ ), whereas a virtual image is formed by diverging rays and has a negative image distance.
2. A concave mirror can form both real and virtual images, depending on object placement, whereas a convex mirror can only form virtual, upright, and reduced images.
3. In a ray diagram, solid lines represent the actual paths of light, whereas dashed lines represent the apparent paths of light extended backward to locate a virtual image.
Change Over Time (CCOT):
- Baseline: An object is placed very far from a concave mirror ( $s_{o} \to \infty$ ). A small, inverted, real image forms at the focal point ( $s_{i} \approx f$ ).
- Change 1: As the object moves from infinity to the center of curvature ( $s_{o} = 2 f$ ), the real image moves from the focal point ( $f$ ) out to the center of curvature ( $2 f$ ), growing in size until it is the same size as the object.
- Change 2: As the object moves from the center of curvature ( $2 f$ ) to the focal point ( $f$ ), the real image moves from $2 f$ out toward infinity, becoming increasingly magnified.
- Continuity: Throughout the object's motion outside the focal point, the image remains real and inverted.

Common Misconceptions & Clarifications

Misconception: Virtual images are not "real" or are just optical illusions.
Clarification: A virtual image is a real, predictable location in space. Your brain processes the diverging light rays exactly as if an object were physically at that location. The term "virtual" simply means that the light rays do not physically converge there, so you cannot place a screen at that location and see the image projected on it.
Misconception: A negative sign in an answer, like $s_{i} = - 20$ cm, means the calculation is wrong.
Clarification: In optics, signs carry critical physical meaning. A negative image distance ( $s_{i} < 0$ ) correctly indicates that the image is virtual and located behind the mirror. A negative magnification ( $M < 0$ ) correctly indicates that the image is inverted relative to the object.
Misconception: Magnification only means "making something bigger."
Clarification: Magnification is the ratio of image size to object size. If the absolute value of magnification is less than one (e.g., $M = 0.5$ ), the image is smaller than the object (often called "minified" or "reduced").

One-Paragraph Summary

Mirrors form images by systematically redirecting light rays according to the law of reflection. We can model this process using two powerful, complementary tools: ray diagrams and algebraic equations. Ray diagrams provide a visual, qualitative understanding of where and how an image forms, distinguishing between real images (where light rays actually converge) and virtual images (where rays appear to diverge from). The mirror and magnification equations provide a precise, quantitative method for calculating an image's location ( $s_{i}$ ), size ( $h_{i}$ ), and orientation. The specific characteristics of the image depend entirely on the mirror's type (concave or convex) and the object's distance ( $s_{o}$ ) from it, allowing for the predictable creation of images that are magnified, reduced, upright, or inverted.

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