Images Formed by Mirrors

Introduction to Mirror Images

Ever wonder why your reflection in a spoon looks upside down on one side but right-side up on the other? Or why the passenger-side mirror on a car says "objects are closer than they appear"? Welcome to the world of spherical mirrors! In this chapter, we will explore how light bounces off different surfaces to create images. Don't worry if the math or the ray diagrams seem intimidating at first—once you learn the "rules of the road" for light, it becomes much like solving a puzzle.

1. Plane Mirrors: The Basics

The simplest mirror is a plane mirror (a flat surface). When you look into a bathroom mirror, you are seeing a virtual image. This means the light rays don't actually pass through the location of the image; your brain just thinks they do because light travels in straight lines.

Key Properties of Plane Mirrors:

• The image is upright (it isn't upside down).
• The image is the same size as the object.
• The object distance \( (s_o) \) is equal to the image distance \( (s_i) \).
• The image is virtual (it appears "behind" the mirror).

2. Spherical Mirrors: Concave and Convex

Spherical mirrors are curved like a slice taken out of a shiny ball. There are two main types you need to know for the AP exam:

Concave Mirrors (Converging)

A concave mirror curves inward like a "cave." These mirrors converge light rays, meaning they bring parallel light rays together toward a single point called the focal point \( (f) \).
Real-world example: The inside of a metal spoon or a makeup mirror.

Convex Mirrors (Diverging)

A convex mirror bulges outward toward the object. These mirrors diverge light rays, making them spread apart as if they are coming from a point behind the mirror.
Real-world example: Security mirrors in stores or passenger-side car mirrors.

Important Geometry: For any spherical mirror, the focal length \( (f) \) is exactly half of the radius of curvature \( (R) \):
\( f = \frac{R}{2} \)

3. Ray Diagrams: Drawing the Path of Light

To find where an image will form, we draw ray diagrams. While light travels in every direction, we only need to draw two or three "principal rays" to find the intersection point. For AP Physics 2, practice drawing these three:

1. The Parallel Ray: A ray that starts parallel to the principal axis and reflects through the focal point \( (f) \).
2. The Focal Ray: A ray that passes through the focal point \( (f) \) and reflects parallel to the principal axis.
3. The Center Ray: A ray that passes through the center of curvature \( (C) \) and reflects straight back on itself.

Quick Tip: If the reflected rays actually meet in front of the mirror, the image is real. If the reflected rays spread apart, you must "trace them back" with dotted lines behind the mirror to find the virtual image.

4. The Math of Mirrors

While diagrams are great for visualizing, we use the Mirror Equation to get precise answers. This is one of the most important formulas in Unit 13:

\( \frac{1}{s_o} + \frac{1}{s_i} = \frac{1}{f} \)

Where:
\( s_o \) = object distance (distance from the object to the mirror)
\( s_i \) = image distance (distance from the image to the mirror)
\( f \) = focal length

Magnification

The magnification \( (m) \) tells us how much larger or smaller the image is compared to the object:

\( m = \frac{h_i}{h_o} = -\frac{s_i}{s_o} \)

Where \( h_i \) is image height and \( h_o \) is object height.
- If \( |m| > 1 \), the image is enlarged.
- If \( |m| < 1 \), the image is reduced.
- If \( m \) is negative, the image is inverted (upside down).

5. The Golden Rules of Sign Conventions

This is where most students make mistakes! You must use the correct plus or minus signs for the math to work.

1. Focal Length \( (f) \):
- Positive \( (+) \) for Concave mirrors.
- Negative \( (-) \) for Convex mirrors.

2. Image Distance \( (s_i) \):
- Positive \( (+) \) for Real images (formed in front of the mirror).
- Negative \( (-) \) for Virtual images (formed behind the mirror).

3. Magnification \( (m) \) and Height \( (h) \):
- Positive \( (+) \) for Upright images.
- Negative \( (-) \) for Inverted images.

Did you know? A convex mirror will always produce an image that is virtual, upright, and smaller than the object. This is why car mirrors show a wider view of the road, but the cars look smaller (and further away) than they actually are!

6. Summary Table for Quick Review

Use this table to predict image characteristics for concave mirrors based on object location:

Common Pitfalls to Avoid

• Forgetting the negative sign in the magnification formula \( m = -s_i/s_o \).
• Confusing \( f \) signs: Always remember: Convex is Negative (the focal point is "inside" the curve, behind the mirror surface).
• Inverting the fraction incorrectly: When solving \( 1/s_i = 1/f - 1/s_o \), calculate the common denominator first before flipping the fraction to find \( s_i \).

Note: For more on how light behaves when it moves through materials rather than bouncing off them, see the next chapter on Refraction.