Reflection and Mirrors

Overview

Reflection occurs when light bounces off a surface. Mirrors use reflection to form images. Understanding mirror geometry is fundamental to optics.

Law of Reflection

$$\theta_i = \theta_r$$

Where:

• $\theta_i$ = angle of incidence (from normal)
• $\theta_r$ = angle of reflection (from normal)

The incident ray, reflected ray, and normal all lie in the same plane.

Types of Reflection

Specular Reflection

• From smooth surfaces
• Parallel rays remain parallel
• Produces clear images (mirrors)

Diffuse Reflection

• From rough surfaces
• Parallel rays scatter in many directions
• Allows us to see non-luminous objects

Plane Mirrors

Image Characteristics

• Virtual (behind mirror)
• Same size as object
• Same distance behind mirror as object is in front
• Laterally inverted (left-right reversed)
• Upright

Image Location

$$d_i = -d_o$$

(Negative indicates virtual image)

Magnification

$$m = 1$$

Spherical Mirrors

Key Points

• Center of Curvature (C): Center of the sphere
• Radius of Curvature (R): Radius of the sphere
• Vertex (V): Point where principal axis meets mirror
• Principal Axis: Line through C and V
• Focal Point (F): Where parallel rays converge/appear to diverge from

Focal Length

$$f = \frac{R}{2}$$

Mirror Equation

$$\frac{1}{d_o} + \frac{1}{d_i} = \frac{1}{f}$$

Where:

• $d_o$ = object distance (positive if in front)
• $d_i$ = image distance (positive if in front for real image)
• $f$ = focal length

Magnification

$$m = \frac{h_i}{h_o} = -\frac{d_i}{d_o}$$

Where:

• $h_i$ = image height
• $h_o$ = object height
• $m > 0$: upright image
• $m < 0$: inverted image
• $\lvert m \rvert > 1$: enlarged
• $\lvert m \rvert < 1$: diminished

Concave Mirrors

$f > 0$ (focal point in front of mirror)

Object Location and Image

Object Position	Image Position	Image Type
Beyond C	Between C and F	Real, inverted, diminished
At C	At C	Real, inverted, same size
Between C and F	Beyond C	Real, inverted, enlarged
At F	At infinity	—
Inside F	Behind mirror	Virtual, upright, enlarged

Convex Mirrors

$f < 0$ (focal point behind mirror)

Image Characteristics

• Always virtual
• Always upright
• Always diminished
• Located between F and V (behind mirror)

Applications: security mirrors, car side mirrors

Ray Diagrams

Rules for Drawing Rays (Concave Mirror)

1. Parallel ray → reflects through F
2. Ray through F → reflects parallel
3. Ray through C → reflects back through C
4. Ray to vertex → reflects at equal angle

Rules for Convex Mirror

1. Parallel ray → reflects as if from F
2. Ray toward F → reflects parallel
3. Ray toward C → reflects back toward C

Examples

Example 1: Plane Mirror

An object is 30 cm in front of a plane mirror. Find image location.

$$d_i = -d_o = -30 \text{ cm (30 cm behind mirror)}$$ $$m = 1 \text{ (same size)}$$

Example 2: Concave Mirror - Real Image

A concave mirror ($f = 15$ cm) has an object at $d_o = 45$ cm.

$$\frac{1}{d_i} = \frac{1}{f} - \frac{1}{d_o} = \frac{1}{15} - \frac{1}{45} = \frac{3}{45} - \frac{1}{45} = \frac{2}{45}$$ $$d_i = 22.5 \text{ cm (real image)}$$ $$m = -\frac{d_i}{d_o} = -\frac{22.5}{45} = -0.5 \text{ (inverted, diminished)}$$

Example 3: Concave Mirror - Virtual Image

Same mirror, object at $d_o = 10$ cm (inside focal point).

$$\frac{1}{d_i} = \frac{1}{15} - \frac{1}{10} = \frac{2}{30} - \frac{3}{30} = -\frac{1}{30}$$ $$d_i = -30 \text{ cm (virtual, behind mirror)}$$ $$m = -\frac{-30}{10} = 3 \text{ (upright, magnified)}$$

Example 4: Convex Mirror

A convex mirror ($f = -20$ cm) has an object at $d_o = 60$ cm.

$$\frac{1}{d_i} = \frac{1}{f} - \frac{1}{d_o} = -\frac{1}{20} - \frac{1}{60} = -\frac{3}{60} - \frac{1}{60} = -\frac{4}{60}$$ $$d_i = -15 \text{ cm (virtual, behind mirror)}$$ $$m = -\frac{-15}{60} = 0.25 \text{ (upright, diminished)}$$

Example 5: Finding Focal Length

An object 40 cm from a concave mirror produces an image at 120 cm.

$$\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} = \frac{1}{40} + \frac{1}{120} = \frac{3}{120} + \frac{1}{120} = \frac{4}{120}$$ $$f = 30 \text{ cm}$$ $$R = 2f = 60 \text{ cm}$$

Example 6: Required Mirror

Design a mirror to produce an image 3× the size of the object, with object at 10 cm.

For real image (inverted): $m = -3$

$$d_i = -m \times d_o = 3 \times 10 = 30 \text{ cm}$$ $$\frac{1}{f} = \frac{1}{10} + \frac{1}{30} = \frac{4}{30}$$ $$f = 7.5 \text{ cm (concave mirror, } R = 15 \text{ cm)}$$

Sign Convention Summary

Quantity	Positive	Negative
$d_o$	Object in front	Object behind
$d_i$	Image in front (real)	Image behind (virtual)
$f$	Concave mirror	Convex mirror
$m$	Upright image	Inverted image
$R$	Concave	Convex