In Reflection, Absorption and Transmission you saw that light can bounce off a surface. Now we make that exact. In geometric optics we draw light as rays, straight lines along which the light travels, and we measure every angle from the normal, the line at right angles to the surface.
Key idea
Law of reflection: the angle of reflection equals the angle of incidence, $\theta_r = \theta_i$, both measured from the normal. A plane mirror makes a virtual, upright image the same size as the object, as far behind the mirror as the object is in front.
Why it works
Why is the image behind the mirror? Rays from a point on the object bounce off with equal angles, so after reflecting they spread out exactly as if they came from a point behind the glass. Equal angles make two congruent triangles, which puts that point the same distance behind the mirror as the object is in front. Your eye traces the rays straight back and sees the object there. No light actually reaches that point, so the image is virtual.
Common mistake
Thinking you can see more of yourself by stepping back from a mirror. The ray from your feet hits the mirror halfway between your feet and your eyes, at any distance, so the part of the mirror you use is always half your height. Step back and your image gets smaller in your view, but the mirror shrinks in your view by the same amount. Try it in the Optics Bench lab, or with a bathroom mirror.
Worked example: Angles from the surface
A ray strikes a mirror at $35^\circ$ to the mirror's surface. Find the angle of reflection.
- Angles are measured from the normal, which is $90^\circ$ from the surface.
- Angle of incidence: $\theta_i = 90^\circ - 35^\circ = 55^\circ$.
- Angle of reflection: $\theta_r = \theta_i = 55^\circ$ from the normal (still $35^\circ$ from the surface on the other side).
Answer: $55^\circ$ from the normal
Worked example: A full-length mirror
A person $1.80\ \text{m}$ tall, with eyes $1.70\ \text{m}$ above the floor, wants to see their whole body in a wall mirror. How tall must the mirror be, and where should its bottom edge be?
- The ray from the feet reflects halfway between the feet and the eyes: at $1.70/2 = 0.85\ \text{m}$ above the floor. That is the bottom edge.
- The ray from the top of the head reflects halfway between head and eyes: at $1.70 + 0.10/2 = 1.75\ \text{m}$.
- Mirror height: $1.75 - 0.85 = 0.90\ \text{m}$, half of $1.80\ \text{m}$.
Answer: $0.90\ \text{m}$ tall, bottom edge $0.85\ \text{m}$ up
Worked example: Walking toward a mirror
You stand $2.0\ \text{m}$ in front of a plane mirror. How far are you from your image? If you walk toward the mirror at $1.0\ \text{m/s}$, how fast does your image approach you?
- The image is $2.0\ \text{m}$ behind the mirror, so it is $2.0 + 2.0 = 4.0\ \text{m}$ from you.
- Each second you get $1.0\ \text{m}$ closer to the mirror and the image does too, from its side.
- So the gap closes by $2.0\ \text{m}$ each second.
Answer: $4.0\ \text{m}$; $2.0\ \text{m/s}$