9.2 Reflection

Properties of reflection

Reflection describes how light bounces off surfaces. Every time you look in a mirror, see a glare on water, or notice a shiny floor, you're observing reflection at work. The core rule is simple: light arrives at a surface, and it leaves at a predictable angle. That predictability is what makes mirrors, telescopes, and fiber optics possible.

Law of reflection

The law of reflection states that the angle of incidence equals the angle of reflection:

$\theta_i = \theta_r$

Both angles are measured from the normal, which is an imaginary line drawn perpendicular to the surface at the point where light strikes it. Not from the surface itself. This is a common mistake on exams, so always draw the normal first.

This law applies to all waves, not just light. Sound waves, water waves, and radio waves all follow the same rule when they bounce off a surface.

Specular vs. diffuse reflection

Specular reflection occurs on smooth surfaces like glass or polished metal. All the reflected rays bounce in the same direction, producing a clear image (think of a calm lake reflecting mountains).

Diffuse reflection occurs on rough surfaces like paper, concrete, or unpolished wood. The surface has tiny irregularities, so each small section has a different normal direction. Light scatters in many directions, which is why you don't see a mirror image in a wall, even though it's still reflecting light.

Most surfaces you encounter are somewhere in between. A slightly scuffed phone screen, for example, shows a dim reflection rather than a perfectly sharp one.

Angle of incidence and reflection

To find these angles in a problem:

1. Identify where the incoming ray hits the surface.
2. Draw the normal (perpendicular to the surface at that point).
3. Measure the angle between the incoming ray and the normal. That's $\theta_i$.
4. The reflected ray leaves on the opposite side of the normal at the same angle, $\theta_r = \theta_i$.

This process is the basis for tracing light paths through any system with reflective surfaces.

Reflection from plane mirrors

Image formation in plane mirrors

A plane (flat) mirror creates a virtual image that appears to be located behind the mirror. The image distance equals the object distance:

$d_i = d_o$

The image is the same size as the object, upright, and located as far behind the mirror as the object is in front. Your brain interprets the diverging reflected rays by tracing them straight back, which is why the image seems to exist behind the glass.

Virtual vs. real images

• A virtual image forms where reflected rays appear to come from but don't actually converge. You can see it by looking into the mirror, but you can't project it onto a screen. Plane mirrors always produce virtual images.
• A real image forms where reflected rays actually intersect. You can project it onto a screen. Curved mirrors can produce real images under certain conditions.

Lateral inversion

When you look in a plane mirror, the image is reversed left-to-right. This is called lateral inversion. Your right hand appears to be the image's left hand. The reversal happens because the mirror flips the image along the axis perpendicular to its surface, not along the vertical axis. That's why text looks backwards in a mirror, but you don't appear upside down.

Curved mirrors

Concave mirrors

A concave mirror curves inward (like the inside of a spoon). Parallel rays hitting a concave mirror converge at the focal point in front of the mirror.

The type of image depends on where the object is placed:

• Beyond the center of curvature: real, inverted, smaller image
• At the center of curvature: real, inverted, same size
• Between the center and the focal point: real, inverted, magnified
• Inside the focal point: virtual, upright, magnified

This is why concave mirrors are used in makeup mirrors (object inside the focal length for magnification) and in car headlights (light source at the focal point to produce parallel beams).

Convex mirrors

A convex mirror curves outward (like the back of a spoon). Parallel rays diverge after reflecting, and they appear to originate from a virtual focal point behind the mirror.

Convex mirrors always produce images that are virtual, upright, and smaller than the object. The tradeoff is a wider field of view, which is why they're used as vehicle side mirrors and security mirrors in stores.

Focal point and focal length

The focal point ($F$) is where parallel rays converge (concave) or appear to diverge from (convex) after reflection. The focal length ($f$) is the distance from the mirror's surface (vertex) to the focal point.

Focal length relates to the radius of curvature ($R$) of the mirror:

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

A shorter focal length means a more strongly curved mirror with greater magnifying power.

Mirror equation

The mirror equation relates object distance, image distance, and focal length:

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

To solve a mirror problem:

1. Identify the known values ($f$, $d_o$, or $d_i$).
2. Apply sign conventions (see below).
3. Plug into the equation and solve for the unknown.
4. Use the magnification equation to find image size and orientation.

Magnification in curved mirrors

Magnification tells you how the image size compares to the object size:

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

• $|m| > 1$: the image is larger than the object
• $|m| < 1$: the image is smaller
• $|m| = 1$: same size
• Negative $m$: the image is inverted
• Positive $m$: the image is upright

Sign conventions

These conventions are essential for getting correct answers. For mirrors (using the standard convention):

Multiple reflections

Kaleidoscopes

Kaleidoscopes use two or more mirrors angled toward each other to create symmetrical, repeating patterns. Each reflection produces another virtual image, and those images get reflected again, generating the complex designs you see when you look through one. The number of images depends on the angle between the mirrors: for two mirrors at angle $\theta$, the number of images is $\frac{360°}{\theta} - 1$ (when $360°/\theta$ is even).

Corner reflectors

A corner reflector uses three mutually perpendicular surfaces (like the inside corner of a cube). Light that enters a corner reflector bounces off each surface and returns parallel to its original direction, regardless of the incoming angle. This property is called retroreflection.

Corner reflectors are used in road signs, bicycle reflectors, and even on the Moon's surface (placed by Apollo astronauts for laser ranging experiments).

Applications of reflection

Telescopes and microscopes

Reflecting telescopes use large concave mirrors instead of lenses to gather and focus light. Mirrors avoid chromatic aberration (the color-fringing problem that lenses have because different wavelengths refract by different amounts). Most major research telescopes, including the James Webb Space Telescope, are reflector designs.

Microscopes use small mirrors to redirect illumination light onto specimens.

Solar concentrators

Solar concentrators use large curved mirrors (often parabolic) to focus sunlight onto a small receiver. By concentrating light from a wide area onto a small point, they can generate extremely high temperatures for power generation. This is a direct application of how concave mirrors converge parallel rays to a focal point.

Retroreflectors

Retroreflectors return light directly back toward its source. They're built from corner reflectors or micro-prism arrays and are found in road signs, safety vests, and cat's-eye road markers. They also enable laser ranging to measure the Earth-Moon distance with centimeter precision.

Reflection of waves

Sound wave reflection

Sound follows the same law of reflection as light. When sound bounces off a hard surface, it can produce an echo (a distinct repeated sound) or reverberation (overlapping reflections in an enclosed space). Sonar technology uses sound reflection underwater to detect objects and map the ocean floor.

Electromagnetic wave reflection

All electromagnetic waves reflect, not just visible light. Radio waves reflect off the ionosphere, which is how AM radio signals travel long distances. Radar systems emit radio or microwave pulses and detect the reflected signals to determine the position and speed of objects like aircraft.

Total internal reflection

Critical angle

Total internal reflection occurs when light traveling through a denser medium (higher refractive index) hits the boundary with a less dense medium at a steep enough angle. Instead of passing through, all the light reflects back.

The minimum angle at which this happens is the critical angle, calculated from Snell's law:

$\sin \theta_c = \frac{n_2}{n_1}$

where $n_1 > n_2$. At any angle of incidence greater than $\theta_c$, the light reflects completely. This is why diamonds sparkle so intensely: diamond has a very high refractive index ($n \approx 2.42$), giving it a small critical angle of about 24.4°, so light bounces around inside before escaping.

Fiber optics

Optical fibers use total internal reflection to transmit light over long distances with very little loss. A fiber consists of a glass or plastic core with a high refractive index surrounded by cladding with a lower refractive index. Light entering the core at a shallow angle keeps bouncing off the core-cladding boundary and travels the length of the fiber.

This technology enables high-speed internet, medical endoscopy, and telecommunications.

Polarization by reflection

Brewster's angle

When unpolarized light reflects off a surface, the reflected light becomes partially polarized. At one specific angle, called Brewster's angle, the reflected light is completely polarized (oscillating only parallel to the surface). This occurs when the reflected and refracted rays are perpendicular to each other.

$\tan \theta_B = \frac{n_2}{n_1}$

For glass with $n = 1.5$, Brewster's angle is about 56.3°.

Polarizing filters

Polarizing filters transmit light oscillating in only one plane and block the rest. They reduce glare from reflected light (which tends to be horizontally polarized) and are used in polarized sunglasses, photography to cut reflections, and LCD screens.

Reflection in everyday life

Mirrors in daily use

Bathroom mirrors, rearview mirrors, and dressing mirrors all rely on plane mirror reflection. Vehicle side mirrors are convex to provide a wider field of view (with the tradeoff that objects appear smaller and farther away than they actually are).

Reflective surfaces in nature

Still water acts as a near-perfect plane mirror, producing clear reflections of landscapes. Mirages occur when light reflects off layers of hot air near the ground (technically refraction, but the visual effect mimics reflection). Many animals have a reflective layer called the tapetum lucidum behind their retinas, which reflects light back through the retina a second time to improve night vision. That's what causes eyeshine when you see animal eyes glowing in headlights.