Chapter 25. Mirrors and the Reflection of Light Our everyday experience that light travels in straight lines is the basis of the ray model of light. Ray optics apply to a variety of situations, including mirrors, lenses, and shiny spoons. Chapter Goal: To understand and apply the ray model of light.

Models of Light The wave model: under many circumstances, light exhibits the same behavior as sound or water waves. We used the wave model in Chapter 24. The ray model: The properties of prisms, mirrors, and lenses are best understood in terms of light rays. The ray model is the basis of ray optics, which we shall study in Chapters 25 and 26. The photon model: In the quantum world, light behaves like neither a wave nor a particle. Instead, light consists of photons that have both wave-like and particle-like properties. This is the quantum theory of light.

The Ray Model – A beam of sunshine or a laser is a good approximation of a light ray, although made up of many parallel rays or a laser

A hemispherical view of a sound wave emitted by a pulsating sphere. 25.1 Wave Fronts and Rays A hemispherical view of a sound wave emitted by a pulsating sphere. The rays are perpendicular to the wave fronts.

At large distances from the source, the wave fronts become less 25.1 Wave Fronts and Rays At large distances from the source, the wave fronts become less and less curved.

The Ray Model Light rays travel in straight lines. Light rays can cross without interacting. An object is a source of an infinite number of light rays. Rays originate from every point and each point sends rays in all directions. Both self-luminous objects and reflective objects are the source of light rays.

The Ray Model To simplify, we usually draw only a few rays from the object.

Specular Reflection Reflected rays are parallel to each other and in the same plane as the normal to the surface. This allows the brain to see virtual images of the source of the light. Opaque object also reflect light, but in an unorganized manner we call diffuse reflection. Specular reflection depends on the reflecting material.

Reflected Light The trees are a light source. Rays that leave the trees are specularly reflected from the still water. Light rays that leave the trees also reflect from the house. This is diffuse reflection and we see no image.

Law of Reflection The incident ray and the reflected ray are in the same plane normal (perpendicular) to the surface, and The angle of reflection equals the angle of incidence: θr = θi

25.2 The Reflection of Light LAW OF REFLECTION For light rays undergoing specular reflection, the incident ray, the reflected ray, and the normal to the surface all lie in the same plane, and the angle of incidence equals the angle of reflection.

How we see an image in the mirror Consider P, a source of light rays. Rays shown are some of those which reflect from the plane mirror. According to the Law of Reflection, the reflected ray is reflected at the same angle as the incident ray. Some of these rays are collected by the eye.

How we see an image in the mirror The brain interprets these rays as having traveled a diverging path from a source behind the mirror. It then reconstructs a “virtual image” based on the light rays. This image is P’. This image looks no different than a real object.

25.3 The Formation of Images by a Plane Mirror A ray of light from the top of the chess piece reflects from the mirror. To the eye, the ray seems to come from behind the mirror. Because none of the rays actually emanate from the image, it is called a virtual image.

The plane mirror and properties of the image The image is upright. The image is horizontally inverted The image is the same size as the object (no magnification). The image is as far behind the mirror as the object is in front.

Geometry used to show that image distance di equals object distance do Consider the triangles ABC and DBC. The two angles labeled θ are equal, according to the Law of Reflection. The angle α is also equal to θ (opposite angles formed by intersecting lines). If α = θ, then β1 = β2 All 3 angles are equal and they share leg BC Therefore, d0 = di

Vanity, thy name is….. To see your full body in a mirror, what must the minimum length be? Half your height Greater than half your height Your full height Greater than full height I hate mirrors

25.3 The Formation of Images by a Plane Mirror Conceptual Example 1 Full-Length Versus Half-Length Mirrors

Two plane mirrors form a right angle Two plane mirrors form a right angle. How many images of the ball can you see in the mirrors? 1 2 3 4 infinite Answer: C

Law of Reflection Problem The tilted mirror in reflects a horizontal laser beam so that the reflected beam is 60˚ from horizontal. What is the angle ɸ (Hint: draw in the normal to the mirror) ? 15° 30° 45 ° 60 °

25.4 Spherical Mirrors If the inside surface of the spherical mirror is reflective, it is a concave mirror. If the outside surface is reflective, is it a convex mirror. The law of reflection applies, just as it does for a plane mirror, with the normal (not shown below) drawn at the point where the incident light ray strikes the mirror.

The radius of the sphere is called the radius of curvature, R. 25.4 Spherical Mirrors The center of curvature is the center of the sphere. It is located at point C. The radius of the sphere is called the radius of curvature, R. The principal axis of the mirror is a normal that passes through the center, C of the sphere.

25.4 Spherical Mirrors – Concave Mirrors Consider a point on the tree lies on the principal axis of the concave mirror. Rays from that point that are near the principal axis (paraxial rays) reflect and then cross the principal axis at the same place, at the image point.

25.4 Spherical Mirrors – Concave Mirrors Light rays diverge from the image point, just as they would from an actual object. Since light rays actually come from the image point, this is a real image, and not a virtual image, as with the plane mirror. In order to see a real image, you need to place an opaque screen at the image point.

25.4 Spherical Mirrors – concave mirrors If the object is infinitely far from the mirror, the light rays are considered to become parallel to the principal axis as they get close to the mirror. In this special case, the image point is called the focal point, F. The focal length f is the distance between the focal point and the mirror. The focal point is not always the image point, just when the object is far away.

The blue triangle is isosceles, so CF = AF. 25.4 Spherical Mirrors Consider a paraxial ray from an object infinitely far away that reflects from the mirror at A. Line AC (a radius) is the normal at the point of incidence, BC is the principal axis (also a radius) and AF is the relected ray. The blue triangle is isosceles, so CF = AF. If θ is small, then AF ≈ FB ≈ ½ R Therefore the focal length, f, is one half the radius of curvature:

We will consider only paraxial rays in our analysis. 25.4 Spherical Mirrors not paraxial almost paraxial paraxial Rays from a distant object that are far from the principal axis do not converge at the focal point. The fact that a spherical mirror does not bring all parallel rays to a single point is known as spherical abberation. We will consider only paraxial rays in our analysis.

25.4 Spherical Mirrors Concave vs Conves concave mirror When paraxial light rays from a distant object strike a concave mirror, a real image is produced at the focal point in front of the mirror. When paraxial light rays from a distant object to the principal axis strike a convex mirror, the rays appear to originate from an focal point behind the mirror. The image is a virtual image. convex mirror

25.4 Spherical Mirrors – Ray Tracing Ray tracing is a graphical procedure using specific rays to determine the location, size and orientation of an image.

25.5 The Formation of Images by Spherical Mirrors RAY TRACING FOR CONCAVE MIRRORS – Use these 3 rays 1. This ray is initially parallel to the principal axis and passes through the focal point. 2. This ray initially passes through the focal point, then emerges parallel to the principal axis. 3. This ray travels along a line that passes through the center.

25.5 The Formation of Images by Spherical Mirrors Image Formation – Concave Mirrors When the object (red arrow) is placed between the center of curvature and the focal point, the resulting image is real, enlarged, and inverted, relative to the object. In order to see a real image, you need to place a screen at the image point

25.5 The Formation of Images by Spherical Mirrors Image Formation – Concave Mirrors When the object (red arrow) is located beyond the center of curvature, the resulting image is real, reduced, and inverted, relative to the object. In order to see a real image, you need to place a screen at the image point.

25.5 The Formation of Images by Spherical Mirrors When an object is located between the focal point and a concave mirror, and enlarged, upright, and virtual image is produced.

Ray diagram applet www.phy.ntnu.edu.tw/java/Lens/lens_e.html

25.5 The Formation of Images by Spherical Mirrors CONVEX MIRRORS Ray 1 is initially parallel to the principal axis and appears to originate from the focal point. Ray 2 heads towards the focal point, emerging parallel to the principal axis. Ray 3 travels toward the center of curvature and reflects back on itself.

25.5 The Formation of Images by Spherical Mirrors The virtual image is diminished in size and upright.

25.6 The Mirror Equation and Magnification (Negative for convex lens) (Negative for virtual images) (Negative for inverted image)

25.6 The Mirror Equation and Magnification Summary of Sign Conventions for Spherical Mirrors

25.6 The Mirror Equation and Magnification These diagrams are used to derive the mirror equation.

25.6 The Mirror Equation and Magnification Example 5 A Virtual Image Formed by a Convex Mirror A convex mirror is used to reflect light from an object placed 66 cm in front of the mirror. The focal length of the mirror is -46 cm. Find the location of the image and the magnification.

25.6 The Mirror Equation and Magnification Example 5 A Virtual Image Formed by a Convex Mirror A convex mirror is used to reflect light from an object placed 66 cm in front of the mirror. The focal length of the mirror is -46 cm. Find the location of the image and the magnification.

Analyzing a concave mirror A 3.0 cm object is located 20 cm from a concave mirror. The radius of curvature is 80 cm. Determine the position, orientation and height of the image.

Analyzing a concave mirror A 3.0 cm-high object is located 20 cm from a concave mirror. The radius of curvature is 80 cm. Determine the position, orientation and height of the image. di = -40 cm, therefore image is virtual (behind mirror). m = +2.0, therefore object is upright and is 6.0 cm high.

A concave mirror of focal length f forms an image of the moon (very distant object). Where is the image located? Almost exactly a distance f behind  the mirror. Almost exactly a distance f in front  of the mirror. At a distance behind the mirror equal to the distance of the moon in front of the mirror. At the mirror’s surface. Answer: B

25.5 The Formation of Images by Spherical Mirrors