Chapters 17 & 18 Mirrors and Lenses

θi = θr 17-1 Plane Mirrors The Law of Reflection When a wave traveling in two dimensions encounters a barrier, the angle of incidence is equal to the angle of reflection of the wave. θi = θr

A smooth surface, such as a mirror, Causes specular reflection, in which Parallel light rays are reflected in parallel. A rough surface, such as jagged ice or A brick wall, causes diffuse reflection, the Scattering of light. The law of reflection applies for both Of the surfaces.

What is the angle of rotation of the reflected ray? Problem... A light ray strikes a plane mirror at an angle of 52.0° to the normal. The mirror then rotates 35.0° around the point where the beam strikes the mirror so that the angle of incidence of the light ray decreases. The axis of rotation is perpendicular to the plane of the incident and the reflected rays. What is the angle of rotation of the reflected ray? Δθr = 52.0° + 35.0° – 17.0° = 70.0° clockwise from the original angle

A plane mirror is a flat, smooth surface from which light is reflected by specular reflection. To understand reflection from a mirror, you must consider the object of the reflection and the type of image that is formed. An object is a source of light rays that are to be reflected by a mirrored surface. An object can be a luminous source, such as a lightbulb, or an illuminated source, such as a girl, as shown in the figure.

The combination of the image points produced by reflected light rays forms the image of the bird. It is a virtual image, which is a type of image formed by diverging light rays. A virtual image is always on the opposite side of the mirror from the object. Images of real objects produced by plane mirrors are always virtual images.

Ray 1, which strikes the mirror at an angle of incidence of 0°, is reflected back on itself, so the sight line is at 90° to the mirror, just as ray 1. Ray 2 is also reflected at an angle equal to the angle of incidence, so the sight line is at the same angle to the mirror as ray 2. The apparent position of the image with respect to the mirror, or the image position, di, has a length equal to the length of do.

There are 2 kinds of curves, 17-2 Curved Mirrors There are 2 kinds of curves, Concave and convex.

Properties of a concave mirror depend on how much it is curved. A concave mirror has a reflective surface, the edges of which curve toward the observer. Properties of a concave mirror depend on how much it is curved. If you turn a spoon around, the outer surface acts as a convex mirror, a reflective surface with edges that curve away from the observer

In a spherical concave mirror, the mirror is shaped as if it were a section of a hollow sphere with an inner reflective surface. The mirror has the same geometric center, C, and radius of curvature, r, as a sphere of radius, r. The line that passes through line segment CM is the principal axis, which is the straight line perpendicular to the surface of the mirror that divides the mirror in half.

the focal point of the mirror, the point where incident light rays that are parallel to the principal axis converge after reflecting from the mirror F is at the halfway point between M and C. The focal length, f, is the position of the focal point with respect to a mirror along the principal axis and can be expressed as f = r/2. The focal length is positive for a concave mirror.

How to draw a ray diagram… Click the video stupid…

Problem... Draw a ray diagram…

Mirror Equation The reciprocal of the focal length of a spherical mirror is equal to the sum of the reciprocals of the image position and the object position

Magnification is a property of spherical mirrors which refers to how much larger or smaller an image is relative to the object. The magnification of an object by a spherical mirror, defined as the image height divided by the object height, is equal to the negative of the image position, divided by the object distance. Magnification Thus, the magnification is negative, which means that the image is inverted compared to the object. Thus, the magnification is negative, which means that the image is inverted.

Magnification +M = upright (imaginary) image ho = Object height hi = Image height +M = upright (imaginary) image -M = inverted (real) image 0<M<1 means image is smaller than object M>1 means image is larger than object

Problem... A concave mirror has a radius of 20.0 cm. A 2.0-cm-tall object is 30.0 cm from the mirror. What is the image position and image height? di hi

When you use the mirror equation to solve problems involving concave mirrors for which an object is between the mirror and the focal point, you will find that the image position is negative. The magnification equation gives a positive magnification greater than 1, which means that the image is upright and larger compared to the object, like the image shown here.

Convex Mirrors Properties of a spherical convex mirror are shown in the figure. Rays reflected from a convex mirror always diverge. Thus, convex mirrors form virtual images. Points F and C are behind the mirror. In the mirror equation, f and di are negative numbers because they are both behind the mirror.

Ray 1 approaches the mirror parallel to the principal axis. The ray diagram in the figure represents how an image is formed by a convex mirror. The figure uses two rays, but remember that there are an infinite number of rays. Ray 1 approaches the mirror parallel to the principal axis. The reflected ray is drawn along a sight line from F through the point where ray 1 strikes the mirror.

Ray 2 approaches the mirror on a path that, if extended behind the mirror, would pass through F. The reflected part of ray 2 and its sight line are parallel to the principal axis. The two reflected rays diverge, and the sight lines intersect behind the mirror at the location of the image.

By forming smaller images, convex mirrors enlarge the area, or field of view, that an observer sees, as shown in the figure. The center of this field of view is visible from any angle of an observer off the principal axis of the mirror; thus, the field of view is visible from a wide perspective. For this reason, convex mirrors often are used in cars as passenger- side rearview mirrors.

The table compares the properties of single-mirror systems with objects that are located on the principal axis of the mirror.

Mirror Comparison Virtual images are always behind the mirror, which means that the image position is negative. When the absolute value of a magnification is between zero and one, the image is smaller than the object. A negative magnification means the image is inverted relative to the object.

Snell’s Law of Refraction 18-1 Refraction of Light Snell’s Law of Refraction When you shine a narrow beam of light at the surface of a piece of glass, it bends as it crosses the boundary from air to glass. The bending of light, called refraction, was first studied by René Descartes and Willebrord Snell around the time of Kepler and Galileo.

Snell found that when light went from air into a transparent substance, the sines of the angles were related by the equation sin θ1/sin θ2 = n. Here, n is a constant that depends on the substance, not on the angles, and is called the index of refraction. Snell’s Law of Refraction According to Snell’s Law of Refraction, the product of the index of refraction of the first medium and the sine of the angle of incidence is equal to the product of the index of refraction of the second medium and the sine of the angle of refraction.

Problem... A light beam in air hits a sheet of crown glass at an angle of 30.0°. At what angle is the light beam refracted? θ1 = 30.0º n1 = 1.00 n2 = 1.52 θ2 = ? θ2 = 19.2º

The frequency of light, f, does not change when it crosses a boundary. The wave relationship for light traveling through a vacuum, λ = c/f, can be rewritten as λ = v/f, where v is the speed of light in any medium, and λ is the wavelength. The frequency of light, f, does not change when it crosses a boundary. That is, the number of oscillations per second that arrive at a boundary is the same as the number that leave the boundary and transmit through the refracting medium. So, the wavelength of light, λ, must decrease when light slows down. Wavelength in a medium is shorter than wavelength in a vacuum.

Index of Refraction The index of refraction of a medium is equal to the speed of light in a vacuum divided by the speed of light in the medium. This definition of the index of refraction can be used to find the wavelength of light in a medium. In a medium with an index of refraction n, the speed of light is given by v = c/n

Click the video stupid… Index of refraction… Click the video stupid…

Total internal reflection causes some curious effects. Suppose that you are looking up at the surface from underwater in a calm pool. You might see an upside-down reflection of another nearby object that also is underwater or a reflection of the bottom of the pool itself. The surface of the water acts like a mirror.

Likewise, when you are standing on the side of a pool, it is possible for things below the surface of the water in the pool to not be visible to you. When a swimmer is underwater, near the surface, and on the opposite side of the pool from you, you might not see him or her. This is because the light from his or her body does not transmit from the water into the air, but is reflected.

The pool, however, disappears as you approach it. Mirages On a hot summer day, as you drive down a road, you see what appears to be the reflection of an oncoming car in a pool of water. The pool, however, disappears as you approach it.

The mirage is the result of the Sun heating the road. The hot road heats the air above it and produces a thermal layering of air that causes light traveling toward the road to gradually bend upward. This makes the light appear to be coming from a reflection in a pool.

Dispersion of Light The speed of light in a medium is determined by interactions between the light and the atoms that make up the medium. Temperature and pressure are related to the energy of particles on the atomic level. The speed of light, and therefore, the index of refraction for a gaseous medium, can change slightly with temperature. In addition, the speed of light and the index of refraction vary for different wavelengths of light in the same liquid or solid medium.

White light separates into a spectrum of colors when it passes through a glass prism. This phenomenon is called dispersion. If you look carefully at the light that passes through a prism, you will notice that violet is refracted more than red. This occurs because the speed of violet light through glass is less than the speed of red light through glass.

Rainbows A prism is not the only means of dispersing light. A rainbow is a spectrum formed when sunlight is dispersed by water droplets in the atmosphere. Sunlight that falls on a water droplet is refracted. Because of dispersion, each color is refracted at a slightly different angle.

Although each droplet produces a complete spectrum, an observer positioned between the Sun and the rain will see only a certain wavelength of light from each droplet. The wavelength depends on the relative positions of the Sun, the droplet, and the observer.

Sometimes, you can see a faint second-order rainbow. The second rainbow is outside the first, is fainter, and has the order of the colors reversed. Light rays that are reflected twice inside water droplets produce this effect.

18-2 Convex & Concave Lenses A lens is a piece of transparent material, such as glass or plastic, that is used to focus light and form an image.

The lens shown in the figure is called a convex lens because it is thicker at the center than at the edges. A convex lens often is called a converging lens because when surrounded by material with a lower index of refraction, it refracts parallel light rays so that the rays meet at a point.

The lens shown in the figure is called a concave lens because it is thinner in the middle than at the edges. A concave lens often is called a diverging lens because when surrounded by material with a lower index of refraction, rays passing through it spread out.

The thin lens equation relates the focal length of a spherical thin lens to the object position and the image position. Same equation as Mirrors BUT SIGNS ARE OPPOSITE! The inverse of the focal length of a spherical lens is equal to the sum of the inverses of the image position and the object position.

The magnification equation for spherical mirrors also can be used for spherical thin lenses. It is used to determine the height and orientation of the image formed by a spherical thin lens. The magnification of an object by a spherical lens, defined as the image height divided by the object height, is equal to the negative of the image position divided by the object position.

Magnification (Same as Mirrors) ho = Object height hi = Image height +M = upright (imaginary) image -M = inverted (real) image 0<M<1 means image is smaller than object M>1 means image is larger than object

The table shows a comparison of the image position, magnification, and type of image formed by single convex and concave lenses when an object is placed at various object positions, do, relative to the lens.

Focal lengths and image positions can be negative. As with mirrors, the distance from the principal plane of a lens to its focal point is the focal length, f. The focal length depends upon the shape of the lens and the index of refraction of the lens material. Focal lengths and image positions can be negative. For lenses, virtual images are always on the same side of the lens as the object, which means that the image position is negative.

The rays of the Sun are almost exactly parallel when they reach Earth. Paper can be ignited by producing a real image of the Sun on the paper. The rays of the Sun are almost exactly parallel when they reach Earth.

The figure shows two focal points, one on each side of the lens. After being refracted by the lens, the rays converge at the focal point, F, of the lens. The figure shows two focal points, one on each side of the lens. You could turn the lens around, and it will work the same.

Problem... An object is placed 32.0 cm from a convex lens that has a focal length of 8.0 cm. Where is the image? 11 cm away on opposite side If the object is 3.0 cm high, how tall is the image? - 1.0 cm What is the orientation of the image?

When an object is placed at the focal point of a convex lens, the refracted rays will emerge in a parallel beam and no image will be seen. When the object is brought closer to the lens, the rays will diverge on the opposite side of the lens, and the rays will appear to an observer to come from a spot on the same side of the lens as the object. This is a virtual image that is upright and larger compared to the object.

A concave lens causes all rays to diverge. The figure shows how such a lens forms a virtual image. Ray 1 approaches the lens parallel to the principal axis, and leaves the lens along a line that extends back through the focal point. Ray 2 approaches the lens as if it is going to pass through the focal point on the opposite side, and leaves the lens parallel to the principal axis.

Because the rays diverge, they produce a virtual image. The sight lines of rays 1 and 2 intersect on the same side of the lens as the object. Because the rays diverge, they produce a virtual image. The image is located at the point from where the two rays apparently diverge. The image also is upright and smaller compared to the object. This is true no matter how far from the lens the object is located. The focal length of a concave lens is negative.

The object position always will be positive. When solving problems for concave lenses using the thin lens equation, you should remember that the sign convention for focal length is different from that of convex lenses. If the focal point for a concave lens is 24 cm from the lens, you should use the value f = −24 cm in the thin lens equation. All images for a concave lens are virtual. Thus, if an image distance is given as 20 cm from the lens, then you should use di = −20 cm. The object position always will be positive.

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