Unit 1, Lesson 8

The speed of light traveling in a vacuum & light traveling in air, in this book, is

3.00 x 10^8 m/s

A good mirror can reflect about

90% of the incident light, but no surface is a perfect reflector

Concave Spherical Mirror

A mirror whose reflecting surface is a segment of the inside of a sphere Used whenever a magnified image of an object is needed One factor that determines where the image will appear in a concave spherical mirror & how large that image will be is the radius of curvature (R) of the mirror; the radius of curvature is the same as the radius of the spherical shell of which the mirror is a small part; R is therefore the distance from the mirror's surface to the center of curvature (C)

Snell's Law

A relationship developed by Willebrord Snell, who, in 1621, experimented with light passing through different media This law can be used to find the angle of refraction for light traveling between any two media ni sin 0i = nr sin 0r index of refraction of first medium x sine of the angle of incidence = index of refraction of second medium x sine of the angle of refraction

Determining If & How Light is Linearly Polarized

By rotating a polarizing substance as a beam of polarized light passes through it, a change in the intensity of the light can be seen; the light is brightest when its plane of polarization is parallel to the transmission axis The larger the angle is between the electric-field waves & the transmission axis, the smaller the component of light that passes through the polarizer will be & the less bright the light will be; when the transmission axis is perpendicular to the plane of polarization for the light, no light passes through

Compound Microscope

Consists of two lenses: an objective lens (near the object) with a focal length of less than 1 cm & an eyepiece with a focal length of a few centimeters The object placed just outside the focal point of the objective lens forms a real, inverted, & enlarged image that is at or just inside the focal point of the eyepiece The eyepiece, which serves as a simple magnifier, uses this enlarged image as its object & produces an even more enlarged virtual image; the image viewed through a microscope is upside-down with respect to the actual orientation of the specimen

Applications of Convex Mirrors

Convex spherical mirrors take the objects in a large field of view & produce a small image, so they're well suited for providing a fixed observer with a complete view of a large area; convex mirrors are often placed in stores to help employees monitor customers & at the intersections of busy hallways so that people in both hallways can tell when others are approaching The side-view mirror on the passenger's side of a car is another application of the convex mirror; this mirror usually carries the warning "objects are closer than they appear"; without this warning, a driver might think they're looking into a flat mirror, which doesn't alter the size of the image; the driver could therefore be fooled into believing that a vehicle is farther away than it is because the image is smaller than the actual object

Mirror Equation 2

If the light bulb is placed very far from the mirror, the object distance (p) is great enough compared with R that 1/p is almost 0; in this case, q is almost R/2, so the image forms about halfway between the center of curvature & the center of the mirror's surface The image point in this special case is called the focal point of the mirror & is denoted by the capital letter F; because the light rays are reversible, the reflected rays from a light source at the focal point will emerge parallel to each other & won't form an image For light emerging from a source very far away from a mirror, the light rays are essentially parallel to one another; in this case, an image forms at the focal point (F), & the image distance is called the focal length (denoted by the lowercase letter f); for a spherical mirror, the focal length is equal to half the radius of curvature of the mirror; the mirror equation can therefore be expressed in terms of the focal length 1/p + 1/q = 1/f 1/object distance + 1/image distance = 1/focal length

Mirror Equation 1

If the object distance & radius of curvature of the mirror are known, you can predict where the image will be Alternatively, the radius of curvature of a mirror can be determined if you know where the image appears for a given object distance The following equation relates object distance (p), image distance (q), & the radius of curvature (R): 1/p + 1/q = 2/R

Combination of Thin Lenses

If two lenses are used to form an image, the system can be treated in the following manner First, the image of the first lens is calculated as though the second lens were not present; the light then approaches the second lens as if it had come from the image formed by the first lens; hence, the image formed by the first lens is treated as the object for the second lens; the image formed by the second lens is the final image of the system The overall magnification of a system of lenses is the product of the magnifications of the separate lenses; if the image formed by the first lens is in back of the second lens, then the image is treated as a virtual object for the second lens (that is, p is negative); the same procedure can be extended to a system of three or more lenses

Light Polarized by Scattering

In addition to reflection & absorption, scattering can also polarize light Scattering (the absorption & reradiation of light by particles in the atmosphere) causes sunlight to be polarized When an unpolarized beam of sunlight strikes air molecules, the electrons in the molecules begin vibrating with the electric field of the incoming wave A horizontally polarized wave is emitted by the electrons as a result of their horizontal motion, & a vertically polarized wave is emitted parallel to Earth as a result of their vertical motion Thus, an observer with their back to the sun will see polarized light when looking up toward the sky

Classical Electromagnetic Wave Theory

Light is considered to be a wave composed of oscillating electric & magnetic fields; these fields are perpendicular to the direction in which the wave moves; therefore, electromagnetic waves are also transverse waves; the electric & magnetic fields are also at right angles to each other Electromagnetic waves are distinguished by their different frequencies & wavelengths; in visible light, these differences in frequency & wavelength account for different colors; the difference in frequencies & wavelengths also distinguishes visible light from invisible electromagnetic radiation, such as X rays The electromagnetic spectrum is, in reality, continuous; there's no sharp division between one kind of wave & the next; some types of waves even have overlapping ranges

Equation for Magnification

M = h'/h = -(q/p) magnification = image height/object height = -(image distance/object distance)

Magnification of a Lens

Magnification (M) is defined as the ratio of image height to object height This equation can be used to calculate the magnification of both converging & diverging lenses M = h'/h = -(q/p) magnification = image height/object height = -(distance from image to lens/distance from object to lens) The magnification will describe the image's size & orientation; when the magnitude of the magnification of an object is less than one, the image is smaller than the object; when the magnitude of the magnification is greater than one, the image is larger than the object A negative sign for the magnification indicates that the image is real & inverted; a positive magnification signifies that the image is upright & virtual

Spherical Mirror

One basic type of curved mirror Has the shape of part of a sphere's surface A spherical mirror with light reflecting from its silvered, concave surface (the inner surface of a sphere) is called a concave spherical mirror

Parabolic Mirrors

Segments of a paraboloid (a three-dimensional parabola) whose inner surface is reflecting All rays parallel to the principal axis converge at the focal point regardless of where on the mirror's surface the rays reflect; thus, a real image forms without spherical aberration Similarly, light rays from an object at the focal point of a parabolic mirror will be reflected from the mirror in parallel rays Parabolic reflectors are ideal for flashlights & automobile headlights

Thin-Lens Equation

The equation that relates object & image distances for a lens Is called this because it's derived using the assumption that the lens is very thin In other words, this equation applies when the lens thickness is much smaller than its focal length 1/p + 1/q = 1/f 1/distance from object to lens + 1/distance from image to lens = 1/focal length This equation can be applied to both converging & diverging lenses if we adhere to a set of sign conventions; under this convention, an image in the back of the lens (a real image) has a positive image distance, & an image in front of the lens (a virtual image) has a negative image distance; a converging lens has a positive focal length & a diverging lens has a negative focal length; therefore, converging lenses are sometimes called positive lenses & diverging lenses are sometimes called negative lenses

Characteristics of Curved Mirrors

The images for objects close to the mirror are larger than the object, whereas the images of objects far from the mirror are smaller & upside down

Light Polarized by Reflection

When light's reflected at a certain angle from a surface, the reflected light is completely polarized parallel to the reflecting surface; if the surface is parallel to the ground, the light is polarized horizontally This is the case with glaring light that reflects at a low angle from roads, bodies of water, & car hoods Because the light that causes glare is in most cases horizontally polarized, it can be filtered out by a polarizing substance whose transmission axis is oriented vertically; this is the case with polarizing sunglasses; the angle between the polarized reflected light & the transmission axis of the polarizer is ninety degrees; thus, none of the polarized light passes through

wavelet

a circular or spherical secondary wave

Each of these point sources produces

a circular or spherical secondary wave, or wavelet

However, if an observer is at an elevated vantage point, such as on an airplane or at the rim of a canyon,

a complete circular rainbow can be seen

converging lens

a lens that's thicker at the middle than it is at the rim

diverging lens

a lens that's thinner at the middle than it is at the rim

ray

a line perpendicular to the wave front

convex spherical mirror

a mirror whose reflecting surface is an outward-curved segment of a sphere; images in this mirror are distorted near the mirror's edges, & the image is smaller than the object; a segment of a sphere that's silvered so that light is reflected from the sphere's outer, convex surface; this type of mirror is also called a diverging mirror because the incoming rays diverge after reflection as though they were coming from some point behind the mirror; the resulting image is therefore always virtual, & the image distance is always negative; because the mirrored surface is on the side opposite the radius of curvature, a convex spherical mirror also has a negative focal length

A typical lens consists of

a piece of glass or plastic ground so that each of its two refracting surfaces is a segment of either a sphere or a plane

crystalline lens

a small lens in the eye that refracts light

lens

a transparent object that refracts light rays such that they converge or diverge to create an image; like mirrors, lenses form images, but lenses do so by refraction rather than reflection; the images formed can be either real or virtual, depending on the type of lens & on the placement of the object

refracting telescope

a type of telescope that uses a combination of lenses to form an image

reflecting telescope

a type of telescope that uses a curved mirror & small lenses to form an image

A flat mirror always forms

a virtual image, which always appears as if it's behind the surface of the mirror; for this reason, a virtual image can never be displayed on a physical surface

electromagnetic wave

a wave that consists of oscillating electric & magnetic fields, which radiate outward from the source at the speed of light

All substances

absorb at least some incoming light & reflect the rest

Pigments rely on colors of light that are

absorbed (or subtracted) from the incoming light (for example, yellow pigment subtracts blue & violet colors from white light & reflects red, orange, yellow, & green light; blue pigment subtracts red, orange, & yellow from the light & reflects green, blue, & violet; when yellow & blue pigments are combined, only green light is reflected)

Because the mirrored surface is on the front side of a concave mirror, its focal point

always has a positive sign

Distances for images that form on the back side of the mirror

always have a negative sign

Light traveling through a uniform substance (whether it's air, water, or a vacuum)

always travels in a straight line

Huygens' principle

an approach to analyzing waves named for physicist Christian Huygens who developed it; involves wavelets in which the line that's tangent to each of the wavelets at some later time determines the new position of the initial wave front

real image

an image formed when rays of light actually pass through a point on the image; unlike the virtual images that appear behind a flat mirror, real images can be displayed on a surface, like the images on a movie screen

virtual image

an image that forms at a point from which light rays appear to come but don't actually come; the image formed by rays that appear to come from the image point behind the mirror but never really do

index of refraction

an important property of transparent substances; the ratio of the speed of light in a vacuum to the speed of light in a given transparent medium; a dimensionless number that's always greater than one because light always travels slower in a substance than in a vacuum; the larger the index of refraction is, the slower light travels in that substance & the more a light ray will bend when it passes from a vacuum into that material

mirage

another phenomenon of nature produced by refraction in the atmosphere; can be observed when the ground is so hot that the air directly above it is warmer than the air at higher elevations

Ray diagrams should be used to obtain

approximate values only; they shouldn't be relied on for the best quantitative results

Although light is an electromagnetic wave, it also shows characteristics of

behaving like a particle

The object & image heights are positive when

both are above the principal axis

Wave Speed Equation

c = f(wavelength) speed of light = frequency x wavelength

Ray diagrams are useful for

checking values calculated from the mirror & magnification equations

Electric-field oscillations of unpolarized light waves can be treated as

combinations of vertical & horizontal electric-field oscillations

Subtractive primary colors filter out all light when

combined

Ray diagrams can be used for

concave spherical mirrors

wave fronts

crests or troughs (lines of particles)

The primary pigments (or primary subtractive colors are)

cyan, magenta, & yellow; these are the same colors that are complementary to the additive primary colors; when any two primary subtractive colors are combined, they produce either red, green, or blue pigments; when the three primary pigments are mixed together in the proper proportions, all of the colors are subtracted from white light, & the mixture is black

Huygens' principle can be used to

derive the properties of any wave (including light) that interacts with matter, but the same results can be obtained by treating the propagating wave as a straight line perpendicular to the wave front; this line is called a ray, & this simplification is called the ray approximation

Rainbows are created by

dispersion of light in water droplets

ray diagrams

drawings that use simple geometry to locate an image formed by a mirror

When pigments are mixed,

each one subtracts certain colors from white light, & the resulting color depends on the frequencies that aren't absorbed

The object & image heights are negative when

either is below the principal axis

Incoming and reflected angles are

equal

transmission axis

for substances that polarize light by transmission, the line along which light is polarized

Lenses are commonly used to

form images in optical instruments, such as cameras, telescopes, & microscopes; in fact, transparent tissue in the front of the human eye acts as a lens, converging light toward the light-sensitive retina, which lines the back of the eye

Electromagnetic waves vary depending on

frequency & wavelength

What are examples of transparent media through which light can pass?

glass, water, ice, diamonds, & quartz; the speed of light in each of these materials is different (for example, the speed of light in water is less than the speed of light in air, and the speed of light in glass is less than the speed of light in water)

The texture of a surface affects

how it reflects light

Magnification relates

image & object signs

Ray diagrams of thin-lens system help identify

image height & location

Luminous flux is a measure of power output but

is weighted to take into account the response of the human eye to light

A surface is considered smooth if

its surface variations are small compared with the wavelength of the incoming light

thin lenses

lenses for which the thickness of the lens is small compared to the radius of curvature of the lens or the distance of the object from the lens

Total internal reflection occurs only when

light moves along a path from a medium of higher index of refraction to a medium of lower index of refraction; if ni were less than nr, this equation would give sin 0c > 1, which is an impossible result because by definition the sine of an angle can never be greater than 1

unpolarized light

light that consists of waves that have electric fields oscillating in random directions

What is one type of electromagnetic waves?

light waves

When two polarizing films are held with the transmission axes parallel,

light will pass through the films

Refraction occurs when

light's velocity changes

Object & image distances have a positive sign when

measured from the center of the mirror to any point on the mirror's front side

Refracted light produces

mirage

The spectrum includes

more than visible light

For simplicity, use the value

n = 1.00 for air when solving problems

Index of Refraction Equation

n = c/v index of refraction = speed of light in vacuum/speed of light in medium

For an image in front of the mirror, M is

negative & the image is upside down (inverted), with respect to the object

When two polarizing films are held with the transmission axes perpendicular to each other,

no light will pass through the films

Magnification by a lens depends on

object & image distances

myopia

occurs either when the eye is longer than normal or when the maximum focal length of the lens is insufficient to produce a clear image on the retina; the person is said to be nearsighted; in this case, light from a distant object is focused in front of the retina; the distinguishing feature of this imperfection is that distant objects aren't seen clearly; myopia can be corrected with a diverging lens

Because a diverging lens has the opposite shape, the chromatic aberration for a diverging lens is

opposite that for a converging lens

Reflecting telescopes use

parabolic mirrors

Because all electromagnetic radiation obeys the law of reflection,

parabolic surfaces can be constructed to reflect & focus electromagnetic radiation of different wavelengths; for instance, a radio telescope consists of a large metal parabolic surface that reflects radio waves in order to receive radio signals from objects in space

Only light waves that are linearly polarized with respect to the transmission axis of the polarizing substance can

pass freely through the substance; all light that's polarized at an angle of ninety degrees to the transmission axis doesn't pass through

A diverging lens creates a virtual image of a real object

placed anywhere with respect to the lens; the image is upright, & the magnification is always less than one (that is, the image size is reduced); additionally, the image appears inside the focal point for any placement of the real object

All the points on the wave front of a plane wave can be treated as

point sources (that is, coming from a source of negligible size)

When the image is behind the mirror, M is

positive & the image is upright with respect to the object

Image location can be predicted with

ray diagrams

Waves can be approximated as

rays

Concave mirrors can produce both

real & virtual images

Concave mirrors can be used to form

real images

Converging lenses can produce

real or virtual images of real objects

Light can be polarized by

reflection & scattering

diffuse reflection

reflection in which light that's reflected from a rough, textured surface (such as paper, cloth, or unpolished wood) is reflected in many different directions

specular reflection

reflection in which light that's reflected from smooth, shiny surfaces (such as a mirror or water in a pond) is reflected in one direction only

What are the two types of telescopes that use visible light?

refracting telescope & reflecting telescope

Objects appear to be in different positions due to

refraction

The path of a light ray that crosses a boundary between two different media is

reversible

If the mirror's large to begin with,

shielding its outer portion will limit how much of the mirror is used & thus will accomplish the same effect; however, many concave mirrors, such as those used in astronomical telescopes, are made large so that they'll collect a large amount of light; therefore, it's not desirable to limit how much of the mirror is used in order to reduce spherical aberration; an alternative approach is to use a mirror that's not a segment of a sphere but still focuses light rays in a manner similar to a small spherical concave mirror; this is accomplished with a parabolic mirror

A contact lens is

simply a lens worn directly over the cornea of the eye; the lens floats on a thin layer of tears

Critical Angle Relationship

sin0c = nr/ni for ni > nr sine (critical angle) = index of refraction of second medium/index of refraction of first medium but only if index of refraction of first medium > index of refraction of second medium

When the second substance is air, the critical angle is

small for substances with large indices of refraction

For our discussion, reflection will be used to mean only

specular reflection

Images created by spherical mirrors suffer from

spherical aberration

Parabolic mirrors eliminate

spherical aberration

linear polarization

the alignment of electromagnetic waves in such a way that the vibrations of the electric fields in each of the waves are parallel to each other

angle of incidence

the angle between a ray that strikes a surface and the line perpendicular to that surface at the point of contact

angle of incidence (0i)

the angle between the incoming ray & the normal

angle of refraction (0r)

the angle between the refracted ray & the normal

angle of reflection

the angle formed by the line perpendicular to a surface and the direction in which a reflected ray moves

critical angle

the angle of incidence at which the refracted light makes an angle of ninety degrees with the normal

Snell's law determines

the angle of refraction

refraction

the bending of a wave front as the wave front passes between two substances in which the speed of the wave differs; the bending of light as it travels from one medium to another

reflection

the change in direction of an electromagnetic wave at a surface that causes it to move away from the surface

total internal reflection

the complete reflection that takes place within a substance when the angle of incidence of light striking the surface boundary is greater than the critical angle; can occur when light moves along a path from a medium with a higher index of refraction to one with a lower index of refraction

Curved surfaces change

the direction of light

light ray

the direction of propagation of the wave; perpendicular to the wave front

focal length (f)

the distance from the focal point to the center of the lens

In an electromagnetic wave,

the electric field is at right angles to both the magnetic field & the direction of propagation

Rainbows are most commonly seen above the horizon, where

the ends of the rainbow disappear into the ground

Spherical aberration occurs for lenses also, and results from

the fact that the focal points of light rays far from the principal axis of a spherical lens are different from the focal points of rays with the same wavelength passing near the axis; rays near the middle of the lens are focused farther from the lens than rays at the edges

chromatic aberration

the focusing of different colors of light at different distances behind a lens; arises from the wavelength dependence of refraction

Focal length is

the image distance for an infinite object distance

Wavelength affects

the index of refraction

The frequency of the light doesn't change when

the light passes from one medium to another

If light travels from one transparent medium to another at any angle other than straight on (normal to the surface),

the light ray changes direction when it meets the boundary; as in the case of reflection, the angles of the incoming & refracted rays are measured with respect to the normal

normal (to the surface)

the line perpendicular to the reflecting surface

The focal point of a converging lens is

the location where the image of an object at an infinite distance from the lens is focused; unlike mirrors, every lens has a focal point on each side of the lens because light can pass through the lens from either side

illuminance (measured in lm/m^2, or lux)

the luminous flux divided by the area of the surface

If an image is larger than the object,

the magnitude of its magnification is greater than 1

If an image is smaller than the object,

the magnitude of its magnification is less than 1

magnification (M)

the measure of how large or small the image is with respect to the original object's size; the ratio of the height of the bulb's image to the actual image; also equals the negative of the ratio of the image distance to the object distance

lux

the measure of the illuminance

lumens (lm)

the measure of the luminous flux, or the rate at which light is emitted from a source

Image location can be predicted with ray diagrams as well as

the mirror equation

The line that is tangent to each of these wavelets at some later time determines

the new position of the initial wave front

The relationship between the object distance from the mirror (which is represented as p) & the image distance (which is represented as q) is such that

the object & image distances are equal; similarly, the image of the object is the same size as the object

The focal point of a diverging lens is

the point from which the diverged rays appear to originate

dispersion

the process of separating polychromatic light into its component wavelengths

luminous flux

the rate at which light is emitted from a source

If the ray moves from a material in which its speed is lower to one in which its speed is higher,

the ray is bent away from the normal

When light moves from a material in which its speed is higher to a material in which its speed is lower (such as from air to glass),

the ray is bent toward the normal

back side of the mirror

the region in which light rays don't exist & where virtual images are formed

front side of the mirror

the region in which light rays reflect & form real images

flat mirror

the simplest mirror

All electromagnetic waves move at

the speed of light

Illuminance decreases as

the square of the distance from the source

cornea

the transparent front of the eye that acts like a lens & directs light rays toward the light-sensitive retina in the back of the eye; most of the refraction of light occurs at the cornea

Chromatic aberration can be greatly reduced by

the use of a combination of converging & diverging lenses made from two different types of glass

Reflection can be explained in terms of

the wave model of light

The amount that light bends when entering a different medium depends on

the wavelength of the light, as well as the speed (thus, a spectrum is produced when white light passes through a prism; each color of light has a different wavelength; therefore, each color of the spectrum is refracted by a different amount)

Light bulbs are rated by

their power input (measured in watts) & their light output

If the incident ray of light is parallel to the normal,

then no refraction (bending) occurs in either case

A variety of forms of radiation (including X rays, microwaves, & radio waves) have many of the same properties as visible light because

they are all examples of electromagnetic waves

paraxial rays

those light rays that are very near the principal axis of the mirror; mirror equation & ray diagrams are valid for these

For spherical mirrors,

three reference rays are used to find the image point; the intersection of any two rays locates the image; the third ray should intersect at the same point & can be used to check the diagram

A polarizing substance can be used not only to linearly polarize light but also

to determine if & how light is linearly polarized

The index of refraction of a material can be used

to figure out how much a ray of light will be refracted as it passes from one medium to another; the greater the index of refraction, the more refraction occurs

A simple way to reduce the effect of spherical aberration is

to use a mirror with a small diameter; that way, the rays are never far from the principal axis

Light can be linearly polarized through

transmission

Light is bent toward the normal when it

travels from a medium with a lower index of refraction to one with a higher index of refraction

ray approximation

treating the propagating wave as a straight line perpendicular to the wave front (the ray)

Compound microscopes use

two converging lenses

Refracting telescopes also use

two converging lenses

Magnification is a

unitless quantity

Certain transparent crystals cause

unpolarized light that passes through them to become linearly polarized; the direction in which the electric fields are polarized is determined by the arrangement of the atoms or molecules in the crystal

Diverging lenses produce

virtual images from real objects

White light passed through a prism produces a

visible spectrum (these colors, in order of decreasing wavelength, are red, orange, yellow, green, blue, and violet)

spherical aberration

when certain rays don't exactly intersect at the image point; particularly noticeable for rays that are far from the principal axis & for mirrors with a small radius of curvature; also occurs with light rays & real spherical mirrors; produces a blurred image

hyperopia

when the eye attempts to produce a focused image of a nearby object but the image position is behind the retina; the person is said to be farsighted; with this defect, distant objects are seen clearly, but near objects are blurred; either the hyperopic eye is too short or the ciliary muscle that adjusts the shape of the lens can't adjust enough to properly focus the image; hyperopia can be corrected with a converging lens

The illuminance decreases as the radius squared when

you move away from a light source