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