Physics Ch 8 Light and Optics
lensmaker's equation
1/f= (n-1) (1/r₁ - 1/r₂) for lenses where thickness cannot be neglected, the focal length is related to the curvature of the lens surfaces and the index of refraction of the lens where n is the index of refraction of the lens material, r₁ is the radius of curvature of the first lens surface and r₂ is the radius of curvature of the second lens surface The eye is a complex refractive instrument that uses real lenses. The cornea acts as the primary source of refractive power bc the change in refractive index from air is so significant. Then, light is passed through an adaptive lens that can change its focal length before reaching the vitreous humor. It is further diffused through layers of retinal tissue to rach the rods and cones. At this point, the image has been focused and minimized significantly, but it is still relatively blurry. Our nervous system processes the remaining errors to provide a crisp view of the world
speed of light
EM waves vary in frequency and wavelength, but in a vacuum, all EM waves travel at the same speed, called the speed of light. This constant is represented by c and is ~3.00×10⁸ m/s. EM waves also travel in air with this speed. in reference to EM waves, the familiar equation v=fλ becomes c=fλ where c is the speed of light in a vacuum and to a first approximation also in air, f is the frequency and λ is the wavelength
sign convention for a single lens
For both lenses and mirrors, positive magnification represents upright images, and negative magnification means inverted images. Also for both, a pos image distance means that the image is real and is located on the real (R) side, whereas a negative image distance means that the image is virtual and located on the virtual (V) side.
multiple lens systems
Lenses in contact are a series of lenses with negligible distances between them. These systems behave as a single lens with equivalent focal length given by [figure] because power is the reciprocal of focal length, the equivalent power is [figure] ex, lenses in contact is corrective contact lens worn directly on the eye. In this case, the lens of the eye (converging lens) is in contact with a contact lens (either converging or diverging, depending on the necessary correction) and their powers would be added
Thin spherical lenses
On the MCAT, lenses generally have negligible thickness. Bc light can travel from either side of a lens, a lens has two focal points, with one on each side. The focal length can be measured in either direction from the center. For thin spherical lenses, the focal lengths are equal, so we speak of just one focal length for the lens as a whole converging lens is always thicker at the cneter, a divering lens is always thinner at the center. The basic formulas for finding image distance and magnification for spherical mirrors also apply to lenses. The object distance o, image distance i, focal length f, and magnification are all related by the equations 1/f= 1/o+1/i=2/r and m=-i/o a) convex lenses b) concave lenses
hyperopia and myopia
People who are nearsighted (can see near objects clearly) need diverging lens while people who are farsighted (can see distant objects clearly) need converging lenses. Bifocal lenses are corrective lenses that have two distinct regions- one that causes convergence of light to correct for farsightedness (*hyperopia*) and a second that causes divergence of light to correct for nearsightedness (*myopia*) in the same lens
Electromagnetic waves
a changing magnetic field can cause a change in an electric field, and changin an electric field can cause a change in the magentic field. bc of the reciprocating nature of these two fields, we can see how electromagnetic waves occur in nature. Each oscillating field causes oscillations on the other field completely independent of matter, so electromagnetic waves can even travel through a vacuum EM waves are transverse waves bc the osillating electric and magnetic filed vectors are perpendicular to the direction of propagation. [Electric field oscillates up and down; magnetic field oscillates into and out of the page]
magnification (m)
a dimensionless value that is the ratio of the image distance to the object distance: m= -i/o by extension, the magnification also gives the ratio of the size of the image to the size of the object. Following the sign convention table, the orientation of the image (inverted or upright) can be determined if |m|<1, the image is smaller than the object (reduced); if |m|>1, the image is larger than the object (enlarged); and if |m|=1 the image is the same size as the object
Chromatic aberration
a dispersive effect w/n a spherical lens. Depending on the thickness and curvature of the lens, there may be significant splitting of the white light, which results in a rainbow halo around images. This phenomenon is corrected for in visual lenses like eyeglasses and car windows with special coatings that have different dispersive qualities than the lens itself
inverted
a negative magnification signifies an inverted image
total internal reflection
a phenomenon in which all light incident on a boundary is reflected back into the original material, results with any angle of incidence greater than the critical angle θc key concept: total internal reflection occurs as the light moves from a medium with a higher refractive index to a medium with a lower one
center of curvature
a point on the optical axis located at a distance equal to the radius of curvature from the vertex of the mirror the center of the curvature would be the center of the spherically shaped mirror if it were a complete sphere
upright
a positive magnification value signifies an upright image
Ray diagrams for convex mirrors
a single divering mirror forms only a virtual, upright, and reduced image, regardless of the position of the object. The further away the object, the smaller the image will be. To quickly remember these rules, recall the convenience store security mirrors mentioned
single slit
although it is usually safe to assume that nonrefracted light travels in a straight line, there are situations where light will not actually travel in a straight line path. When light passes through a narrow opening (an opening with a size that is on the order of light wavelengths), the light waves seem to spread out (diffract), as the slit is narrowed, the light spreads out more light emerges from a narrow slit in a wide arc, not a narrow beam
Key concept: an object at the focal point of a converging mirror
any time an object is at the focal point of a converging mirror, the reflected rays will be parallel, and thus the image will be at infinity
spherical mirrors
come in two varieties: concave and convex. The word spherical implies that the mirror can be considered a spherical cap or dome taken from a much larger spherically shaped mirror. Spherical mirrors have an associated center of curvature (C) and a radius of curvature (r)
Diffraction gratings
consist of multiple slits arranged in patterns. Diffraction grating can create colorful patterns similar to a prism as the different wavelengths interfere in characteristic patterns. For example, the organization of the grooves on a CD or DVD act like a diffraction grating, creating an iridescent rainbow pattern on the surface of the disc. This films may also cause interference patterns bc light waves reflecting off the external surface of the film interfere with light waves reflecting off the internal surface of the film. Common examples of thin films: soap bubbles, oil puddles. Note that the interference is not between defracted rays but between reflected rays interference patterns occur as light waves reflecting off the external surface of the film interfere with light waves reflecting off the internal surface of the film. Note that there would be a small degree of refraction as well.
Key concept: Concave mirrors are ________________________ mirrors Convex mirrors are _________________________ mirrors The _______________is true for lenses
converging; diverging; reverse
focal length and radii
for both mirrors and lenses, converging species have a positive focal lengths and radii of curvature, and diverging species have negative focal lengths and radii of curvature. Remember that lenses have two focal lengths and two radii of curvature bc they have two surfaces. For a thin lens where thickness is negligible, the sign of the focal length and radius of curvature are generally given based on the first surface the light passes through
magnification of a multiple lens system
for lenses not in contact, the image of one lens becomes the object of another lens. The image from the last lens is considered the image of the system. Microscopes and telescopes are good examples of these systems. The magnification for the system is m=m₁×m₂×m₃×...mⁿ
Slit-lens system
if a lens is placed between a narrow slit and a screen, a pattern is observed consisting of bright central fringe with alternating dark and bright fringes on each side. The central bright fringe (maximum) is twice as wide as the bright fringes on the sides, and as the slit becomes narrower, the central maximum becomes wider. The location of the dark fringes (minima) is given by the formula aSinθ=nλ where a is the width of the slit, θ is the angle between the line drawn from the center of the lens to the dark fringe and the axis of the lens, n is an integer indicating the number of the fringe, and λ is the wavelength of the incident wave. Note that bright fringes are halfway between dark fringes
real image
if the light actually converges at the position of the image
virtual image
if the light only appears to be coming from the position of the image but does not actually converge there. One of the distinguishing features of real images is the ability of the image to be projected onto a screen
concave
if we were to look from the inside of a sphere to its surface we would see a concave surface mnemonic: concave is like looking into a cave the center of curvature and the radius of curvature are located in front of the mirror concave mirrors= *converging* *mirrors*, they cause parallel incident light rays to converge
convex
if we were to look from the outside of the sphere we would see a convex surface the center of the curvature and the radius of the curvature are behind the mirror. convex mirrors = *diverging* *mirrors*, they cause parallel incident light rays to diverge
electromagnetic spectrum
includes *radio* waves on one end (long wavelength, low frequency, low energy) and *gamma* rays on the other (short wavelength, high frequency high energy) Between the two extremes we find in order from lowest to highest energy, *microwaves*, *infrared*, *visible* light, *ultraviolet* and *x-rays* visible light (400nm to 700nm) describes the full range of frequencies and wavelengths of E< waves. Wavlengths are often given in the following units: mm (10⁻³m), µm (10⁻⁶m), nm (10⁻⁹m) and angstom, 10⁻¹⁰ m) the full spectrum is broken up into many regions, which in descending order of wavelength are radio (10⁹-1m), microwave (1m-1mm), infrared (1mm-700nm), visible light (700-400nm), UV (400-50nm), x-ray (50-10²nm) and γ-rays (less than 10⁻²nm) Order of colors: Roy G Biv
normal
is a line drawn perpendicular to the boundary of a medium; all angles in optics are measured from the normal, not the surface of the medium
key concept: mirrors and lenses
it is important to realize that concave mirrors and convex lenses are both converging and thus have similar properties. convex mirrors and concave lenses are both diverging and also have similar properties
lenses
lenses refract light mirrors reflect. When working with lenses, there are two surfaces that affect the light path. ex a person wearing glasses sees light that travels from an object through the air into the glass lens (first surface). Then the light travels through the glass until it reaches the other side, where again it travels out of the glass and into the air (second surface). The light is refracted twice as it passes from air to lens and from lens back to air.
plane-polarized light
light in which the electric fileds of all the waves are oriented in the same direction (that is their electric field vectors are parallel) it follows that their magnetic filed vectors are also parallel but convention dictates that the plane of the electric field identifies the plane of polarization. Un polarized light has a random orientation of its electric field vectors; sunlight and lamp light are exs. Application to stereosiomers in ochem: the optical activity of a compound, due to the presence of chiral centers, causes plane polarized light to rotate clockwise or counterclockwise by a given number of degrees relative to its concentration (its *specific* *rotation*). Remember that enantiomers, as nonsuperimposable mirror images will have opposite specific rotations there are filters called polarizers, often used in cameras and sunglasses, which allow only light with an electric filed pointing in a particular direction to pass through. If one passes a beam of light through a polarizer, it will only let through that portion of the light parallel to the axis of the polarizer. If a second polarizer is then held up to the first, the angle between the polarizers axes will determine how much light passes through. When polarizers are aligned, all the light that passes through the first polarizer also passes through the second. When the second polarizer is turned so that its axis is perpendicular, no light gets through it at all
sign convention for a single mirror
on the MCAT, for almost all problems involving mirrors, the object will be placed in front of the mirror. Thus the object distance o is almost always positive key concept: the focal length of converging mirrors (and converging lenses) will always be positive. The focal length of diverging mirrors (and diverging lenses) will always be negative Mnemonic: image types with a single lens or mirror (assuming o is positive): *IR* and *UV* *I*nverted images are always *r*eal *u*pright images are always *v*irtual
image distance
on the mcat you will most often use the optics equation to calculate the image distance for all types of mirrors and lenses. If the image has a positive distance (i>0), it is a real image which implies that the image is in front of the mirror. If the image has a negative distance (i<0) it is virtual and thus located behind the mirror plane mirrors can be thought of as spherical mirrors with infinitely large focal distances. As such, for a plane mirror r=f=∞ and the equation becomes 1/o +1/i= 0 or i=-o this can be interpreted as saying the virtual image is at a distance behind the mirror equal to the distance the object is infront of the mirror
Power (P)
optometrists often describe a lens in terms of its power (P) this is measured in *diopters* where f (the focal length) is in meters and is given by the equation P=1/f P has the same sign as f and is therefore, positive for a converging lens and negative for a diverging lens. People who are nearsighted (can see near objects clearly) need diverging lens while people who are farsighted (can see distant objects clearly) need converging lenses. Bifocal lenses are corrective lenses that have two distinct regions- one that causes convergence of light to correct for farsightedness (*hyperopia*) and a second that causes divergence of light to correct for nearsightedness (*myopia*) in the same lens
plane mirrors
parallel incident light rays remain parallel after reflection from a plane mirror; that is, plane mirrors- being relatively flat surfaces- cause neither convergence nor divergence of reflected light rays. Bc the light does not converge at all, plane mirrors always create virtual images. In a plane mirror, the image appears to be the same distance behind the mirror as the object is in front of it. In other words, plane mirrors create the appearance of light rays originating behind the mirrored surface. Bc the reflected light remains in front of the mirror, but the image appears behind the mirror, the image is virtual. Plane mirrors include most of the common mirrors found in our homes. To assist in our discussion of spherical mirrors, plane mirrors can be conceptualized as spherical mirrors with an infinite radius of curvature. all incident angles ( θ) are equal to their respective reflected angles (θ')
circular polarization
rarely seen natural phenomenon that results from the interaction of light with certain pigments or highly specialized filters. circularly polarized light has a uniform amplitude but a continuously changing direction, which causes a helical orientation in the propagating wave. The helix has avg electrical field vectors and magnetic field vectors that lie perpendicular to one another like other waves with maxima that fall on the outer border of the helix
key conept: to find where the image is (for a mirror), draw the following rays and find a point where any two intersect. This point of intersection marks the tip of the image. If the rays you draw do not appear to intersect, extend them to the other side of the mirror, creating a virtual image
ray parallel to axis→ reflects back through focal point Ray through focal point→ reflects back parallel to axis Ray to center of mirror→ reflects back at same angle relative to normal
to find where the image is (for a lens), draw the following rays and find a point where any two intersect. This point of intersection marks the tip of the image. If the rays you draw do not appear to intersect, extend them to the same side of the lens from which the light came, creating a virtual image.
ray parallel to axis→ refracts through focal point of front face of the lens Ray through or toward focal point before reaching lens→ refracts parallel to axis Ray to center of lens→ continues straight through with no refraction
reflection
rebounding of incident light waves at the boundary of a medium. Light waves that are reflected are not absorbed into the second medium; rather they bounce off of the boundary and travel back through the first medium. The law of reflection is θ₁=θ₂ where θ₁ is the incident angle and θ₂ is the reflected angle, both measured from the normal. the *normal* is a line drawn perpendicular to the boundary of a medium; all angles in optics are measured from the normal, not the surface of the medium
blackbody
refers to an ideal absorber of all wavelengths of light, which would appear completely black if it were at a lower temperature than its surroundings
diffraction
refers to the speading out of light as it passes through a narrow opening or around an obstacle. Interference between diffracted light rays lead to characteristic fringes in slit-lens and double-slit systems. Diffraction and interference are significant evidence for the wave theory of light
Snell's Law
refracted rays of light obey Snell's law as they pass from one medium to another: n₁sinθ₁= n₂sinθ₂ where n₁ and θ₁ refer to the medium from which the light is coming and n₂ and θ₂ refer to the medium into which light is entering. Note that θ is once again measured with respect to the normal from snell's law we can see that when light enters a medium with a higher index of refraction (n₂>n₁), it bends toward the normal (sinθ₂<sinθ₁; therefore, θ₂<θ₁). conversely, if the light travels into a medium where the index of refraction is smaller (n₂<n₁) the light will bend away from the normal (sinθ₂>sinθ₁; therefore, θ₂>θ₁)
key concept: refraction
remember that when light enters a medium with a higher index of refraction, it bends toward the normal. When light enters a medium with a lower index of refraction, it bends away from the normal
aberrations
spherical mirrors and lenses are imperfect. They are therefore subject to specific types of errors or aberrations
Refraction
the bending of light as it passes from one medium to another and changes speed. The speed of light through any medium is always less than its speed through a vacuum. Remember that the speed of light in a vacuum, c, is equal to 3*10^8 m/s.
Spherical aberration
the blurring of the periphery of an image as a result of inadequate reflection of parallel beams at the edge of a mirror or inadequate refraction of parallel beams at the edge of a lens. This creates an area of multiple images with very slightly different image distances at the edge of the image, which appears blurry. chromatic aberration is predominantly seen in spherical lenses
real and virtual
the designations or real and virtual are often a point of confusion bc they are on opposite sides when comparing mirrors and lenses. To identify the real side (R), remember that the real side is where light actually goes after interacting with the lens or mirror. For mirrors, light is reflected and therefore, stays infront of the mirror. Hence for a mirror the real side is in front of the mirror and the virtual side is behind the mirror for lenses, the convention is different: bc light travels through the lens and comes out on the other side, the real side is on the opposite side of the lens from the original light source, and the virtual side is on the same side of the lens as the original light source. Although the object of a single lens is on the virtual side, this does not make the object virtual. Objects are real, with a positive object distance, unless they are placed in certain multiple lens systems in which the image of one lens becomes the object for another (a scenario which is very rarely encountered on the MCAT)
Key concept: the electric filed of unpolarized light waves exist in all three dimensions:
the direction of the waves propagation is surrounded by electric fields in every plane perpendicular to that direction. Polarizing light limits the electric field's oscillation to only two dimensions.
focal length (f)
the distance between the focal point (F) and the mirror. for all spherical mirrors, f=r/2 where the radius of the curvature (r) is the distance between C and the mirror. The distance between the object and the mirror is o; the distance between the image and mirror is i. There is a simple relationship between these three distances: 1/f=1/o+1/i=2/r while it is not important which units of distance are used in this equation, it is important that all values used have the same units as each other
visible region
the only part of the EM spectrum that is perceived as light by the human eye within this region, differing wavelengths are perceived as different colors, with violet at one end of the visible spectrum (400nm) and red at the other (700nm) Light that contains all the colors in equal intensity is perceived as white. The color of an object that does not emit its own light is dependent on the color of that it refelcts. Thus an object that appears red is one that absorbs all colors of light except red. This implies that a red object under green illumination will appear black bc it absorbs the green light and has no light to reflect
ray diagram
useful for getting an approximation of where an image is should be used with caution; under the pressure of test day it can be easy to draw them incorrectly. Therefore, it is important to practice drawing ray diagrams to avoid careless errors on test day, and it is also important to be familiar with how to solve optics questions mathematically When drawing a ray diagram there 3 important rays to draw. For concave mirror, a ray that strikes the mirror parallel to the *axis* (the normal passing through the center of the mirror) is reflected back through the focal point (green lines). A ray that passes through the focal point before reaching the mirror is reflected back parallel to the axis (red lines). A ray that strikes the mirror at the point of intersection with the axis is reflected back with the same angle measured from the normal (blue lines). The object is placed between F and C, and the image produced is real, inverted and magnified in 8.6b, the object is placed at F and no image is formed bc the reflected light rays are parallel to each other. In terms of the mirror equation we can say the image distance i=∞ here. For the scenario in Figure 8.6c, the object is placed between F and the mirror, and the image produced is virtual, upright and magnified.
X-ray diffraction
uses the bending of light rays to create a model of molecules. X-ray diffraction is often combined with protein crystallography during protein analysis. Dark and light fringes do not take on a linear appearance, but rather a complex two dimensional image.
Index of refraction
when light is in any medium besides a vacuum, its speed is less than c. For a given medium, n=c/v where c is the speed of light in a vacuum, v is the speed of light in the medium, and n is a dimensionless quantity called the *index* *of* *refraction* of the medium the index of refraction will be greater than 1. For air, n is essentially equal to 1 bc the speed of light in air is extremely close to c.
critical angel
when light travels from a medium with a higher index of refraction (such as water) to a medium with a lower index of refraction (such as air), the refracted angle is larger than the incident angle ( θ₂>θ₁); that is, the refracted light ray bends away from the normal. As the incident angle is increased the refracted angle also increases and eventually a special incident called the critical angle is reached, for which the refracted angle θ₂ equals 90 degrees. at the critical angle, the refracted light rays passes along the interface between the two media. The critical angle can be derived from Snell's law if θ₂= 90°, such that θc= sin⁻¹(n₂/n₁)
rectilinear propagation
when light travels through a homogeneous medium, it travels in a straight line. The behavior of light at the boundary of a medium or interface between two media is described by the theory of geometrical optics- explains reflection and refraction as well as the applications of mirrors as lenses.
dispersion
when various wavelengths of light separate from each other. ex splitting of white light into its component colors using a prism. if a source of white light is incident on one of the faces of a prism, the light emerging from the prism is spread out into a fan shaped beam. This occurs bc violet light has a smaller wavelength than red light and so is bent to a greater extent. Bc red experiences the least amount of refraction, it is always on top of the spectrum; violet, having experienced the greatest amount of refraction is always on the bottom of the spectrum. Note that as light enters a medium with a different index of refraction, the wavelength changes but the frequency of the light does not due to their different speeds while inside the prism, the various wavelengths of light are refracted to different degrees
interference
when waves interact with each other, the displacements of waves add together in a process called interference. In his famous double slit experiment, Thomas Young showed that the diffracted rays of light emerging from two parallel slits can interfere with one another. This was a landmark finding that contributed to understanding of light as a wave set up for Young's double slit experiment: monochromatic light (light of only one wavelenght) passes through the slits, an interference pattern is observed on a screen played behind the slits. Regions of constructive interference between the two light waves appear as bright fringes (maxima) on the screen. Conversely, in regions where the light waves interfere destructively, dark fringes (minima occur) the positions of dark fringes (maxima) on the screen can be found from the equation dSinθ= (n+½)λ where d is the distance between the two slits, θ is the angle between the line drawn from the midpoint between the two slits to the dark fringe and the normal, n is an integer indicating the number of the fringe, and λ is the wavelength of the incident wave. Note that bright fringes are halfway between dark fringes
