Chapter 22 Reflection and Refraction ofLight

A Brief History of Light 1000 AD 1000 AD It was proposed that light consisted of tiny particles It was proposed that light consisted of tiny particles Newton Newton Used this particle model to explain reflection and refraction Used this particle model to explain reflection and refraction Huygens Huygens Explained many properties of light by proposing light was wave-like Explained many properties of light by proposing light was wave-like

A Brief History of Light, cont Young Young Strong support for wave theory by showing interference Strong support for wave theory by showing interference Maxwell Maxwell Electromagnetic waves travel at the speed of light Electromagnetic waves travel at the speed of light

A Brief History of Light, final Planck Planck EM radiation is quantized EM radiation is quantized Implies particles Implies particles Explained light spectrum emitted by hot objects Explained light spectrum emitted by hot objects Einstein Einstein Particle nature of light Particle nature of light Explained the photoelectric effect Explained the photoelectric effect

Dual Nature of Light Experiments can be devised that will display either the wave nature or the particle nature of light Experiments can be devised that will display either the wave nature or the particle nature of light Nature prevents testing both qualities at the same time Nature prevents testing both qualities at the same time

The Nature of Light “Particles” of light are called photons “Particles” of light are called photons Each photon has a particular energy Each photon has a particular energy E = h ƒ E = h ƒ h is Planck’s constant h is Planck’s constant h = 6.63 x J s h = 6.63 x J s Encompasses both natures of light Encompasses both natures of light Interacts like a particle Interacts like a particle Has a given frequency like a wave Has a given frequency like a wave

Geometric Optics – Using a Ray Approximation Light travels in a straight-line path in a homogeneous medium until it encounters a boundary between two different media Light travels in a straight-line path in a homogeneous medium until it encounters a boundary between two different media The ray approximation is used to represent beams of light The ray approximation is used to represent beams of light A ray of light is an imaginary line drawn along the direction of travel of the light beams A ray of light is an imaginary line drawn along the direction of travel of the light beams

Ray Approximation A wave front is a surface passing through points of a wave that have the same phase and amplitude A wave front is a surface passing through points of a wave that have the same phase and amplitude The rays, corresponding to the direction of the wave motion, are perpendicular to the wave fronts The rays, corresponding to the direction of the wave motion, are perpendicular to the wave fronts

Reflection of Light A ray of light, the incident ray, travels in a medium A ray of light, the incident ray, travels in a medium When it encounters a boundary with a second medium, part of the incident ray is reflected back into the first medium When it encounters a boundary with a second medium, part of the incident ray is reflected back into the first medium This means it is directed backward into the first medium This means it is directed backward into the first medium

Specular Reflection Specular reflection is reflection from a smooth surface Specular reflection is reflection from a smooth surface The reflected rays are parallel to each other The reflected rays are parallel to each other All reflection in this text is assumed to be specular All reflection in this text is assumed to be specular

Diffuse Reflection Diffuse reflection is reflection from a rough surface Diffuse reflection is reflection from a rough surface The reflected rays travel in a variety of directions The reflected rays travel in a variety of directions Diffuse reflection makes the road easy to see at night Diffuse reflection makes the road easy to see at night

QUICK QUIZ 22.1 Which part of the figure below shows specular reflection of light from the roadway?

QUICK QUIZ 22.1 ANSWER (a). In part (a), you can see clear reflections of the headlights and the lights on the top of the truck. The reflection is specular. In part (b), although bright areas appear on the roadway in front of the headlights, the reflection is not as clear and no separate reflection of the lights from the top of the truck is visible. The reflection in part (b) is mostly diffuse.

Law of Reflection The normal is a line perpendicular to the surface The normal is a line perpendicular to the surface It is at the point where the incident ray strikes the surface It is at the point where the incident ray strikes the surface The incident ray makes an angle of θ 1 with the normal The incident ray makes an angle of θ 1 with the normal The reflected ray makes an angle of θ 1 ’ with the normal The reflected ray makes an angle of θ 1 ’ with the normal

Law of Reflection, cont The angle of reflection is equal to the angle of incidence The angle of reflection is equal to the angle of incidence θ 1 = θ 1 ’ θ 1 = θ 1 ’

Refraction of Light When a ray of light traveling through a transparent medium encounters a boundary leading into another transparent medium, part of the ray is reflected and part of the ray enters the second medium When a ray of light traveling through a transparent medium encounters a boundary leading into another transparent medium, part of the ray is reflected and part of the ray enters the second medium The ray that enters the second medium is bent at the boundary The ray that enters the second medium is bent at the boundary This bending of the ray is called refraction This bending of the ray is called refraction

Refraction of Light, cont The incident ray, the reflected ray, the refracted ray, and the normal all lie on the same plane The incident ray, the reflected ray, the refracted ray, and the normal all lie on the same plane The angle of refraction, θ 2, depends on the properties of the medium The angle of refraction, θ 2, depends on the properties of the medium

Following the Reflected and Refracted Rays Ray  is the incident ray Ray  is the incident ray Ray  is the reflected ray Ray  is the reflected ray Ray is refracted into the lucite Ray is refracted into the lucite Ray  is internally reflected in the lucite Ray  is internally reflected in the lucite Ray is refracted as it enters the air from the lucite Ray is refracted as it enters the air from the lucite

More About Refraction The angle of refraction depends upon the material and the angle of incidence The angle of refraction depends upon the material and the angle of incidence The path of the light through the refracting surface is reversible The path of the light through the refracting surface is reversible

QUICK QUIZ 22.2 If beam 1 is the incoming beam in the figure below, which of the other four beams are reflected and which are refracted?

QUICK QUIZ 22.2 ANSWER Beams 2 and 4 are reflected; beams 3 and 5 are refracted.

Refraction Details, 1 Light may refract into a material where its speed is lower Light may refract into a material where its speed is lower The angle of refraction is less than the angle of incidence The angle of refraction is less than the angle of incidence The ray bends toward the normal The ray bends toward the normal

Refraction Details, 2 Light may refract into a material where its speed is higher Light may refract into a material where its speed is higher The angle of refraction is greater than the angle of incidence The angle of refraction is greater than the angle of incidence The ray bends away from the normal The ray bends away from the normal

The Index of Refraction When light passes from one medium to another, it is refracted because the speed of light is different in the two media When light passes from one medium to another, it is refracted because the speed of light is different in the two media The index of refraction, n, of a medium can be defined The index of refraction, n, of a medium can be defined

Index of Refraction, cont For a vacuum, n = 1 For a vacuum, n = 1 For other media, n > 1 For other media, n > 1 n is a unitless ratio n is a unitless ratio

Frequency Between Media As light travels from one medium to another, its frequency does not change As light travels from one medium to another, its frequency does not change Both the wave speed and the wavelength do change Both the wave speed and the wavelength do change The wavefronts do not pile up, nor are created or destroyed at the boundary, so ƒ must stay the same The wavefronts do not pile up, nor are created or destroyed at the boundary, so ƒ must stay the same

Index of Refraction Extended The frequency stays the same as the wave travels from one medium to the other The frequency stays the same as the wave travels from one medium to the other v = ƒ λ v = ƒ λ The ratio of the indices of refraction of the two media can be expressed as various ratios The ratio of the indices of refraction of the two media can be expressed as various ratios

Snell’s Law of Refraction n 1 sin θ 1 = n 2 sin θ 2 n 1 sin θ 1 = n 2 sin θ 2 θ 1 is the angle of incidence θ 1 is the angle of incidence 30.0° in this diagram 30.0° in this diagram θ 2 is the angle of refraction θ 2 is the angle of refraction

QUICK QUIZ 22.3 A material has an index of refraction that increases continuously from top to bottom. Of the three paths shown in the figure below, which path will a light ray follow as it passes through the material?

QUICK QUIZ 22.3 ANSWER (b). When light goes from one material into one having a higher index of refraction, it refracts toward the normal line of the boundary between the two materials. If, as the light travels through the new material, the index of refraction continues to increase, the light ray will refract more and more toward the normal line.

QUICK QUIZ 22.4 As light travels from vacuum (n = 1) to a medium such as glass (n > 1), which of the following properties remains the same: (a) wavelength, (b) wave speed, or (c) frequency?

QUICK QUIZ 22.4 ANSWER (c). Both the wave speed and the wavelength decrease as the index of refraction increases. The frequency is unchanged.

Dispersion The index of refraction in anything except a vacuum depends on the wavelength of the light The index of refraction in anything except a vacuum depends on the wavelength of the light This dependence of n on λ is called dispersion This dependence of n on λ is called dispersion Snell’s Law indicates that the angle of refraction when light enters a material depends on the wavelength of the light Snell’s Law indicates that the angle of refraction when light enters a material depends on the wavelength of the light

Variation of Index of Refraction with Wavelength The index of refraction for a material usually decreases with increasing wavelength The index of refraction for a material usually decreases with increasing wavelength Violet light refracts more than red light when passing from air into a material Violet light refracts more than red light when passing from air into a material

Refraction in a Prism The amount the ray is bent away from its original direction is called the angle of deviation, δ The amount the ray is bent away from its original direction is called the angle of deviation, δ Since all the colors have different angles of deviation, they will spread out into a spectrum Since all the colors have different angles of deviation, they will spread out into a spectrum Violet deviates the most Violet deviates the most Red deviates the least Red deviates the least

Prism Spectrometer A prism spectrometer uses a prism to cause the wavelengths to separate A prism spectrometer uses a prism to cause the wavelengths to separate The instrument is commonly used to study wavelengths emitted by a light source The instrument is commonly used to study wavelengths emitted by a light source

Using Spectra to Identify Gases All hot, low pressure gases emit their own characteristic spectra All hot, low pressure gases emit their own characteristic spectra The particular wavelengths emitted by a gas serve as “fingerprints” of that gas The particular wavelengths emitted by a gas serve as “fingerprints” of that gas Some uses of spectral analysis Some uses of spectral analysis Identification of molecules Identification of molecules Identification of elements in distant stars Identification of elements in distant stars Identification of minerals Identification of minerals

The Rainbow A ray of light strikes a drop of water in the atmosphere A ray of light strikes a drop of water in the atmosphere It undergoes both reflection and refraction It undergoes both reflection and refraction First refraction at the front of the drop First refraction at the front of the drop Violet light will deviate the most Violet light will deviate the most Red light will deviate the least Red light will deviate the least

The Rainbow, 2 At the back surface the light is reflected At the back surface the light is reflected It is refracted again as it returns to the front surface and moves into the air It is refracted again as it returns to the front surface and moves into the air The rays leave the drop at various angles The rays leave the drop at various angles The angle between the white light and the violet ray is 40° The angle between the white light and the violet ray is 40° The angle between the white light and the red ray is 42° The angle between the white light and the red ray is 42°

Observing the Rainbow If a raindrop high in the sky is observed, the red ray is seen A drop lower in the sky would direct violet light to the observer The other colors of the spectra lie in between the red and the violet

Huygen’s Principle Huygen assumed that light is a form of wave motion rather than a stream of particles Huygen assumed that light is a form of wave motion rather than a stream of particles Huygen’s Principle is a geometric construction for determining the position of a new wave at some point based on the knowledge of the wave front that preceded it Huygen’s Principle is a geometric construction for determining the position of a new wave at some point based on the knowledge of the wave front that preceded it

Huygen’s Principle, cont All points on a given wave front are taken as point sources for the production of spherical secondary waves, called wavelets, which propagate in the forward direction with speeds characteristic of waves in that medium All points on a given wave front are taken as point sources for the production of spherical secondary waves, called wavelets, which propagate in the forward direction with speeds characteristic of waves in that medium After some time has elapsed, the new position of the wave front is the surface tangent to the wavelets After some time has elapsed, the new position of the wave front is the surface tangent to the wavelets

Huygen’s Construction for a Plane Wave At t = 0, the wave front is indicated by the plane AA’ At t = 0, the wave front is indicated by the plane AA’ The points are representative sources for the wavelets The points are representative sources for the wavelets After the wavelets have moved a distance cΔt, a new plane BB’ can be drawn tangent to the wavefronts After the wavelets have moved a distance cΔt, a new plane BB’ can be drawn tangent to the wavefronts

Huygen’s Construction for a Spherical Wave The inner arc represents part of the spherical wave The inner arc represents part of the spherical wave The points are representative points where wavelets are propagated The points are representative points where wavelets are propagated The new wavefront is tangent at each point to the wavelet The new wavefront is tangent at each point to the wavelet

Huygen’s Principle and the Law of Reflection The Law of Reflection can be derived from Huygen’s Principle The Law of Reflection can be derived from Huygen’s Principle AA’ is a wave front of incident light AA’ is a wave front of incident light The reflected wave front is CD The reflected wave front is CD

Huygen’s Principle and the Law of Reflection, cont Triangle ADC is congruent to triangle AA’C Triangle ADC is congruent to triangle AA’C θ 1 = θ 1 ’ θ 1 = θ 1 ’ This is the Law of Reflection This is the Law of Reflection

Huygen’s Principle and the Law of Refraction In time Δt, ray 1 moves from A to B and ray 2 moves from A’ to C In time Δt, ray 1 moves from A to B and ray 2 moves from A’ to C From triangles AA’C and ACB, all the ratios in the Law of Refraction can be found From triangles AA’C and ACB, all the ratios in the Law of Refraction can be found n 1 sin θ 1 = n 2 sin θ 2 n 1 sin θ 1 = n 2 sin θ 2

Total Internal Reflection Total internal reflection can occur when light attempts to move from a medium with a high index of refraction to one with a lower index of refraction Total internal reflection can occur when light attempts to move from a medium with a high index of refraction to one with a lower index of refraction Ray 5 shows internal reflection Ray 5 shows internal reflection

Critical Angle A particular angle of incidence will result in an angle of refraction of 90° A particular angle of incidence will result in an angle of refraction of 90° This angle of incidence is called the critical angle This angle of incidence is called the critical angle

Critical Angle, cont For angles of incidence greater than the critical angle, the beam is entirely reflected at the boundary For angles of incidence greater than the critical angle, the beam is entirely reflected at the boundary This ray obeys the Law of Reflection at the boundary This ray obeys the Law of Reflection at the boundary Total internal reflection occurs only when light attempts to move from a medium of higher index of refraction to a medium of lower index of refraction Total internal reflection occurs only when light attempts to move from a medium of higher index of refraction to a medium of lower index of refraction

Fiber Optics An application of internal reflection An application of internal reflection Plastic or glass rods are used to “pipe” light from one place to another Plastic or glass rods are used to “pipe” light from one place to another Applications include Applications include medical use of fiber optic cables for diagnosis and correction of medical problems medical use of fiber optic cables for diagnosis and correction of medical problems Telecommunications Telecommunications

Chapter 23 Mirrors and Lenses

Notation for Mirrors and Lenses The object distance is the distance from the object to the mirror or lens The object distance is the distance from the object to the mirror or lens Denoted by p Denoted by p The image distance is the distance from the image to the mirror or lens The image distance is the distance from the image to the mirror or lens Denoted by q Denoted by q The lateral magnification of the mirror or lens is the ratio of the image height to the object height The lateral magnification of the mirror or lens is the ratio of the image height to the object height Denoted by M Denoted by M

Types of Images for Mirrors and Lenses A real image is one in which light actually passes through the image point A real image is one in which light actually passes through the image point Real images can be displayed on screens Real images can be displayed on screens A virtual image is one in which the light does not pass through the image point A virtual image is one in which the light does not pass through the image point The light appears to diverge from that point The light appears to diverge from that point Virtual images cannot be displayed on screens Virtual images cannot be displayed on screens

More About Images To find where an image is formed, it is always necessary to follow at least two rays of light as they reflect from the mirror To find where an image is formed, it is always necessary to follow at least two rays of light as they reflect from the mirror

Flat Mirror Simplest possible mirror Simplest possible mirror Properties of the image can be determined by geometry Properties of the image can be determined by geometry One ray starts at P, follows path PQ and reflects back on itself One ray starts at P, follows path PQ and reflects back on itself A second ray follows path PR and reflects according to the Law of Reflection A second ray follows path PR and reflects according to the Law of Reflection

Properties of the Image Formed by a Flat Mirror The image is as far behind the mirror as the object is in front The image is as far behind the mirror as the object is in front q = p q = p The image is unmagnified The image is unmagnified The image height is the same as the object height The image height is the same as the object height h’ = h and M = 1 h’ = h and M = 1 The image is virtual The image is virtual The image is upright The image is upright It has the same orientation as the object It has the same orientation as the object There is an apparent left-right reversal in the image There is an apparent left-right reversal in the image

QUICK QUIZ 23.1 In the overhead view of the figure below, the image of the stone seen by observer 1 is at C. Where does observer 2 see the image––at A, at B, at C, at E, or not at all?

QUICK QUIZ 23.1 ANSWER Observer 2 sees the image at C.

Application – Day and Night Settings on Auto Mirrors With the daytime setting, the bright beam of reflected light is directed into the driver’s eyes With the nighttime setting, the dim beam of reflected light is directed into the driver’s eyes, while the bright beam goes elsewhere

Spherical Mirrors A spherical mirror has the shape of a segment of a sphere A spherical mirror has the shape of a segment of a sphere A concave spherical mirror has the silvered surface of the mirror on the inner, or concave, side of the curve A concave spherical mirror has the silvered surface of the mirror on the inner, or concave, side of the curve A convex spherical mirror has the silvered surface of the mirror on the outer, or convex, side of the curve A convex spherical mirror has the silvered surface of the mirror on the outer, or convex, side of the curve

Concave Mirror, Notation The mirror has a radius of curvature of R The mirror has a radius of curvature of R Its center of curvature is the point C Its center of curvature is the point C Point V is the center of the spherical segment Point V is the center of the spherical segment A line drawn from C to V is called the principle axis of the mirror A line drawn from C to V is called the principle axis of the mirror

Image Formed by a Concave Mirror Geometry shows the relationship between the image and object distances Geometry shows the relationship between the image and object distances This is called the mirror equation This is called the mirror equation

Image Formed by a Concave Mirror Geometry can also be used to determine the magnification of the image Geometry can also be used to determine the magnification of the image h’ is negative when the image is inverted with respect to the object h’ is negative when the image is inverted with respect to the object

Spherical Aberration Rays are generally assumed to make small angles with the mirror Rays are generally assumed to make small angles with the mirror When the rays make large angles, they may converge to points other than the image point When the rays make large angles, they may converge to points other than the image point This results in a blurred image This results in a blurred image

Focal Length If an object is very far away, then p  and 1/p  0 If an object is very far away, then p  and 1/p  0 Incoming rays are essentially parallel Incoming rays are essentially parallel In this special case, the image point is called the focal point In this special case, the image point is called the focal point The distance from the mirror to the focal point is called the focal length The distance from the mirror to the focal point is called the focal length The focal length is ½ the radius of curvature The focal length is ½ the radius of curvature

Focal Point and Focal Length, cont The focal point is dependent solely on the curvature of the mirror, not by the location of the object The focal point is dependent solely on the curvature of the mirror, not by the location of the object f = R / 2 f = R / 2 The mirror equation can be expressed as The mirror equation can be expressed as

Focal Length Shown by Parallel Rays

Convex Mirrors A convex mirror is sometimes called a diverging mirror A convex mirror is sometimes called a diverging mirror The rays from any point on the object diverge after reflection as though they were coming from some point behind the mirror The rays from any point on the object diverge after reflection as though they were coming from some point behind the mirror The image is virtual because it lies behind the mirror at the point where the reflected rays appear to originate The image is virtual because it lies behind the mirror at the point where the reflected rays appear to originate In general, the image formed by a convex mirror is upright, virtual, and smaller than the object In general, the image formed by a convex mirror is upright, virtual, and smaller than the object

Image Formed by a Convex Mirror

Ray Diagrams A ray diagram can be used to determine the position and size of an image A ray diagram can be used to determine the position and size of an image They are graphical constructions which tell the overall nature of the image They are graphical constructions which tell the overall nature of the image They can also be used to check the parameters calculated from the mirror and magnification equations They can also be used to check the parameters calculated from the mirror and magnification equations

Drawing A Ray Diagram To make the ray diagram, you need to know To make the ray diagram, you need to know The position of the object The position of the object The position of the center of curvature The position of the center of curvature Three rays are drawn Three rays are drawn They all start from the same position on the object They all start from the same position on the object The intersection of any two of the rays at a point locates the image The intersection of any two of the rays at a point locates the image The third ray serves as a check of the construction The third ray serves as a check of the construction

The Rays in a Ray Diagram Ray 1 is drawn parallel to the principle axis and is reflected back through the focal point, F Ray 1 is drawn parallel to the principle axis and is reflected back through the focal point, F Ray 2 is drawn through the focal point and is reflected parallel to the principle axis Ray 2 is drawn through the focal point and is reflected parallel to the principle axis Ray 3 is drawn through the center of curvature and is reflected back on itself Ray 3 is drawn through the center of curvature and is reflected back on itself

Notes About the Rays The rays actually go in all directions from the object The rays actually go in all directions from the object The three rays were chosen for their ease of construction The three rays were chosen for their ease of construction The image point obtained by the ray diagram must agree with the value of q calculated from the mirror equation The image point obtained by the ray diagram must agree with the value of q calculated from the mirror equation

Ray Diagram for Concave Mirror, p > R The object is outside the center of curvature of the mirror The image is real The image is inverted The image is smaller than the object

Ray Diagram for a Concave Mirror, p < f The object is between the mirror and the focal point The image is virtual The image is upright The image is larger than the object

Ray Diagram for a Convex Mirror The object is in front of a convex mirror The image is virtual The image is upright The image is smaller than the object

Notes on Images With a concave mirror, the image may be either real or virtual With a concave mirror, the image may be either real or virtual When the object is outside the focal point, the image is real When the object is outside the focal point, the image is real When the object is at the focal point, the image is infinitely far away When the object is at the focal point, the image is infinitely far away When the object is between the mirror and the focal point, the image is virtual When the object is between the mirror and the focal point, the image is virtual With a convex mirror, the image is always virtual and upright With a convex mirror, the image is always virtual and upright As the object distance increases, the virtual image gets smaller As the object distance increases, the virtual image gets smaller

Sign Conventions for Mirrors Quantity Positive When Negative When Object location (p) Object is in front of the mirror Object is behind the mirror Image location (q) Image is behind mirror Image is in front of mirror Image height (h’) Image is upright Image is inverted Focal length (f) and radius (R) Mirror is concave Mirror is convex Magnification (M) Image is upright Image is inverted

Images Formed by Refraction Rays originate from the object point, O, and pass through the image point, I Rays originate from the object point, O, and pass through the image point, I When n 2 > n 1, When n 2 > n 1, Real images are formed on the side opposite from the object Real images are formed on the side opposite from the object

Sign Conventions for Refracting Surfaces Quantity Positive When Negative When Object location (p) Object is in front of surface Object is in back of surface Image location (q) Image is in back of surface Image is in front of surface Image height (h’) Image is upright Image is inverted Radius (R) Center of curvature is in back of surface Center of curvature is in front of surface

Flat Refracting Surface The image formed by a flat refracting surface is on the same side of the surface as the object The image formed by a flat refracting surface is on the same side of the surface as the object The image is virtual The image is virtual The image forms between the object and the surface The image forms between the object and the surface The rays bend away from the normal since n 1 > n 2 The rays bend away from the normal since n 1 > n 2

QUICK QUIZ 23.2 A person spear fishing from a boat sees a fish located 3 m from the boat at an apparent depth of 1 m. To spear the fish, should the person aim (a) at, (b) above, or (c) below the image of the fish?

QUICK QUIZ 23.2 ANSWER (c). Since n water > n air, the virtual image of the fish formed by refraction at the flat water surface is closer to the surface than is the fish. See Equation 23.9.

QUICK QUIZ 23.3 True or false? (a) The image of an object placed in front of a concave mirror is always upright. (b) The height of the image of an object placed in front of a concave mirror must be smaller than or equal to the height of the object. (c) The image of an object placed in front of a convex mirror is always upright and smaller than the object.

QUICK QUIZ 23.3 ANSWER (a) False. A concave mirror forms an inverted image when the object distance is greater than the focal length. (b) False. The magnitude of the magnification produced by a concave mirror is greater than 1 if the object distance is less than the radius of curvature. (c) True

Atmospheric Refraction There are many interesting results of refraction in the atmosphere There are many interesting results of refraction in the atmosphere Sunsets Sunsets Mirages Mirages

Atmospheric Refraction and Sunsets Light rays from the sun are bent as they pass into the atmosphere Light rays from the sun are bent as they pass into the atmosphere It is a gradual bend because the light passes through layers of the atmosphere It is a gradual bend because the light passes through layers of the atmosphere Each layer has a slightly different index of refraction Each layer has a slightly different index of refraction The Sun is seen to be above the horizon even after it has fallen below it The Sun is seen to be above the horizon even after it has fallen below it

Atmospheric Refraction and Mirages A mirage can be observed when the air above the ground is warmer than the air at higher elevations A mirage can be observed when the air above the ground is warmer than the air at higher elevations The rays in path B are directed toward the ground and then bent by refraction The rays in path B are directed toward the ground and then bent by refraction The observer sees both an upright and an inverted image The observer sees both an upright and an inverted image

Thin Lenses A thin 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 A thin 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 Lenses are commonly used to form images by refraction in optical instruments Lenses are commonly used to form images by refraction in optical instruments

Thin Lens Shapes These are examples of converging lenses These are examples of converging lenses They have positive focal lengths They have positive focal lengths They are thickest in the middle They are thickest in the middle

More Thin Lens Shapes These are examples of diverging lenses These are examples of diverging lenses They have negative focal lengths They have negative focal lengths They are thickest at the edges They are thickest at the edges

Focal Length of Lenses The focal length, ƒ, is the image distance that corresponds to an infinite object distance The focal length, ƒ, is the image distance that corresponds to an infinite object distance This is the same as for mirrors This is the same as for mirrors A thin lens has two focal points, corresponding to parallel rays from the left and from the right A thin lens has two focal points, corresponding to parallel rays from the left and from the right A thin lens is one in which the distance between the surface of the lens and the center of the lens is negligible A thin lens is one in which the distance between the surface of the lens and the center of the lens is negligible

Focal Length of a Converging Lens The parallel rays pass through the lens and converge at the focal point The parallel rays can come from the left or right of the lens

Focal Length of a Diverging Lens The parallel rays diverge after passing through the diverging lens The focal point is the point where the rays appear to have originated

Lens Equations The geometric derivation of the equations is very similar to that of mirrors The geometric derivation of the equations is very similar to that of mirrors

Lens Equations The equations can be used for both converging and diverging lenses The equations can be used for both converging and diverging lenses A converging lens has a positive focal length A converging lens has a positive focal length A diverging lens has a negative focal length A diverging lens has a negative focal length

Sign Conventions for Thin Lenses Quantity Positive When Negative When Object location (p) Object is in front of the lens Object is in back of the lens Image location (q) Image is in back of the lens Image is in front of the lens Image height (h’) Image is upright Image is inverted R 1 and R 2 Center of curvature is in back of the lens Center of curvature is in front of the lens Focal length (f) Converging lens Diverging lens

Focal Length for a Lens The focal length of a lens is related to the curvature of its front and back surfaces and the index of refraction of the material The focal length of a lens is related to the curvature of its front and back surfaces and the index of refraction of the material This is called the lens maker’s equation This is called the lens maker’s equation

Ray Diagrams for Thin Lenses Ray diagrams are essential for understanding the overall image formation Ray diagrams are essential for understanding the overall image formation Three rays are drawn Three rays are drawn The first ray is drawn parallel to the first principle axis and then passes through (or appears to come from) one of the focal lengths The first ray is drawn parallel to the first principle axis and then passes through (or appears to come from) one of the focal lengths The second ray is drawn through the center of the lens and continues in a straight line The second ray is drawn through the center of the lens and continues in a straight line The third ray is drawn from the other focal point and emerges from the lens parallel to the principle axis The third ray is drawn from the other focal point and emerges from the lens parallel to the principle axis There are an infinite number of rays, these are convenient There are an infinite number of rays, these are convenient

Ray Diagram for Converging Lens, p > f The image is real The image is inverted

Ray Diagram for Converging Lens, p < f The image is virtual The image is upright

Ray Diagram for Diverging Lens The image is virtual The image is upright

QUICK QUIZ 23.4 A plastic sandwich bag filled with water can act as a crude converging lens in air. If the bag is filled with air and placed under water, is the effective lens (a) converging or (b) diverging?

QUICK QUIZ 23.4 ANSWER (b). In this case, the index of refraction of the lens material is less than that of the surrounding medium. Under these conditions, a biconvex lens will be divergent.

QUICK QUIZ 23.5 In the figure below, the blue object arrow is replaced by one that is much taller than the lens. How many rays from the object will strike the lens?

QUICK QUIZ 23.5 ANSWER Although a ray diagram only uses 2 or 3 rays (those whose direction is easily determined using only a straight edge), an infinite number of rays leaving the object will always pass through the lens.

QUICK QUIZ 23.6 An object is placed to the left of a converging lens. Which of the following statements are true and which are false? (a) The image is always to the right of the lens. (b) The image can be upright or inverted. (c) The image is always smaller or the same size as the object. Justify your answers with ray diagrams.

QUICK QUIZ 23.6 ANSWER (a) False. A virtual image is formed on the left side of the lens if p f. (c) False. A magnified, real image is formed if 2f > p > f, and a magnified, virtual image is formed if p > f.

Problem Solving Strategy Be very careful about sign conventions Be very careful about sign conventions Do lots of problems for practice Do lots of problems for practice Draw confirming ray diagrams Draw confirming ray diagrams

Combinations of Thin Lenses The image produced by the first lens is calculated as though the second lens were not present The image produced by 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 of the first lens The light then approaches the second lens as if it had come from the image of the first lens The image of the first lens is treated as the object of the second lens The image of the first lens is treated as the object of the second lens The image formed by the second lens is the final image of the system The image formed by the second lens is the final image of the system

Combination of Thin Lenses, 2 If the image formed by the first lens lies on the back side of the second lens, then the image is treated at a virtual object for the second lens If the image formed by the first lens lies on the back side of the second lens, then the image is treated at a virtual object for the second lens p will be negative p will be negative The overall magnification is the product of the magnification of the separate lenses The overall magnification is the product of the magnification of the separate lenses

Combination of Thin Lenses, example

Lens and Mirror Aberrations One of the basic problems is the imperfect quality of the images One of the basic problems is the imperfect quality of the images Largely the result of defects in shape and form Largely the result of defects in shape and form Two common types of aberrations exist Two common types of aberrations exist Spherical aberration Spherical aberration Chromatic aberration Chromatic aberration

Spherical Aberration Results from the focal points of light rays far from the principle axis are different from the focal points of rays passing near the axis Results from the focal points of light rays far from the principle axis are different from the focal points of rays passing near the axis For a mirror, parabolic shapes can be used to correct for spherical aberration For a mirror, parabolic shapes can be used to correct for spherical aberration

Chromatic Aberration Different wavelengths of light refracted by by a lens focus at different points Different wavelengths of light refracted by by a lens focus at different points Violet rays are refracted more than red rays Violet rays are refracted more than red rays The focal length for red light is greater than the focal length for violet light The focal length for red light is greater than the focal length for violet light Chromatic aberration can be minimized by the use of a combination of converging and diverging lenses Chromatic aberration can be minimized by the use of a combination of converging and diverging lenses