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Transcript
Lecture 24 1 Friday, November 13, 15 Reminder: Snell’s law 2 Friday, November 13, 15 Mirrors 3 Friday, November 13, 15 Curved Mirrors (1) § When light is reflected from the surface of a curved mirror, the light rays follow the law of reflection at each point on the surface § However, unlike the plane mirror, the surface of a curved mirror is not flat § Thus light rays that are parallel before they strike the mirror are reflected in different directions depending on the part of the mirror that they strike § The light rays can be focused or made to diverge Soar Telescope 11 Friday, November 13, 15 Concave Spherical Mirrors (1) § We can abstract this sphere to a two-dimensional semicircle § The optical axis of the mirror is a line through the center of the sphere, represented in this drawing by a horizontal dashed line § A horizontal light ray above the optical axis is incident on the surface of the mirror § At the point the light ray strikes the mirror, the law of reflection applies • θi = θr § The normal to the surface is a radius line that points to the center of the sphere marked as C 13 Friday, November 13, 15 Concave Spherical Mirror (2) § Now let us suppose that we have many horizontal light rays incident on this spherical mirror as shown § Each light ray obeys the law of reflection at each point § All the reflected rays cross the optical axis at the same point called the focal point F § Point F is half way between point C and the surface of the mirror 14 Friday, November 13, 15 Concave Spherical Mirror (3) § C is located at the center of the sphere so the distance of C from the surface of the mirror is just the radius of the mirror, R § Thus the focal length, f, of a spherical mirror is 15 Friday, November 13, 15 Forming an Image with a Concave Mirror Start with convex mirror and optical axis 18 Friday, November 13, 15 Forming an Image with a Concave Mirror Start with convex mirror and optical axis Add object to be imaged 18 Friday, November 13, 15 Forming an Image with a Concave Mirror Start with convex mirror and optical axis Add object to be imaged Draw focal length and radius of mirror 18 Friday, November 13, 15 Forming an Image with a Concave Mirror Start with convex mirror and optical axis Add object to be imaged Draw focal length and radius of mirror Object has height ho and distance do 18 Friday, November 13, 15 Forming an Image with a Concave Mirror Start with convex mirror and optical axis Add object to be imaged Draw focal length and radius of mirror Object has height ho and distance do Draw first ray through center of circle, reflecting on itself 18 Friday, November 13, 15 Forming an Image with a Concave Mirror Start with convex mirror and optical axis Draw second ray parallel to the optical axis Add object to be imaged Draw focal length and radius of mirror Object has height ho and distance do Draw first ray through center of circle, reflecting on itself 18 Friday, November 13, 15 Forming an Image with a Concave Mirror Start with convex mirror and optical axis Add object to be imaged Draw focal length and radius of mirror Draw second ray parallel to the optical axis Reflect the ray through the focal point Object has height ho and distance do Draw first ray through center of circle, reflecting on itself 18 Friday, November 13, 15 Forming an Image with a Concave Mirror Start with convex mirror and optical axis Add object to be imaged Draw focal length and radius of mirror Object has height ho and distance do Draw second ray parallel to the optical axis Reflect the ray through the focal point Image is formed at the intersection of the two rays Draw first ray through center of circle, reflecting on itself 18 Friday, November 13, 15 Forming an Image with a Concave Mirror Start with convex mirror and optical axis Add object to be imaged Draw focal length and radius of mirror Object has height ho and distance do Draw first ray through center of circle, reflecting on itself Draw second ray parallel to the optical axis Reflect the ray through the focal point Image is formed at the intersection of the two rays Image has height hi 18 Friday, November 13, 15 Forming an Image with a Concave Mirror Start with convex mirror and optical axis Add object to be imaged Draw focal length and radius of mirror Object has height ho and distance do Draw first ray through center of circle, reflecting on itself Draw second ray parallel to the optical axis Reflect the ray through the focal point Image is formed at the intersection of the two rays Image has height hi Image has distance di 18 Friday, November 13, 15 Image with Concave Mirror § The reconstruction of this special case produces a real image • Defined by the fact that the image is on the same side of the mirror as the object • Not behind the mirror § The image has a height hi located a distance di from the surface of the mirror on the same side of the mirror as the object with height ho and distance do from the mirror § The image is inverted and reduced in size relative to the object that produced the image § The image is called real when you are able to place a screen at the image location and obtain a sharp projection of the image on the screen at that point § For a virtual image, it is impossible to place a screen at the location of the image 19 Friday, November 13, 15 Concave Mirror, do < f (2) § To determine the location of the image, we must extrapolate the reflected rays to the other side of the mirror § These two rays intersect a distance di from the surface of the mirror producing an image with height hi C F 21 Friday, November 13, 15 Conventions for Mirror Distances (1) § In this case, we have the image formed on the opposite side of the mirror from the object, a virtual image § To an observer, the image appears to be behind the mirror § The image is upright and larger than the object. These results are very different than those for do > f § To treat all possible cases for convex mirrors, we must define some conventions for distances and heights § We define • Distances on the same side of the mirror as the object to be positive • Distances on the far side of the mirror from the object to be negative 22 Friday, November 13, 15 Conventions for Mirror Distances (2) § Thus f and do are positive for concave mirrors § In the case of real images, di is positive § In cases where the image is virtual, we define di to be negative § If the image is upright, then hi is positive and if the image is inverted, hi is negative 23 Friday, November 13, 15 The mirror Equation § Using these sign conventions we can express the mirror equation in terms of the object distance, do, and the image distance, di, and the focal length f of the mirror § The magnification m of the mirror is defined to be 24 Friday, November 13, 15 Possible Cases for the Mirror Equation We can summarize the image characteristics of concave mirrors 25 Friday, November 13, 15 Mirror Equation and Concave Mirrors d>2f (real, inverted, reduced) d<f (virtual, upright enlarged) d<f (real, upright, image at infinity) f<d<2f (real, inverted, enlarged) 26 Friday, November 13, 15 Spherical Aberration § The equations we have derived for spherical mirrors apply only to light rays that are close to the optical axis § If the light rays are far from the optical axis, they will not be focused through the focal point of the mirror § Thus we will see a distorted image § In the drawing several light rays are incident on a spherical concave mirror § You can see that the rays far from the optical axis are reflected such that they cross the optical axis closer to the mirror than the focal point § As the rays approach the optical axis, they are reflected through points closer and closer to the focal point 34 Friday, November 13, 15 Parabolic Mirror § Parabolic mirrors have a surface that reflects light to the focal point from anywhere on the mirror § Thus the full size of the mirror can be used to collect light and form images § In the drawing to the right, horizontal light rays are incident on a parabolic mirror § All rays are reflected through the focal point of the mirror § Parabolic mirrors are more difficult to produce than spherical mirrors and are accordingly more expensive § Most large telescopes use parabolic mirrors 35 Friday, November 13, 15 Convex Spherical Mirrors (1) § Suppose we have a spherical mirror where the reflecting surface is on the outside of the sphere § Thus we have a convex reflecting surface and the reflected rays will diverge § The optical axis of the mirror is a line through the center of the sphere, represented in this drawing by a horizontal dashed line § Imagine that a horizontal light ray above the optical axis is incident on the surface of the mirror § At the point the light ray strikes the mirror, the law of reflection applies, θi = θr § The normal to the surface is a radius line that points to the center of the sphere marked as C § Example: car right-hand rearview mirror 28 Friday, November 13, 15 Convex Spherical Mirrors (2) § In contrast to the concave mirror, the normal points away from the center of the sphere § When we extrapolate the normal through the surface of the sphere, it intersects the optical axis of the sphere at the center, marked C § When we observe the reflected ray, it seems to be coming from inside the sphere R f 29 Friday, November 13, 15 Convex Spherical Mirrors (3) § Now let us suppose that we have many horizontal light rays incident on this spherical mirror as shown § Each light ray obeys the law of reflection at each point § You can see that the rays diverge and do not seem to form any kind of image § However, if we extrapolate the reflected rays through the surface of the mirror they all intersect the optical axis at one point § This point is the focal point of this convex spherical mirror 30 Friday, November 13, 15 Images from Convex Mirrors (1) § Now let us discuss images formed by convex mirrors § Again we use three rays • The first ray establishes that the tail of the arrow lies on the optical axis • The second ray starts from the top of the object traveling parallel to the optical axis and is reflected from the surface of the mirror such that its extrapolation crosses the optical axis a distance from the surface of the mirror equal to the focal length of the mirror • The third ray begins at the top of the object and is directed so that its extrapolation would intersect the center of the sphere • This ray is reflected back on itself 31 Friday, November 13, 15 Images from Convex Mirrors (2) § We can see that we form an upright, reduced image on the side of the mirror opposite the object § This image is virtual because it cannot be projected § These characteristics are valid for all cases of a convex mirror § In the case of a convex mirror, we define the focal length f to be negative because the focal point of the mirror is on the opposite side of the object § We always take the object distance do to be positive § We recall the mirror equation 32 Friday, November 13, 15 Images from Convex Mirrors (3) § We can rearrange the mirror equation to get § do is always positive and f is always negative, we can see that di will always be negative § Applying the equation for the magnification we find that it is always positive § Looking at our diagram of the ray construction for the convex mirror will also convince you that the image will always be reduced in size § Thus, for a convex mirror, we always will obtain a virtual, upright, and reduced image 33 Friday, November 13, 15 Lenses 24 Friday, November 13, 15 Lenses § When light is refracted crossing a curved boundary between two different media, the light rays follow the law of refraction at each point on the boundary § The angle at which the light rays cross the boundary is different along the boundary, so the refracted angle is different at different points along the boundary § A curved boundary between two optically transparent media is called a lens § Light rays that are initially parallel before they strike the boundary are refracted in different directions depending on the part of the lens they strike § Depending of the shape of the lens, the light rays can be focused or caused to diverge 1 Friday, November 13, 15 Focal lengths converging lens f diverging lens f 26 Friday, November 13, 15 Lens Focal Lengths § Unlike mirrors, lenses have a focal length on both sides § Light can pass through lenses while mirrors reflect the light allowing no light on the opposite side Convex (converging) lens § The focal length of a convex (converging) lens is defined to be positive § The focal length of a concave (diverging) lens is defined to be negative positive negative Concave (diverging) lens 4 Friday, November 13, 15 Focal lengths converging lens f>0 diverging lens f<0 28 Friday, November 13, 15 Convex Lenses (1) § A convex lens is shaped such that parallel rays will be focused by refraction at the focal distance f from the center of the lens § In the drawing on the right, a light ray is incident on a convex glass lens § At the surface of the lens, the light ray is refracted toward the normal § When the ray leaves the lens, it is refracted away from the normal 5 Friday, November 13, 15 Thin Lenses (1) § The rays enter the lens, get refracted at the surface, traverse the lens in a straight line, and get refracted when they exit the lens § In the third frame, we represent the thin lens approximation by drawing a black dotted line at the center of the lens § Instead of following the detailed trajectory of the light rays inside the lens, we draw the incident rays to the centerline and then on to the focal point 7 Friday, November 13, 15 Images with Convex Lenses (1) § Convex lenses can be used to form images § We show the geometric construction of the formation of an image using a convex lens with focal length f § We place an object standing on the optical axis represented by the green arrow § This object has a height ho and is located a distance do from the center of the lens such that do > f 9 Friday, November 13, 15 Images with Convex Lenses (2) § We start with a ray along the optical axis of the lens that passes straight through the lens that defines the bottom of the image. § A second ray is then drawn from the top of the object parallel to the optical axis • This ray is focused through the focal point on the other side of the lens § A third ray is drawn through the center of the lens that is not refracted in the thin lens approximation § A fourth ray is drawn from the top of the object through the focal point on the same side of the lens that is then directed parallel to the optical axis 10 Friday, November 13, 15 Images with Convex Lenses (4) § Now let us consider the image formed by an object with height ho placed a distance do from the center of the lens such that do < f § The first ray again is drawn from the bottom of the object along the optical axis § § § The second ray is drawn from the top of the object parallel to the optical axis and is focused through the focal point on the opposite side A third ray is drawn through the center of the lens A fourth line is drawn such that it originated from the focal point on the same side of the lens and is then focused parallel to the optical axis • These three rays are diverging § A virtual image is located on the same side of the lens as the object 12 Friday, November 13, 15 do f f 34 Friday, November 13, 15 do f f 34 Friday, November 13, 15 do f f 34 Friday, November 13, 15 do f f 34 Friday, November 13, 15 do f f 34 Friday, November 13, 15 do f f 34 Friday, November 13, 15 do f f 34 Friday, November 13, 15 do f f 34 Friday, November 13, 15 di do f f 34 Friday, November 13, 15 di do f f § A virtual image is located on the same side of the lens as the object 34 Friday, November 13, 15 35 Friday, November 13, 15