Reflection and Plane Mirrors 2 The law of reflection is given by r i Physics for Scientists Engineers 2 the focal length f of a spherical mirror with radius R is Spring Semester 2005 Lecture 37 f March 26 2005 Physics for Scientists Engineers 2 1 March 26 2005 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 2 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 1 1 1 d o di f The optical axis of the mirror is a line through the center of the sphere represented in this drawing by a horizontal dashed line The magnification m of the mirror is d h m i i do ho Imagine that a horizontal light ray above the optical axis is incident on the surface of the mirror We can summarize the image characteristics of concave mirrors March 26 2005 Physics for Scientists Engineers 2 Convex Spherical Mirrors Review 2 Case do f d f f do 2 f do 2 f do 2 f R 2 Type Virtual Real Real Real Real Direction Upright Upright Inverted Inverted Inverted At the point the light ray strikes the mirror the law of reflection applies i r Magnification Enlarged Image at infinity Enlarged Same size Reduced Physics for Scientists Engineers 2 The normal to the surface is a radius line that points to the center of the sphere marked as C 3 March 26 2005 Physics for Scientists Engineers 2 4 Convex Spherical Mirrors 2 Convex Spherical Mirrors In contrast to the concave mirror the normal points away from the center of the sphere Now let us suppose that we have many horizontal light rays incident on this spherical mirror as shown When we extrapolate the normal through the surface of the sphere it intersects the optical axis of the sphere at the center of the sphere marked C 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 When we observe the reflected ray it seems to be coming from inside the sphere March 26 2005 Physics for Scientists Engineers 2 This point called the focal point of this convex spherical mirror 5 Images from Convex Mirrors March 26 2005 Physics for Scientists Engineers 2 6 Images from Convex Mirrors 2 Now let us discuss images formed by convex mirrors starting with the case of the do f 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 Again we use three rays The first ray establishes that the tail of the arrow lies on the optical axis These characteristics are valid for all cases for a convex mirror 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 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 The third ray begins at the top of the object and is directed so that its extrapolation would intersect the center of the sphere Physics for Scientists Engineers 2 We recall the mirror equation 1 1 1 d o di f This ray is reflected back on itself March 26 2005 We always take the object distance do to be positive 7 March 26 2005 Physics for Scientists Engineers 2 8 Images from Convex Mirrors 3 Compare Concave and Convex We can rearrange the mirror equation to get d f di o do f If 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 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 Concave Mirror do 2f Thus for a convex mirror we always will obtain a virtual upright and reduced image March 26 2005 Physics for Scientists Engineers 2 9 March 26 2005 Spherical Aberration Convex Mirror Physics for Scientists Engineers 2 Parabolic Mirror The equations we have derived for spherical mirrors apply only to light rays that are close to the optical axis Parabolic mirrors have a surface that reflects light to the focal point from anywhere on the mirror If the light rays are far from the optical axis they will not be focused through the focal point of the mirror Thus the full size of the mirror can be used to collect light and form images Thus we will see a distorted image In the drawing to the right horizontal light rays are incident on a parabolic mirror In the drawing several light rays are incident on a spherical concave mirror All rays are reflected through the focal point of the 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 that rays that are incident closer to the optical axis Parabolic mirrors are more difficult to produce than spherical mirrors and are accordingly more expensive As the rays approach the optical axis they are reflected through points closer and closer to the focal point March 26 2005 Physics for Scientists Engineers 2 10 Most large telescopes use parabolic mirrors 11 March 26 2005 Physics for Scientists Engineers 2 12 Refraction Refraction 2 When light crosses the boundary between two transparent materials it is refracted Light travels at different speeds in different optically transparent materials The ratio of the speed of light in a material divided by the speed of light in vacuum is called the index of refraction of the material Refraction means that the light rays do not travel in a straight line across the boundary The index of refraction n is given by n c v where c is the speed of light in a vacuum and v is the speed of light in the medium Thus the index of refraction of a material is always greater than or equal to one and by definition the index of refraction of vacuum is one March 26 2005 Material Air Water Ice Ethyl alcohol Quartz glass Linseed oil Typical glass Typical oil Diamond Physics for Scientists Engineers 2 Index of Refraction 1 00029 1 333 1 310 1 362 1 459 1 486 1 5 1 5 2 417 When light crosses a boundary from a medium with a lower index of refraction n1 to a …
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