March 26, 2005 Physics for Scientists&Engineers 2 1Physics for Scientists &Physics for Scientists &EngineersEngineers 22Spring Semester 2005Lecture 37March 26, 2005 Physics for Scientists&Engineers 2 2Reflection and Plane Mirrors (2)Reflection and Plane Mirrors (2)! The law of reflection is given by! the focal length, f, of a spherical mirror with radius R is!r=!if =R2March 26, 2005 Physics for Scientists&Engineers 2 3Review (2)Review (2)! 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! We can summarize the image characteristics of concave mirrors1do+1di=1fm = !dido= !hihoCase Type Direction Magnification do< f Virtual Upright Enlarged d = f Real Upright Image at infinity f < do< 2 f Real Inverted Enlarged do= 2 f Real Inverted Same size do> 2 f Real Inverted Reduced March 26, 2005 Physics for Scientists&Engineers 2 4Convex Spherical MirrorsConvex Spherical Mirrors! Suppose we have a spherical mirror where thereflecting surface is on the outside of the sphere! Thus we have a convex reflecting surface andthe reflected rays will diverge! The optical axis of the mirror is a line throughthe center of the sphere, represented in thisdrawing by a horizontal dashed line! Imagine that a horizontal light ray above theoptical axis is incident on the surface of the mirror! At the point the light ray strikes the mirror, the law of reflectionapplies, !i = !r! The normal to the surface is a radius line that points to the center ofthe sphere marked as CMarch 26, 2005 Physics for Scientists&Engineers 2 5! In contrast to the concave mirror, the normalpoints away from the center of the sphere! When we extrapolate the normal through thesurface of the sphere, itintersects the optical axisof the sphere at the centerof the sphere marked C! When we observe thereflected ray, it seems tobe coming from inside the sphereConvex Spherical Mirrors (2)Convex Spherical Mirrors (2)March 26, 2005 Physics for Scientists&Engineers 2 6Convex Spherical MirrorsConvex Spherical Mirrors! Now let us suppose that we havemany horizontal light rays incidenton this spherical mirror as shown! Each light ray obeys the law ofreflection at each point! You can see that the rays divergeand do not seem to form any kind ofimage! However, if we extrapolate thereflected rays through the surfaceof the mirror they all intersect theoptical axis at one point! This point called the focal point ofthis convex spherical mirrorMarch 26, 2005 Physics for Scientists&Engineers 2 7Images from Convex MirrorsImages from Convex Mirrors! Now let us discuss images formedby convex mirrors starting withthe case of the do > f! Again we use three rays• The first ray establishes that thetail of the arrow lies on the opticalaxis• The second ray starts from thetop of the object traveling parallelto the optical axis and is reflectedfrom the surface of the mirrorsuch that its extrapolation crossesthe optical axis a distance from the surface of the mirror equal to the focallength of the mirror• The third ray begins at the top of the object and is directed so that itsextrapolation would intersect the center of the sphere• This ray is reflected back on itselfMarch 26, 2005 Physics for Scientists&Engineers 2 8Images from Convex Mirrors (2)Images from Convex Mirrors (2)! We can see that we form an upright, reduced image on theside of the mirror opposite the object! This image is virtual because it cannot be projected! These characteristics are valid for all cases for a convexmirror! In the case of a convex mirror, we define the focal length fto be negative because the focal point of the mirror is onthe opposite side of the object! We always take the object distance do to be positive! We recall the mirror equation1do+1di=1fMarch 26, 2005 Physics for Scientists&Engineers 2 9Images from Convex Mirrors (3)Images from Convex Mirrors (3)! We can rearrange the mirror equation to get! If do is always positive and f is always negative, we can seethat di will always be negative! Applying the equation for the magnification we find that isalways positive! Looking at our diagram of the ray construction for theconvex mirror will also convince you that the image willalways be reduced in size! Thus, for a convex mirror, we always will obtain a virtual,upright, and reduced imagedi=dofdo! fMarch 26, 2005 Physics for Scientists&Engineers 2 10Compare Concave and ConvexCompare Concave and ConvexConcave Mirrordo > 2fConvex MirrorMarch 26, 2005 Physics for Scientists&Engineers 2 11Spherical AberrationSpherical Aberration! The equations we have derived for spherical mirrors apply only to lightrays that are close to the optical axis! If the light rays are far from the optical axis, they will not be focusedthrough the focal point of the mirror! Thus we will see a distorted image! In the drawing several light rays areincident on a spherical concave mirror! You can see that the rays far from theoptical axis are reflected such that theycross the optical axis closer to the mirrorthat rays that are incident closer to the optical axis! As the rays approach the optical axis, they are reflected throughpoints closer and closer to the focal pointMarch 26, 2005 Physics for Scientists&Engineers 2 12Parabolic MirrorParabolic Mirror! Parabolic mirrors have a surface that reflects light to the focalpoint from anywhere on the mirror! Thus the full size of the mirror can be used to collect light andform images! In the drawing to the right, horizontal lightrays are incident on a parabolic mirror! All rays are reflected through the focalpoint of the mirror! Parabolic mirrors are more difficult toproduce than spherical mirrors and areaccordingly more expensive! Most large telescopes use parabolic mirrorsMarch 26, 2005 Physics for Scientists&Engineers 2 13RefractionRefraction! Light travels at different speeds in different optically transparentmaterials! The ratio of the speed of light in a material divided by the speed oflight in vacuum is called the index of refraction of the material! The index of refraction, n, is given by! where c is the speed of light in avacuum and v is the speed of lightin the medium! Thus the index of refraction of amaterial is always greater than orequal to one, and by definition theindex of refraction of vacuum is onen =cvMaterial Index of Refraction Air 1.00029 Water 1.333 Ice 1.310 Ethyl alcohol 1.362 Quartz glass 1.459 Linseed
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