474 lines
15 KiB
HTML
474 lines
15 KiB
HTML
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<!DOCTYPE HTML PUBLIC "-//w3c//dtd html 4.0 transitional//en">
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<html>
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<head>
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<meta http-equiv="Content-Type"
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content="text/html; charset=ISO-8859-1">
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<meta name="GENERATOR"
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content="Mozilla/4.7 [en] (X11; U; OSF1 V4.0 alpha) [Netscape]">
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<meta name="Author" content="C. L. Davis">
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<title>Light and Optics - Spherical Mirrors - Physics 299</title>
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</head>
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<body style="color: rgb(0, 0, 0); background-color: rgb(255, 255, 255);"
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alink="#ff0000" link="#0000ee" vlink="#551a8b">
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<center>
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<h1><img src="ULPhys1.gif" align="texttop" height="50" width="189"></h1>
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</center>
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<center>
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<h1>Spherical Mirrors</h1>
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</center>
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<center><img src="celticbar.gif" height="22" width="576">
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<br>
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<br>
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<font color="#ff0000"><i>"A modern compter hovers between the
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obsolescent and the non existent"</i></font><br>
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Sydney Brenner<br>
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</center>
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<img src="netbar.gif" align="middle" height="40" width="100%">
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<br>
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<ul>
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<li>There are two kinds of spherical mirrors, <i><b>concave</b></i>
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and <b><i>convex</i></b>.<img src="lo_sm_cc.gif" alt="Concave convex"
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align="top" height="130" width="202"> </li>
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<li><img src="lo_sm_parallel.gif" alt="Concave parallel" align="left"
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height="99" width="128"> The focal point (F) of a concave mirror is
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the point at which a parallel beam of light is "focussed" after
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reflection in the mirror. For a convex mirror the focal point is
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the point from which light appears to have originated after reflection
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from the mirror. The centre of curvature (C) is the centre of the
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circle (sphere) of which the mirror is an arc.<br clear="all">
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<br>
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</li>
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<li><img src="lo_sm_defs.gif" alt="Concave definitions" align="right"
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height="209" width="274"> The focal length (f) and radius of curvature
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(R) are defined in the diagram at the right. It can be shown that R =
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2f. "A" in the diagram is known as the <i>"vertex"</i> (often
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labeled V).<br clear="all">
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<br>
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</li>
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<br>
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<li><u><b>Image formation</b></u> in spherical mirrors is defined by
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certain
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"characteristic" rays whose behaviour is governed by the law
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of reflection.</li>
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<ul>
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<li>Rays parallel to the principal axis are reflected through the
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focal point - <b>concave</b> (or as if they came from the focal point
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- <b> convex</b> ).</li>
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<li>Rays passing through the focal point are reflected parallel to
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the principal axis - <b>concave</b> (for <b>convex</b> mirrors a ray
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that would have passed through the focal point is reflected parallel to
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the axis).</li>
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<li>Rays passing through the centre of curvature are reflected back
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along their original path -<b>concave</b> (or a ray which would have
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passed through the centre of curvature is reflected back along itself -
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<b> convex</b> ) </li>
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</ul>
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<br>
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<img src="lo_sm_anim.gif" alt="Concave animation" align="right"
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height="222" width="322">
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<li><u><b>Concave mirror.</b></u> In
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the animation the first two rays from the object are examples of the
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first two characteristic rays described above. Only two rays are
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needed to define the position of the image. The paths of the
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other rays in the animation are
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defined since the image position is already known.<br>
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The general characteristics of the image depend on the location of the
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object with respect to the centre of
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curvature and the focal point. Animations of image formation
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in a concave mirror for the five possible object positions can be
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observed by choosing from the following options.<br>
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<ul>
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<li><a href="lo_sm_anim2.gif">Object beyond C</a> </li>
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<li><a href="lo_sm_anim3.gif">Object at C</a> </li>
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<li><a href="lo_sm_anim4.gif">Object between C and F</a> </li>
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<li><a href="lo_sm_objatF.gif">Object at F</a> </li>
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<li><a href="lo_sm_anim5.gif">Object between F and the vertex<br>
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</a> </li>
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<br>
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</ul>
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</li>
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<li><u><b> Convex mirror.</b></u> The general characteristics
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of images in convex mirrors are independent of the location of the
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object. Three examples are shown below.
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<div align="center"><img src="lo_sm_convex.gif" alt="Convex images"
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align="middle" height="112" width="361"> </div>
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</li>
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<br>
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<li>Note that for the convex mirror the reflected rays <i><b>DIVERGE</b></i>
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(this is also the case for the concave mirror when the object is closer
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than the focal point to the mirror). In these cases the image
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formed is <b>virtual</b> - light rays do not pass though it, but to an
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observer "<i> appear</i>" to come from it.</li>
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</ul>
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<ul>
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<li>Perhaps the most common everyday experience of a convex mirror is
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the passenger side rear-view mirror in your car. As can be seen
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from the above ray diagrams, the image you see in a convex mirror
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is always smaller than the object. You "know" the typical size of
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a car or truck, so much so that the "size" it appears tells you how far
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away it is (if you see a tiny car in the mirror you "know" it is far
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away). Thus, seeing a vehicle "looking smaller" due to the convex
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mirror, will make you think the vehicle is further away from the mirror
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than it actually is, hence the warning, "<a
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href="http://www.physlink.com/Education/AskExperts/ae449.cfm"><span
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style="font-style: italic; font-weight: bold;">objects mirror are
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closer than they appear</span></a>".<br>
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</li>
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<br>
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<li><u><b>Mirror equation</b></u>. For objects placed close to
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the
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principle axis the distance of the object from the vertex of the mirror
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(p), the distance of the image from the vertex of the mirror (q) and
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the focal length (f) are related by the following equation,<br>
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<br>
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<div align="center"><b><img alt="" src="lo_spmirror_eqn1.gif"
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style="width: 68px; height: 44px;"> <br>
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</b>
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<div align="left"><br>
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</div>
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</div>
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</li>
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<li><u><b>Magnification</b></u>. The magnification (m) is
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defined by,<br>
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<br>
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<div align="center"><b><img alt="" src="lo_spmirror_eqn2.gif"
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style="width: 244px; height: 44px;"> </b><b><br>
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</b></div>
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</li>
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<br>
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<li><u><b>Sign Conventions.</b></u> In order to make use of the
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above formulae the following sign conventions must be followed. <br>
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<br>
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<center>
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<table bgcolor="#99ffff" border="1" cellpadding="2" cellspacing="2"
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width="50%">
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<tbody>
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<tr>
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<td valign="top">
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<div align="center"><b>SIGN</b><br>
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</div>
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</td>
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<td valign="top">
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<div align="center"><b>+</b><br>
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</div>
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</td>
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<td valign="top">
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<div align="center"><b>-</b><br>
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</div>
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</td>
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</tr>
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<tr>
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<td valign="top">
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<div align="center">f - focal length<br>
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</div>
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</td>
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<td valign="top">
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<div align="center">Concave<br>
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</div>
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</td>
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<td valign="top">
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<div align="center">Convex<br>
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</div>
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</td>
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</tr>
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<tr>
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<td valign="top">
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<div align="center">p - object distance<br>
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</div>
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</td>
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<td valign="top">
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<div align="center">Real <br>
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</div>
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</td>
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<td valign="top">
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<div align="center">Virtual<br>
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</div>
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</td>
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</tr>
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<tr>
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<td valign="top">
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<div align="center">q - image distance<br>
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</div>
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</td>
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<td valign="top">
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<div align="center">Real <br>
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</div>
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</td>
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<td valign="top">
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<div align="center">Virtual<br>
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</div>
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</td>
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</tr>
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<tr>
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<td valign="top">
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<div align="center">m - magnification<br>
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</div>
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</td>
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<td valign="top">
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<div align="center">Upright image<br>
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</div>
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</td>
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<td valign="top">
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<div align="center">Inverted image<br>
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</div>
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</td>
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</tr>
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</tbody>
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</table>
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</center>
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</li>
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<br>
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<li><u><b>Image properties.</b></u> There are four basic properties,
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dependent on the position of the object, as indicated in the table
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below. These properties can be verified either graphically or by
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using the mirror
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equation and the definition of magnification. <br>
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<div align="center"><br>
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</div>
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<center>
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<table bgcolor="#ffcccc" border="1" cellpadding="2" cellspacing="2"
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width="80%">
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<tbody>
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<tr>
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<td bgcolor="#ffff99" valign="top">
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<div align="center"><b>Mirror</b><br>
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</div>
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</td>
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<td bgcolor="#ffff99" valign="top">
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<div align="center"><b>Object location</b><br>
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</div>
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</td>
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<td bgcolor="#ffff99" valign="top">
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<div align="center"><b>Image location</b><br>
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</div>
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</td>
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<td colspan="1" bgcolor="#ffff99" valign="top">
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<div align="center"><b>Type</b><br>
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</div>
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</td>
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<td colspan="1" bgcolor="#ffff99" valign="top">
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<div align="center"><b>Orientation</b><br>
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</div>
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</td>
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<td bgcolor="#ffff99" valign="top">
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<div align="center"><b>Relative size</b><br>
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</div>
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</td>
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</tr>
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<tr>
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<td bgcolor="#ffff99" valign="top">
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<div align="center">CONCAVE<br>
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</div>
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</td>
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<td valign="top">
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<div align="center">At infinity<br>
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</div>
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</td>
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<td valign="top">
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<div align="center">At F<br>
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</div>
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</td>
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<td valign="top">
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<div align="center">Real<br>
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</div>
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</td>
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<td valign="top">
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<div align="center">Inverted<br>
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</div>
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</td>
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<td valign="top">
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<div align="center">Smaller<br>
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</div>
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</td>
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</tr>
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<tr>
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<td bgcolor="#ffff99" valign="top">
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<div align="center">CONCAVE<br>
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</div>
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</td>
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<td valign="top">
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<div align="center">Beyond C<br>
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</div>
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</td>
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<td valign="top">
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<div align="center">Between F and C<br>
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</div>
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</td>
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<td valign="top">
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<div align="center">Real<br>
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</div>
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</td>
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<td valign="top">
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<div align="center">Inverted<br>
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</div>
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</td>
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<td valign="top">
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<div align="center">Smaller<br>
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</div>
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</td>
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</tr>
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<tr>
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<td bgcolor="#ffff99" valign="top">
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<div align="center">CONCAVE<br>
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</div>
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</td>
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<td valign="top">
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<div align="center">At C<br>
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</div>
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</td>
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<td valign="top">
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<div align="center">At C<br>
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</div>
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</td>
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<td valign="top">
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<div align="center">Real<br>
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</div>
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</td>
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<td valign="top">
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<div align="center">Inverted<br>
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</div>
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</td>
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<td valign="top">
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<div align="center">Same size<br>
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</div>
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</td>
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</tr>
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<tr>
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<td bgcolor="#ffff99" valign="top">
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<div align="center">CONCAVE<br>
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</div>
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</td>
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<td valign="top">
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<div align="center">Between C and F<br>
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</div>
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</td>
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<td valign="top">
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<div align="center">Beyond C<br>
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</div>
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</td>
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<td valign="top">
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<div align="center">Real<br>
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</div>
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</td>
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<td valign="top">
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<div align="center">Inverted<br>
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</div>
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</td>
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<td valign="top">
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<div align="center">Larger<br>
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</div>
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</td>
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</tr>
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<tr>
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<td bgcolor="#ffff99" valign="top">
|
||
|
|
<div align="center">CONCAVE<br>
|
||
|
|
</div>
|
||
|
|
</td>
|
||
|
|
<td valign="top">
|
||
|
|
<div align="center">At F<br>
|
||
|
|
</div>
|
||
|
|
</td>
|
||
|
|
<td valign="top">
|
||
|
|
<div align="center">At infinity<br>
|
||
|
|
</div>
|
||
|
|
</td>
|
||
|
|
<td valign="top">
|
||
|
|
<div align="center">No image<br>
|
||
|
|
</div>
|
||
|
|
</td>
|
||
|
|
<td valign="top">
|
||
|
|
<div align="center">No image<br>
|
||
|
|
</div>
|
||
|
|
</td>
|
||
|
|
<td valign="top">
|
||
|
|
<div align="center">No image<br>
|
||
|
|
</div>
|
||
|
|
</td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td bgcolor="#ffff99" valign="top">
|
||
|
|
<div align="center">CONCAVE<br>
|
||
|
|
</div>
|
||
|
|
</td>
|
||
|
|
<td valign="top">
|
||
|
|
<div align="center">Closer than F<br>
|
||
|
|
</div>
|
||
|
|
</td>
|
||
|
|
<td valign="top">
|
||
|
|
<div align="center">Behind the mirror<br>
|
||
|
|
</div>
|
||
|
|
</td>
|
||
|
|
<td valign="top">
|
||
|
|
<div align="center">Virtual<br>
|
||
|
|
</div>
|
||
|
|
</td>
|
||
|
|
<td valign="top">
|
||
|
|
<div align="center">Upright<br>
|
||
|
|
</div>
|
||
|
|
</td>
|
||
|
|
<td valign="top">
|
||
|
|
<div align="center">Larger<br>
|
||
|
|
</div>
|
||
|
|
</td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td colspan="6" valign="top">
|
||
|
|
<div align="center"><br>
|
||
|
|
</div>
|
||
|
|
</td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td bgcolor="#ffff99" valign="top">
|
||
|
|
<div align="center">CONVEX<br>
|
||
|
|
</div>
|
||
|
|
</td>
|
||
|
|
<td valign="top">
|
||
|
|
<div align="center">Anywhere<br>
|
||
|
|
</div>
|
||
|
|
</td>
|
||
|
|
<td valign="top">
|
||
|
|
<div align="center">Behind the mirror<br>
|
||
|
|
</div>
|
||
|
|
</td>
|
||
|
|
<td valign="top">
|
||
|
|
<div align="center">Virtual<br>
|
||
|
|
</div>
|
||
|
|
</td>
|
||
|
|
<td valign="top">
|
||
|
|
<div align="center">Upright<br>
|
||
|
|
</div>
|
||
|
|
</td>
|
||
|
|
<td valign="top">
|
||
|
|
<div align="center">Smaller<br>
|
||
|
|
</div>
|
||
|
|
</td>
|
||
|
|
</tr>
|
||
|
|
</tbody>
|
||
|
|
</table>
|
||
|
|
</center>
|
||
|
|
</li>
|
||
|
|
</ul>
|
||
|
|
<img src="netbar.gif" height="40" width="100%">
|
||
|
|
<center><span style="font-style: italic; color: rgb(255, 0, 0);">What's
|
||
|
|
a light-year? </span><br
|
||
|
|
style="font-style: italic; color: rgb(255, 0, 0);">
|
||
|
|
<span style="font-style: italic; color: rgb(255, 0, 0);">One-third less
|
||
|
|
calories than a regular year.</span> <br>
|
||
|
|
(Very Punny)<br>
|
||
|
|
<br>
|
||
|
|
<img src="celticbar.gif" height="22" width="576"><br>
|
||
|
|
|
||
|
|
<p><i>Dr. C. L. Davis</i><br>
|
||
|
|
<i>Physics Department</i><br>
|
||
|
|
<i>University of Louisville</i><br>
|
||
|
|
<i>email</i>: <a href="mailto:c.l.davis@louisville.edu">c.l.davis@louisville.edu</a>
|
||
|
|
<br>
|
||
|
|
</p>
|
||
|
|
<p><img src="header-index.gif" height="51" width="92">
|
||
|
|
</p>
|
||
|
|
</center>
|
||
|
|
<p><br>
|
||
|
|
</p>
|
||
|
|
</body>
|
||
|
|
</html>
|