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<title>Light and Optics - Spherical Mirrors - Physics 299</title>
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<center>
<h1><img src="ULPhys1.gif" align="texttop" height="50" width="189"></h1>
</center>
<center>
<h1>Spherical Mirrors</h1>
</center>
<center><img src="celticbar.gif" height="22" width="576">
<br>
<br>
<font color="#ff0000"><i>"A modern compter hovers between the
obsolescent and the non existent"</i></font><br>
Sydney Brenner<br>
</center>
<img src="netbar.gif" align="middle" height="40" width="100%">
<br>
<ul>
<li>There are two kinds of spherical mirrors, <i><b>concave</b></i>
and <b><i>convex</i></b>.<img src="lo_sm_cc.gif" alt="Concave convex"
align="top" height="130" width="202"> </li>
<li><img src="lo_sm_parallel.gif" alt="Concave parallel" align="left"
height="99" width="128"> The focal point (F) of a concave mirror is
the point at which a parallel beam of light is "focussed" after
reflection in the mirror. &nbsp;For a convex mirror the focal point is
the point from which light appears to have originated after reflection
from the mirror. &nbsp;The centre of curvature (C) is the centre of the
circle (sphere) of which the mirror is an arc.<br clear="all">
<br>
</li>
<li><img src="lo_sm_defs.gif" alt="Concave definitions" align="right"
height="209" width="274"> The focal length (f) and radius of curvature
(R) are defined in the diagram at the right. It can be shown that R =
2f. &nbsp;"A" in the diagram is known as the <i>"vertex"</i> (often
labeled V).<br clear="all">
<br>
</li>
<br>
<li><u><b>Image formation</b></u> in spherical mirrors is defined by
certain
"characteristic" rays whose behaviour is governed by the law
of reflection.</li>
<ul>
<li>Rays parallel to the principal axis are reflected through the
focal point - <b>concave</b> (or as if they came from the focal point
- <b> convex</b> ).</li>
<li>Rays passing through the focal point are reflected parallel to
the principal axis - <b>concave</b> (for <b>convex</b> mirrors a ray
that would have passed through the focal point is reflected parallel to
the axis).</li>
<li>Rays passing through the centre of curvature are reflected back
along their original path -<b>concave</b> (or a ray which would have
passed through the centre of curvature is reflected back along itself -
<b> convex</b> ) </li>
</ul>
<br>
<img src="lo_sm_anim.gif" alt="Concave animation" align="right"
height="222" width="322">
<li><u><b>Concave mirror.</b></u> &nbsp;In
the animation the first two rays from the object are examples of the
first two characteristic rays described above. &nbsp;Only two rays are
needed to define the position of the image. &nbsp;The paths of the
other rays in the animation are
defined since the image position is already known.<br>
The general characteristics of the image depend on the location of the
object with respect to the centre of
curvature and the focal point. &nbsp;Animations of image formation
in a concave mirror for the five possible object positions can be
observed by choosing from the following options.<br>
<ul>
<li><a href="lo_sm_anim2.gif">Object beyond C</a> </li>
<li><a href="lo_sm_anim3.gif">Object at C</a> </li>
<li><a href="lo_sm_anim4.gif">Object between C and F</a> </li>
<li><a href="lo_sm_objatF.gif">Object at F</a> </li>
<li><a href="lo_sm_anim5.gif">Object between F and the vertex<br>
</a> </li>
<br>
</ul>
</li>
<li><u><b> Convex mirror.</b></u> &nbsp;The general characteristics
of images in convex mirrors are independent of the location of the
object. &nbsp;Three examples are shown below.
<div align="center"><img src="lo_sm_convex.gif" alt="Convex images"
align="middle" height="112" width="361"> </div>
</li>
<br>
<li>Note that for the convex mirror the reflected rays <i><b>DIVERGE</b></i>
(this is also the case for the concave mirror when the object is closer
than the focal point to the mirror). &nbsp;In these cases the image
formed is <b>virtual</b> - light rays do not pass though it, but to an
observer "<i> appear</i>" to come from it.</li>
</ul>
<ul>
<li>Perhaps the most common everyday experience of a convex mirror is
the passenger side rear-view mirror in your car.&nbsp; As can be seen
from the above ray diagrams, the image you see&nbsp; in a convex mirror
is always smaller than the object.&nbsp; You "know" the typical size of
a car or truck, so much so that the "size" it appears tells you how far
away it is (if you see a tiny car in the mirror you "know" it is far
away).&nbsp; Thus, seeing a vehicle "looking smaller" due to the convex
mirror, will make you think the vehicle is further away from the mirror
than it actually is, hence the warning, "<a
href="http://www.physlink.com/Education/AskExperts/ae449.cfm"><span
style="font-style: italic; font-weight: bold;">objects mirror are
closer than they appear</span></a>".<br>
</li>
<br>
<li><u><b>Mirror equation</b></u>. &nbsp;For objects placed close to
the
principle axis the distance of the object from the vertex of the mirror
(p), the distance of the image from the vertex of the mirror (q) and
the focal length (f) are related by the following equation,<br>
<br>
<div align="center"><b><img alt="" src="lo_spmirror_eqn1.gif"
style="width: 68px; height: 44px;"> <br>
</b>
<div align="left"><br>
</div>
</div>
</li>
<li><u><b>Magnification</b></u>. &nbsp;The magnification (m) is
defined by,<br>
<br>
<div align="center"><b><img alt="" src="lo_spmirror_eqn2.gif"
style="width: 244px; height: 44px;"> </b><b><br>
</b></div>
</li>
<br>
<li><u><b>Sign Conventions.</b></u> &nbsp;In order to make use of the
above formulae the following sign conventions must be followed. <br>
<br>
<center>
<table bgcolor="#99ffff" border="1" cellpadding="2" cellspacing="2"
width="50%">
<tbody>
<tr>
<td valign="top">
<div align="center"><b>SIGN</b><br>
</div>
</td>
<td valign="top">
<div align="center"><b>+</b><br>
</div>
</td>
<td valign="top">
<div align="center"><b>-</b><br>
</div>
</td>
</tr>
<tr>
<td valign="top">
<div align="center">f - focal length<br>
</div>
</td>
<td valign="top">
<div align="center">Concave<br>
</div>
</td>
<td valign="top">
<div align="center">Convex<br>
</div>
</td>
</tr>
<tr>
<td valign="top">
<div align="center">p - object distance<br>
</div>
</td>
<td valign="top">
<div align="center">Real <br>
</div>
</td>
<td valign="top">
<div align="center">Virtual<br>
</div>
</td>
</tr>
<tr>
<td valign="top">
<div align="center">q - image distance<br>
</div>
</td>
<td valign="top">
<div align="center">Real <br>
</div>
</td>
<td valign="top">
<div align="center">Virtual<br>
</div>
</td>
</tr>
<tr>
<td valign="top">
<div align="center">m - magnification<br>
</div>
</td>
<td valign="top">
<div align="center">Upright image<br>
</div>
</td>
<td valign="top">
<div align="center">Inverted image<br>
</div>
</td>
</tr>
</tbody>
</table>
</center>
</li>
<br>
<li><u><b>Image properties.</b></u> There are four basic properties,
dependent on the position of the object, as indicated in the table
below. &nbsp;These properties can be verified either graphically or by
using the mirror
equation and the definition of magnification. <br>
<div align="center"><br>
</div>
<center>
<table bgcolor="#ffcccc" border="1" cellpadding="2" cellspacing="2"
width="80%">
<tbody>
<tr>
<td bgcolor="#ffff99" valign="top">
<div align="center"><b>Mirror</b><br>
</div>
</td>
<td bgcolor="#ffff99" valign="top">
<div align="center"><b>Object location</b><br>
</div>
</td>
<td bgcolor="#ffff99" valign="top">
<div align="center"><b>Image location</b><br>
</div>
</td>
<td colspan="1" bgcolor="#ffff99" valign="top">
<div align="center"><b>Type</b><br>
</div>
</td>
<td colspan="1" bgcolor="#ffff99" valign="top">
<div align="center"><b>Orientation</b><br>
</div>
</td>
<td bgcolor="#ffff99" valign="top">
<div align="center"><b>Relative size</b><br>
</div>
</td>
</tr>
<tr>
<td bgcolor="#ffff99" valign="top">
<div align="center">CONCAVE<br>
</div>
</td>
<td valign="top">
<div align="center">At infinity<br>
</div>
</td>
<td valign="top">
<div align="center">At F<br>
</div>
</td>
<td valign="top">
<div align="center">Real<br>
</div>
</td>
<td valign="top">
<div align="center">Inverted<br>
</div>
</td>
<td valign="top">
<div align="center">Smaller<br>
</div>
</td>
</tr>
<tr>
<td bgcolor="#ffff99" valign="top">
<div align="center">CONCAVE<br>
</div>
</td>
<td valign="top">
<div align="center">Beyond C<br>
</div>
</td>
<td valign="top">
<div align="center">Between F and C<br>
</div>
</td>
<td valign="top">
<div align="center">Real<br>
</div>
</td>
<td valign="top">
<div align="center">Inverted<br>
</div>
</td>
<td valign="top">
<div align="center">Smaller<br>
</div>
</td>
</tr>
<tr>
<td bgcolor="#ffff99" valign="top">
<div align="center">CONCAVE<br>
</div>
</td>
<td valign="top">
<div align="center">At C<br>
</div>
</td>
<td valign="top">
<div align="center">At C<br>
</div>
</td>
<td valign="top">
<div align="center">Real<br>
</div>
</td>
<td valign="top">
<div align="center">Inverted<br>
</div>
</td>
<td valign="top">
<div align="center">Same size<br>
</div>
</td>
</tr>
<tr>
<td bgcolor="#ffff99" valign="top">
<div align="center">CONCAVE<br>
</div>
</td>
<td valign="top">
<div align="center">Between C and F<br>
</div>
</td>
<td valign="top">
<div align="center">Beyond C<br>
</div>
</td>
<td valign="top">
<div align="center">Real<br>
</div>
</td>
<td valign="top">
<div align="center">Inverted<br>
</div>
</td>
<td valign="top">
<div align="center">Larger<br>
</div>
</td>
</tr>
<tr>
<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>
&nbsp;
<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>
&nbsp; </p>
<p><img src="header-index.gif" height="51" width="92">
</p>
</center>
<p><br>
</p>
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