Lecture #5.3 Mirrors
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- Erika Foster
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1 Lecture #5.3 Mrrrs We ave already dscussed reflectn f EM waves. Ts penmenn fnds useful applcatn n te devces we use n ur everyday lfe. Tday we sall talk abut w reflectn wrks n rder t prduce mages n dfferent types f mrrrs. We sall cntnue lkng at lgt nt as EM wave but rater as beam f rays, wc prpagates alng te stragt lne untl t cmes t a bundary between te tw dfferent substances. As t was already mentned, te part f pyscs wc studes lgt frm ts stand pnt s called Gemetrcal Optcs. We already knw tat f te ray f lgt ts te mrrr (surface f te cnductr) at sme angle relatve t te nrmal (angle f ncdence), ten te angle wc te reflected ray makes wt te nrmal t te mrrr (angle f reflectn) s te same as te angle f ncdence. Ts s knwn as te law f reflectn wc s (5.3.1) r Als te reflected ray s n te same plane as te ncdent ray. Ts law allws us t explan w te mages are frmed n a plane mrrr. Te fllwng pcture depcts te uprgt bject f egt at dstance n frnt f te plane mrrr Te lgt ray wc s emtted frm any pnt f te bject (fr nstance frm te tp pnt as n te drawng) ges t te mrrr and ten t s reflected back n suc a way tat
2 te angle f ncdence s equal t te angle f reflectn. If ne cntnues te reflected rays bend te mrrr and fnds te pnt were tey crss eac ter, te mage f te bject can be btaned. As t can be seen, by cnsderng trangles n te pcture, te mage bend te mrrr s at te same dstance = as te bject n frnt f te mrrr. Te mage as te same sze and te same uprgt rentatn as te bject. Ts, wever, s a vrtual mage, snce te rays d nt actually crss te surface f te mrrr. If te mage was real we wuld be able t nstall a screen smewere and see ts mage n ts screen, but n ur stuatn t can nt be prjected nt any screen, snce t always appears bend te mrrr. Te plane mrrr s nt te nly type f mrrrs used n practce. Spercal cncave and cnvex mrrrs are ften used t. Tese mrrrs ave fcal pnts. If te beam f parallel rays s drected t te surface f suc cncave mrrr, te rays wll be reflected and crss eac ter at te fcal pnt, wc (n te case f te small curvature) s at dstance f frm te surface f te mrrr. Here s te radus f curvature fr ts mrrr wc s te radus f te spercal sell f wc te mrrr s a selectn. Ts als means tat f yu extend te nrmal t te surface f te mrrr at any pnt f ts spercal mrrr, all te nrmal lnes wll crss at te same pnt at dstance frm te surface f te mrrr wc s te center f curvature fr a gven mrrr. Dependng n weter yu ave cnvex r cncave mrrr, te lgt rays wll beave dfferently. Fr cnvex mrrrs, te real reflected rays f parallel beam are nt crssng eac ter. But f yu cntnue tse rays bend te mrrr, t lks lke tey are cmng frm te same (fcal) pnt. Snce ts pnt s n te ter sde f te mrrr, te fcal lengt fr suc mrrr cnsdered t be negatve, wc s fcnvex. (5.3.) Fr cncave mrrr, te reflected parallel rays d ndeed crss eac ter at te fcal pnt n te same sde f te mrrr, s fcncave (5.3.3) In rder t see w an mage s frmed, we ave t trace several mprtant rays as tey are reflected by te mrrr. We already knw tat te ray wc ges parallel t te
3 ptcal axs f te mrrr wll be reflected n suc a way tat eter te ray tself (cncave mrrr) r ts extensn (cnvex mrrr) passes trug te fcal pnt f te mrrr. In a same way te ray wc rgnally passes (r ts extensn passes) trug te fcal pnt wll be reflected parallel t te axs f te mrrr. Fnally te ray wc ts te surface f te mrrr perpendcular t ts surface (eter te ray r ts extensn passng trug te center f curvature) wll be reflected back alng te same lne as t came t te mrrr. In te case f te cnvex mrrr (snce reflected rays are always dvergng) te mage can nly be te vrtual mage (bend te mrrr), t s always uprgt and reduced. Te typcal stuatn s swn n te pcture belw Te stuatn s mre cmplcated n te case f cncave mrrr. In fact, tere are tree dstnct stuatns n tat case. 1) Te bject s far away (mre tan ) frm te mrrr ten te ray dagram lks lke te fllwng
4 As we can see frm te drawng, te mage s gng t be real reduced and upsde dwn. ) Fr te secnd case ne can cnsder a stuatn, wen te bject s between te center f curvature and te fcal pnt, ten we ave te fllwng drawng It sws tat we ave real, magnfed upsde dwn mage. 3) Te fnal case ccurs f te bject s between te mrrr and te fcal pnt ten te fllwng ray dagram can be used. S, n ts case t s vrtual, magnfed, uprgt mage. All te ray dagrams wc we just saw can be summarzed n terms f matematcal equatn knwn as te mrrr equatn. We can pck any f te dagrams n rder t derve te equatn. Ten we need t cnsder a set f te smlar trangles (swed by red clr n ne f te dagrams). If s te egt f te bject, s te egt f te mage (can be pstve r negatve, dependng n weter r nt te mage s uprgt r upsde dwn), s te dstance frm te bject t te mrrr and s te
5 dstance frm te mage t te mrrr (pstve f te mage s real and n te same sde f te mrrr as te bject, negatve f te mage s vrtual and n te ter sde f te mrrr) ten we ave,,,, f Te last equatn s te mrrr equatn (5.3.4) (5.3.5) f Even tug we ave derved ts equatn, usng ne f te pctures fr cncave mrrr, t s als vald fr a cnvex mrrr wt accunt f negatve fcal lengt. Anter quantty wc s ften used s te magnfcatn f te mrrr m (5.3.6)
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