Physics 3 (PHYF144) Chap 9: Image Formation - 6. h o h h

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1 Pysics 3 (PHY44) a 9: mage ormatio mages ormed by lat mirrors A real image is ormed we ligt rays reac te image oit. A virtual image is ormed we te ligt rays do ot reac te image oit but aear to diverge rom tat oit. mage ormed by a lae mirror is virtual, ad is located P as ar beid te mirror as te object is i rot o te mirror. Q P So, te triagles PQ ad P Q are cogruet, o o bject mage Te egative is itroduced because te image is virtual. By sig covetio, image distace is a egative value or virtual image.*te sig covetio (ositive/egative) o all uatities used i tis cater is summarized i te table o age. Te lateral magiicatio: M mage eigt bject eigt o : object distace : image distace image urigt =>, M +ve. M >, image is magiied. Te image ormed by lat mirror is urigt (M > 0), ad is te same size as te object (M =). Examle : Multile images ormed by two mirrors Two lat mirrors are eredicular to eac oter, as sow i te igure, ad a object is laced at oit. tis situatio, multile images are ormed. Locate te ositios o tese images. Te image o te object is at i mirror ad at i mirror. Mirror Te tird image 3 is te image o, or euivaletly, image o. Tat is, te image (or ) serves as te object or 3. Mirror 3 9. mages ormed by serical mirrors A serical mirror is a mirror, wic as te sae o a sectio cut out o a serical surace. Tere are two tyes o serical mirrors: cocave mirrors, ad covex mirrors. Te most commoly occurrig examles o cocave mirrors are savig mirrors ad makeu mirrors, were objects (i.e., a ace) laced close to tem are magiied. Te most commoly occurrig examles o covex mirrors are te asseger-side wig mirrors o cars. Tese tyes o mirrors ave wider ields o view ta euivalet lat mirrors, but objects aear smaller (ad, tereore, arter away) ta tey actually are. ocave mirror comig ligt Pricial axis utgoig V Trimester 3, 05/06

2 Pysics 3 (PHY44) a 9: mage ormatio - 7 Te ormal to te ceter o te mirror is called te ricial axis. Te oit V at wic te ricial axis touces te surace o te mirror is called te vertex. Te oit, o te ricial axis, wic is euidistat rom all oits o te relectig surace o te mirror, is called te ceter o curvature. Te distace alog te ricial axis rom oit to oit V is called te radius o curvature o te mirror,. Te rays tat arallel ad ear to te ricial axis are relected by te mirror troug a commo oit, wic is called te ocal oit (or ocus) o te mirror. Te distace alog te ricial axis rom te ocus to te vertex is called te ocal legt o te mirror,. = / Te at o a ligt-ray is comletely reversible. Tus, a ligt source laced at te ocus yields a arallel beam o ligt, ater te ligt as relected o te surace o te mirror. Te receivig dises o radio telescoes relect te icomig radio waves rom (very) distat astroomical sources ad brig tem to a ocus at a sigle oit, were a detector is laced. A car eadligt cosists o a ligt-bulb laced at te ocus, wic eables te eadligt to cast a very straigt beam o ligt aead o te car. Locatig a image Tere are two alterative metods o locatig te image ormed by serical mirrors: Graical metod (ay diagram) Algebraic aalysis (Mirror euatio) ay Diagram 3 4 V, object, image Te image is real, iverted, ad magiied. mage roduced by a cocave mirror ca be located usig just our simle rules:. A icidet ray, wic is arallel to te ricial axis, is relected troug te ocus o te mirror.. A icidet ray, wic strikes te mirror at its vertex V, is relected suc tat its agle o icidece wit resect to te ricial axis is eual to its agle o relectio. 3. A icidet ray, wic asses troug te ocus o te mirror is, relected arallel to te ricial axis. 4. A icidet ray, wic asses troug te ceter o curvature o te mirror, is relected back alog its ow at (sice it is ormally icidet o te mirror). (ly two rays are eeded to locate te image) Trimester 3, 05/06

3 Pysics 3 (PHY44) a 9: mage ormatio - 8 Aalytic Metod rom te largest triagle wose base V, rom te blue-saded rigt triagle, iverted. Tus, te magiicatio: M ta. ta mage eigt bject eigt, te egative sig is itroduced because te image is te same ormula or lat mirror. Pricial axis V rom oter triagles, ta, ad ta So, ta rom revious euatio: Simle algebra reduces tis to. (Mirror euatio) or cocave mirror, is +ve. or covex mirror, is ve Examle : Te image ormed by a cocave mirror Assume tat a certai serical mirror as a ocal legt o +0.0 cm. Locate ad describe te image or object distace o 5.0 cm Sice te ocal legt is ositive, te serical mirror is a cocave mirror. 5.0 cm 0.0 cm Te magiicatio o te image is 6.7 cm (ositive,tus,realimage) Trimester 3, 05/06

4 Pysics 3 (PHY44) a 9: mage ormatio cm M (egative,tus,ivertedimage; M,dimiised) 5.0 cm Exercise: A detist uses a cocave mirror (ocal legt cm) to examie some teet. te distace rom te object to te mirror is cm, wat is te magiicatio o te toot? Aswer: ovex mirror Te deiitios o te ricial axis, ceter o curvature, radius o curvature, ad te vertex, o a covex mirror are aalogous to te corresodig deiitios or a cocave mirror. We arallel ligt rays strike a covex mirror tey are relected suc tat tey aear to emaate rom a sigle oit located beid te mirror. Te oit is called te virtual ocus o te mirror. comig ligt utgoig V Pricial axis ay Diagram 4 3 Te image is virtual, urigt, ad dimiised. V Similarly, te image roduced by a covex mirror ca be located by drawig a ray diagram accordig to our simle rules:. A icidet ray, wic is arallel to te ricial axis, is relected as i it came rom te virtual ocus o te mirror.. A icidet ray, wic strikes te mirror at its vertex V, is relected suc tat its agle o icidece wit resect to te ricial axis is eual to its agle o relectio. 3. A icidet ray, wic is directed towards te ceter o curvature o te mirror, is relected back alog its ow at (sice it is ormally icidet o te mirror). 4. A icidet ray, wic is directed towards te virtual ocus o te mirror, is relected arallel to te ricial axis. (ly two rays are eeded to locate te image) Note: ovex mirror always orms virtual image regardless o te ositio o te real object. Trimester 3, 05/06

5 Pysics 3 (PHY44) a 9: mage ormatio - 0 Aalytic Metod: Te same mirror euatio: summary, te ormatio o a image by a serical mirror ivolves te crossig o ligt-rays emitted by te object ad relected o te mirror. te ligt-rays actually cross (i rot o te mirror) te te image is real. te ligt rays do ot actually cross, but aear to cross we rojected backwards beid te mirror, te te image is virtual. A real image ca be rojected oto a scree, a virtual image caot. However, bot tyes o images ca be see by a observer ad otograed by a camera. < Sig covetio or mirror, reractig surace, ad les > +ve ve bject is o te side o icomig ligt. terwise (virtual object) (real object) mage is o te side o outgoig ligt terwise (virtual image) (real image) is o te side o outgoig ligt terwise real ocus virtual ocus, M mage is urigt mage is iverted Te icomig ligt (to mirror/reractig surace/les) ad outgoig ligt (rom to mirror/reractig surace/les) are reerrig to te actual ray rom te real object, ot virtual ray tat exteded rom te actual ray used to locate virtual image. determiig te sig or,, or, te coditio or ositive sig must be cecked irst. Exercise: Determie te sides o icomig ad outgoig ligts or lat ad curved mirrors. te calculated image distace is egative, were is te locatio o te image ormed by te mirror? a a covex mirror orm a real image or a real object? Aswer: rom te ligt o te object; Not o te side o outgoig ligt; o. Examle 3: Te image ormed by a covex mirror A ati-solitig mirror sows a image o a woma wo is located 3.0 m rom te mirror. Te ocal legt o te mirror is 0.5 m. id te ositio o er image ad te magiicatio. 0.5m 0.3m Te magiicatio o te image is M 3.0 m (egative,tus, virtualimage) 0.3m 0.077(ositive,tus,urigt image; M,dimiised) 3.0 m Exercise: A serical mirror as a ocal legt o 0 cm. Wat is te ositio o te resultig image (i cm) i te image is urigt ad our times smaller ta te object? Aswer:5 Trimester 3, 05/06

6 Pysics 3 (PHY44) a 9: mage ormatio mages ormed by reractio Te reractive idex o te medium o icomig ligt to reractig surace is labeled as, ad tat o outgoig ligt rom te reractig surace is labeled as. Te igure sows tat a sigle ray leavig oit ad ocusig at oit. Assume tat te agles ad are small (tus, si ). Sell s law alied to tis reracted ray gives si si comig ligt utgoig ligt P d rom triagle P : rom triagle P : [A exterior agle o a triagle euals te sum o te two oosite iterior agles.] ombie te last tree exressios, ad elimiate ad, we id ) ( Agai, i small agle aroximatio, ta, so we ca write d d d Substitute tese ito revious exressio gives Examle 4: Gaze ito te crystal ball A coi is embedded i a lastic sere o radius 3.0 cm. Te reractive idex o te lastic is =.50. Te coi is located.0 cm rom te edge o te sere. id te ositio o te image o te coi. Sice te object is iside sere, te ceter o curvature is ot o te side o outgoig ligt. Tus, is egative cm cm.7 cm (egative,tus, virtualimage) Exercise: Bottles o erume sometimes ave tick glass ( =.5) walls wic gives te imressio te volume is larger ta it is. Assume a cylidrical bottle as a ier radius o cm ad outer radius o cm. How tick (i cm) would te wall aear i you could see a mark o te iside surace? Aswer: 0.8 cm Trimester 3, 05/06

7 Pysics 3 (PHY44) a 9: mage ormatio Ti leses Leses are commoly used to orm images by reractio i otical istrumets, suc as cameras, telescoes, ad microscoes. Te essetial idea i locatig te ial image o a les is to use te image ormed by oe reractig surace as te object or te secod surace. Note: is or reractig surace closer to te object; is or te oter surace o te les. = Surace Surace t Wit ad assumig = because te les is surrouded by air, te image ormed by surace satisies te euatio: Similarly wit does ot exist, satisies te euatio: were () ad = ad =, te image ormed by surace, as i surace, t (te egative sig is iduced because te image is virtual). or ti les, we ca eglect t. Hece,.() ()+(), ( ). or ti les, we ca omit te subscrits o ad ad called te object distace ad te image distace. Hece, ( ) (3) We ow deie te ocal legt o a ti les as te image distace tat corresods to a iiite object distace, as we did i mirrors; i.e., i, te. Trimester 3, 05/06

8 Pysics 3 (PHY44) a 9: mage ormatio - 3 Tereore, ( ) (Les maker s euatio) A les as two suraces. is te radius o curvature or te surace closer to te object. However, oly deeds o, ad (i.e. te geometry o te les). t does ot deed o locatio o te object. calculatig te ocal legt o a les, simly assumig te object o te let or rigt side o te les will lead to te same aswer. ad (idetical to te mirror euatio) te les is laced i a medium wit reractive idex m, te euatio (3) becomes m m ( m ) or ( m ) overgig les: as ositive ocal legt ad is tickest at te middle. ay Diagram comig ligt utgoig ligt 3 Te image is real, iverted, ad dimiised. Te image roduced by a covergig les ca be located usig just tree simle rules:. A icidet ray, wic is arallel to te ricial axis, is reracted by te les ad asses troug te ocus (or te ocal oit) i te back o te les.. A icidet ray, wic asses troug te ceter o te les, is ot reracted at all. 3. A icidet ray, wic asses troug te ocus i te rot te les, is reracted arallel to te ricial axis. As te igures o te covergig leses above are reerred, te irst les is amed bicovex (or simly covex), d amed cocave-covex, ad 3 rd amed lao-covex. As te igures o te divergig leses below are reerred, st is amed bicocave (or simly), d also amed cocave-covex, ad 3 rd amed alo-cocave. Trimester 3, 05/06

9 Pysics 3 (PHY44) a 9: mage ormatio - 4 Divergig les: as egative ocal legt ad is tickest at te edges. ay Diagram comig ligt 3 utgoig ligt Te image is virtual, urigt, ad dimiised. Te image roduced by a divergig les ca be located usig te ollowig tree rules:. A icidet ray, wic is arallel to te ricial axis, is reracted as i it came rom te ocus o te les.. A icidet ray, wic asses troug te ceter o te les, is ot reracted at all. 3. A icidet ray, wic is directed towards te ocal oit i te back o te les is reracted arallel to te ricial axis. Note: Divergig les always orms virtual image regardless o te ositio o te real object. Examle 5: mage ormed by covergig les A object o eigt = 7.00 cm is laced a distace = 5.0 cm i rot o a ti covergig les o ocal legt = 35.0 cm. Wat are te eigt, locatio, ad ature o te image? 35.0 cm 87.5 cm 5.0 cm (egative,tus, virtualimage) Te image is virtual, o side o icomig ligt, ad located 87.5 cm i rot o te les. Te magiicatio o te image is 87.5 cm M 3.50(ositive,tus,urigt image; M, magiied) 5.0 cm Tus, te image is urigt ad magiied by a actor o t meas tat te eigt o te image is M (3.50)(7.00cm) 4.5 cm Exercise: Suose tat te object is moved to a ew locatio a distace = 90.0 cm i rot o te les. Wat ow are te eigt, locatio, ad ature o te image? Aswer: = 57.7 cm, M = 0.636, = 4.45 cm; a real, iverted, ad dimiised image Trimester 3, 05/06

10 Pysics 3 (PHY44) a 9: mage ormatio - 5 Examle 6: Te absolute values o te radii o curvature o te les suraces i te igure are, resectively, 0.0 cm ad 30.0 cm. Te reractive idex o te les is.5. A object is laced 0.0 cm i rot o te surace o smaller radius o curvature. Neglect te tickess o te les. (a) Wat is te ocal legt o te les. 0.0 cm, 30.0 cm, ( ) (.5) 0.0 cm 3.077cm 3.cm 30.0 cm.5 (b) Locate te ositio o te image ormed by te les. 0.0 cm 3.077cm 50cm (c) Suose tat te object is moved to a ew locatio a distace = 60.0 cm i rot o te les, locate te ositio o te ial image cm 3.077cm 37.5 cm Lateral magiicatio, Les maker s e., ormula i use ormula M o = / or Mirror & Les Mirror Mirror & Les eractig surace ( ) Les Trimester 3, 05/06