Chapter 25 The Reflection of Light: Mirrors. The content contained in all sections of chapter 25 of the textbook is included on the AP Physics B exam.

Transcription

1 Chapter 25 The Reflectn f Lght: Mrrrs Chapter 25 THE REFLECTION OF LIGHT: MIRRORS PREVIEW The ray mel f lght states that lght may be represente by a straght lne alng the rectn f mtn, an ray ptcs s the stuy f lght usng the ray mel. Lght can be reflecte frm a surface such as a mrrr. In ths chapter, we ll stuy reflectn frm plane mrrrs an sphercal mrrrs. As lght rays are reflecte, they may frm an mage, whch can be real r vrtual, epenng n the stance frm the mrrr t the bject whch s the surce f the lght rays. Ray agrams can be rawn t shw the benng f a lght ray r t lcate an mage frme by a mrrr. The cntent cntane n all sectns f chapter 25 f the textbk s nclue n the AP Physcs B exam. QUICK REFERENCE Imprtant Terms angle f ncence the angle between the nrmal lne t a surface an the ncent ray r wave angle f reflectn the angle between the nrmal lne t a surface an the reflecte ray r wave cnvergng mrrr a mrrr whch cnverges lght rays reflectng frm t; als knwn as a cncave mrrr vergng mrrr a mrrr whch verges lght rays reflectng frm t; als knwn as a cnvex mrrr fcal length the stance between the center f a lens r mrrr t the pnt at whch the rays cnverge at the fcal pnt fcal pnt the pnt at whch lght rays cnverge r appear t rgnate mage repructn f an bject usng lenses r mrrrs magnfcatn rat f the sze f an ptcal mage t the sze f the bject bject (ptcs) the surce f vergng lght rays plane mrrr smth, flat surface that reflects lght regularly 28

2 Chapter 25 The Reflectn f Lght: Mrrrs prncpal axs the lne cnnectng the center f curvature f a curve mrrr wth ts gemetrcal vertex; the lne perpencular t the plane f a lens passng thrugh ts center raus f curvature the raus f a sphercal mrrr whch s equal t twce the fcal length f the mrrr ray mel f lght lght may be represente by a straght lne alng the rectn f mtn ray ptcs stuy f lght usng the ray mel real mage an mage that can be prjecte nt a screen vrtual mage an mage that cannt be prjecte nt a screen; pnt at whch vergng lght rays appear t rgnate Equatns an Symbls θ = θ r f = R 2 h = h m = + = f where θ r = angle f reflectn as measure frm the nrmal lne θ = angle f ncence as measure frm the nrmal lne f = fcal length f a mrrr R = raus f curvature f a cnvex r cncave mrrr h = heght f the bject h = heght f the mage pruce by a mrrr = stance frm the center f the mrrr t the bject = stance frm the center f the mrrr t the mage m = magnfcatn Ten Hmewrk Prblems Chapter 25 Prblems 2, 5, 0,, 4, 6, 7, 23, 33,

3 Chapter 25 The Reflectn f Lght: Mrrrs DISCUSSION OF SELECTED SECTIONS 25.3 The Frmatn f Images by a Plane Mrrr Any wave that bunces ff f a barrer fllws the law f reflectn: the angle f ncence s equal t the angle f reflectn as measure frm a lne nrmal (perpencular) t the barrer (Fgure A). In the case f lght, the barrer s ften a mrrr. Ths s why f smene can see yu n a mrrr, yu can see hm n the mrrr as well. θ Nrmal θ r Fgure A Fgure B A plane mrrr s smply a flat mrrr. Frm yur everyay experence yu knw that a plane mrrr always pruces an mage whch s the same sze as the bject (whch cul be yu n the mrnng), left-rght reverse, an the same stance behn the surface f the mrrr as the bject s n frnt f the mrrr. We say that the mage frme by a plane mrrr s vrtual, snce we cannt place a screen behn the mrrr an see the mage prjecte n the screen. A vrtual mage s ne that cannt be prjecte nt a screen. We can lcate the mage frme by a plane mrrr by tracng tw reflecte rays an then extenng them backwar (Fgure B). The pnt at whch the tw rays seem t meet s where the vrtual mage s frme Sphercal Mrrrs, The Frmatn f Images by Sphercal Mrrrs, an The Mrrr Equatn an the Magnfcatn Equatn We wll scuss tw types f sphercal (curve) mrrrs: vergng (cnvex) an cnvergng (cncave), althugh the AP Physcs B exam usually fcuses n cncave mrrrs. A vergng mrrr s smetmes referre t as a cnvex mrrr. Yu may have seen ths type f mrrr n the crner f a cnvenence stre. The mrrr verges the rays f lght whch strke t, allwng the clerk at the stre t see practcally the entre stre n ne 283

4 Chapter 25 The Reflectn f Lght: Mrrrs mrrr. Lght rays cmng nt the mrrr parallel t ts prncpal axs wll verge, r sprea apart: 3f 2f f f 2f 3f Ntce that the rays appear t rgnate frm a pnt behn the mrrr. Ths pnt s calle the vrtual fcus, an the stance between the surface f the mrrr at ts center an the fcal pnt s calle the fcal length f. A cnvergng mrrr s smetmes referre t as a cncave mrrr. If a sphercallyshape cncave mrrr s small cmpare t ts raus f curvature R, then lght rays cmng n parallel t the prncpal axs f the mrrr wll cnverge t a fcal pnt. 3f 2f =R f f 2f =R 3f Fr ths kn f mrrr, the fcal length f an the raus f curvature R f the mrrr are relate by the equatn f = ½ R. If yu lk nt a cnvergng mrrr, yu wll at frst see an mage f yurself whch s nverte (upse-wn), but then as yu mve clser t the mrrr yu wll see yur mage turn uprght as yu pass the fcal pnt f the mrrr. Satellte shes act as cnvergng mrrrs fr ra an TV waves, gatherng them at a etectr lcate at the fcal pnt f the sh. Mst research telescpes als use cnvergng mrrrs rather than lenses t ntally fcus ncmng lght an stuy mages. The mage frme by a cnvergng mrrr epens n the lcatn f the bject n relatn t the fcal length an ts rentatn, an can be real, whch means t can be prjecte nt a screen, r vrtual. The mage can als be uprght r nverte (upsewn), an smaller, larger, r the same sze as the bject. The bject stance s the stance frm the center f the surface f the mrrr t the bject, an the mage stance s the stance frm the center f the surface f the mrrr t the mage. 284

5 Chapter 25 The Reflectn f Lght: Mrrrs Fr the mage frme, the rat f the bject heght h t the mage heght h s equal t the rat f the bject stance t the mage stance: h = h an the magnfcatn f the mage s gven by h m = =. h Magnfcatn tells us hw many tmes larger r smaller an mage s than the bject. The negatve sgn s nserte as a cnventn. The bject heght h s always taken as pstve, an the mage heght h s pstve f the mage s uprght an negatve f nverte. The bject an mage stances are pstve f the mage an bject are n the reflectng se f the mrrr. If ether the bject r the mage s behn the mrrr, the crrespnng bject r mage stance s negatve. In the en, the magnfcatn m s pstve fr an uprght mage an negatve fr an nverte mage. The relatnshp between the bject stance, mage stance, an fcal length s f = + where f = ½ R. Example A cnvergng (cncave) mrrr has a fcal length f 20.0 cm. A 5.0-cm tall canle s place at a stance f 50.0 cm n frnt f the mrrr. (a) By rawng a ray agram, fn the lcatn f the mage frme by the cnvergng mrrr. State whether the mage s real r vrtual, uprght r nverte, an larger, smaller, r the same sze as the bject (canle). (b) Usng the mrrr an magnfcatn equatns, verfy yur results frm part (a). Slutn (a) T fn the mage frme by the mrrr, we wll raw tw rays: () ne ray frm the flame whch strkes the mrrr parallel t the prncpal axs an reflectng thrugh the fcal pnt, an (2) anther ray frm the flame whch ges thrugh the fcal pnt an reflects back parallel t the prncpal axs. The mage s frme at the lcatn f the ntersectn f these tw rays: 285

6 Chapter 25 The Reflectn f Lght: Mrrrs = 60 cm 2 3f 2f f f 2f 3f Nte that when the bject s place at a stance greater than twce the fcal length, the mage s nverte an smaller than the bject. If we place a screen at the lcatn f the mage, we wul see that the mage s real. Fr a cncave mrrr, f the mage s frme n the same se f the mrrr as the bject the mage s real. If the mage s frme n the ppste se f the mrrr as the bject, t s vrtual. Recall that a plane mrrr wll nly pruce an mage n the ppste se t the bject, an thus s always vrtual. If we ve rawn ur agram t scale we can smply measure the mage stance frm the center f the mrrr. Let s say ur measurement s = 32.5 cm, an the heght f the mage s 3.2 cm. (b) Usng the mrrr an magnfcatn equatns, we can slve the mage stance: = + f = + 20cm 50cm = 33.3cm Nte that ur measurement f the mage stance s very clse t the calculate value wthn reasnable expermental errr. The magnfcatn f the canle can be fun by h m = = h ( 5.0 cm)( 33.3cm) h h = = = 3.3cm 50 cm Agan, the negatve sgn ncates the mage s nverte. 286

7 Chapter 25 The Reflectn f Lght: Mrrrs Example 2 A cnvergng (cncave) mrrr has a fcal length f 20.0 cm. A 5.0-cm tall canle s place at a stance f 0.0 cm n frnt f the mrrr. (a) By rawng a ray agram, fn the lcatn f the mage frme by the cnvergng mrrr. State whether the mage s real r vrtual, uprght r nverte, an larger, smaller, r the same sze as the bject (canle). (b) Usng the mrrr an magnfcatn equatns, verfy yur results frm part (a). Slutn If we place the canle at a stance less than the fcal length, the reflecte rays verge, an the mage s frme at the pnt frm whch the rays seem t rgnate. Drawng ur tw prncpal rays: C 3f 2f f f 2f 3f Nte that ths tme ur secn ray s rawn as f t rgnates at the center f curvature C, whch s lcate at 2f. A ray that passes thrugh the center f curvature wll reflect back n tself, an therefre f we exten t behn the mrrr t cntnues n a straght lne. We see that the mage f the canle s uprght, larger, an behn the mrrr, an therefre s vrtual. (b) Usng the mrrr an magnfcatn equatns, we can slve the mage stance: = + f = + 20cm 0cm = 20cm The mage stance s negatve, ncatng t s frme behn the mrrr. Nte that the mage stance shwn n the agram s slghtly less than 20 cm ue t measurement errr. The heght f the mage s h m = = h ( 5.0cm)( 20.0cm) h h = = = + 0.0cm 0.0cm The pstve sgn ncates that the mage s uprght. 287

8 Chapter 25 The Reflectn f Lght: Mrrrs The mage frme by a cncave mrrr when the bject s place nse the fcal length s vrtual uprght an enlarge. Thus f yu want t see an uprght mage f yur face n a cncave mrrr, yu must mve the mrrr t a stance less than ne fcal length frm yur face. Α summary f the mages frme by a cncave mrrr (an a cnvex lens) s lste n the table belw. Object place at: Image stance real r vrtual uprght r nverte larger r smaller > 2f + real nverte smaller = 2f + real nverte same sze f < < 2f + real nverte larger = f N mage N mage N mage N mage < f - vrtual uprght larger Nte that all real mages are nverte. Practce Verfy the results n the table abve by rawng the ray agrams belw. 3f 2f f f 2f 3f 3f 2f f f 2f 3f 3f 2f f f 2f 3f 288

9 Chapter 25 The Reflectn f Lght: Mrrrs CHAPTER 25 REVIEW QUESTIONS Fr each f the multple-chce questns belw, chse the best answer. N In the fgure shwn the angle f ncence s 2. Whch angle(s) s/are the angle(s) f reflectn? (A) (B) 3 an 4 (C) 3 (D) 4 (E), 3, an A plane mrrr wll pruce a vrtual mage (A) when the bject stance s greater than the mage stance. (B) when the bject stance s less than the mage stance. (C) when the bject s n the prncpal axs f the mrrr. (D) when the rays cnverge at the fcal pnt f the mrrr (E) at all stances frm the mrrr. 3. Whch f the fllwng mrrrs verge parallel lght rays? (A) plane (B) cnvex (C) cncave (D) nverte (E) uprght 4. A canle s place n the prncpal axs f a cncave mrrr at a stance f 30 cm frm the mrrr. The fcal length f the mrrr s 0 cm. The mage frme wll be (A) real, uprght, an enlarge (B) real, nverte, an enlarge (C) real, nverte, an smaller (D) vrtual, uprght, an enlarge (E) vrtual, uprght, an smaller 5. A canle s place n the prncpal axs f a cncave mrrr at a stance f 0 cm frm the mrrr. The fcal length f the mrrr s 20 cm. The mage frme wll be (A) real, uprght, an enlarge (B) real, nverte, an enlarge (C) real, nverte, an smaller (D) vrtual, uprght, an enlarge (E) vrtual, uprght, an smaller 6. A canle s place n the prncpal axs f a cncave mrrr at a stance f 20 cm frm the mrrr. The mage frme s magnfe 3 tmes. The mage stance s (A) 7 cm (B) 20 cm (C) 60 cm (D) 90 cm (E) 20 cm 289

10 Chapter 25 The Reflectn f Lght: Mrrrs Free Respnse Questn Drectns: Shw all wrk n wrkng the fllwng questn. The questn s wrth 5 pnts, an the suggeste tme fr answerng the questn s abut 5 mnutes. The parts wthn a questn may nt have equal weght.. (5 pnts) cm The cncave mrrr shwn abve has a fcal length f 30.0 centmeters. Yu are gven a canle 5.0 centmeters hgh. Yu wsh t pruce an mage n a screen whch s 5.0 cm hgh. (a) State whether yu wul place the canle n the left se f the mrrr shwn abve r the rght se. Explan yur chce. (b) State whether yu wul place the screen n the left se f the mrrr shwn abve r the rght se n rer t see the mage frme by the mrrr. Explan yur chce. (c) Chse an apprprate bject stance, an calculate the crrespnng stance frm the mrrr at whch the screen shul be place n rer t see the 5.0 cm mage. () Usng an apprprate scale, raw a ray agram n the fgure abve whch verfes yur calculatn. 290

11 Chapter 25 The Reflectn f Lght: Mrrrs ANSWERS AND EXPLANATIONS TO CHAPTER 25 REVIEW QUESTIONS Multple Chce. C The angles f ncence an reflectn are bth measure frm the nrmal lne. 2. E A plane (flat) mrrr wll reflect lght an pruce an mage at any stance frm the bject. 3. B A cnvex mrrr s als calle a vergng mrrr. 4. C The canle s place at a stance greater than twce the fcal length, an s the mage frme wll be real, nverte, an smaller than the canle. 5. D The canle s place at a stance less than the fcal length, an s the mage frme wll be vrtual, uprght, an enlarge. 6. C Snce Magnfcat n =, then = (M)( ) = (3)(20 cm) = 60 cm. Free Respnse Questn Slutn (a) 2 pnts The canle must be place n the left se f the mrrr, snce the left se s the cncave (cnvergng) se f the mrrr. (b) 2 pnts In rer t see an mage n a screen, the mage must be real. Fr a cncave mrrr, real mages are frme n the same se f the mrrr as the bject. (c) 5 pnts In rer t pruce an enlarge real mage, the canle wul nee t be place at an bject stance between f an 2f. Chsng 40 cm as the bject stance, we can fn the crrespnng mage stance. 29

12 Chapter 25 The Reflectn f Lght: Mrrrs f = + = + 30cm 40cm = 20cm The screen shul be place 20 cm t the left f the mrrr. () 6 pnts Drawng ne prncpal ray parallel t the prncpal axs an reflectng thrugh the fcal pnt, an ne prncpal ray passng thrugh the center f curvature (ther prncpal rays cul have been use): cm 292