SPH3UW Unt 7.3 Sphercal Concave Mrrors Page 1 of 1 Notes Physcs Tool box Concave Mrror If the reflectng surface takes place on the nner surface of the sphercal shape so that the centre of the mrror bulges away from the vewer. Focus (Focal Pont) where the reflected rays from the sphercal mrrors nearly cross each other. Prncpal Axs The straght lne perpendcular to the curved surface at ts centre lne. Focal Length The dstance between the focal pont and the centre of the mrror. Sphercal Aberraton The effect caused by the rays not comng to a perfect focus. Ths defect s called the sphercal Aberraton. h0 d0 d0 f h d f ; Books lke to use d, for dstance n ths formula. R f 1 1 1 ; The varables that most text books use for the poston of the s s f y s m y s object s s and the poston of the mage s s A plane mrror produces an mage that s the same sze as the object. But, there are many stuatons where we requre the mage sze and object sze to be dfferent. We wll consder the specal (and easly analyzed) case of mage formaton by a sphercal mrror. There are two types of curved mrrors, convergng (whch cause parallel rays to come to a focus) and dvergng (whch cause parallel rays to spread apart from a focus). Both types of mrrors have common terms. Common Terms Centre of Curvature (C) : Is the centre of a sphere from whch the mrror s made. Radus of Curvature (R) : Is the dstance from the centre of curvature to the mrror Vertex (V ) : s the geometrc centre of the mrror. Where the prncpal axs meets the mrror. Prncpal Axs (PA ) : Is the lne drawn through the vertex perpendcular to the mrror. Prncpal Focus (F) : The pont where lght rays parallel to the prncpal axs ether converge or appear to dverge. Focal Length (f) : Is the dstance from the vertex to the prncpal focus.
SPH3UW Unt 7.3 Sphercal Concave Mrrors Page of 1 R C f V PA The followng concave fgure represents a sphercal mrror wth radus of curvature R, wth ts concave sde facng the ncdent lght. The centre of curvature of the surface s at C and the vertex of the mrror s at V. The lne CV s called the Optc axs or Prncpal Axs. Pont P s an object pont that les on the optc axs. Rays from P that strke the mrror are reflected back through the pont P. Thus pont P wll be the mage of object P, therefore P s a real mage The object dstance, measured from the vertex V, s s. The mage dstance, also measured from V, s s. The sgns of s, s, and the radus of curvature are determned by the sgn rules. The object pont P s on the same sde as the ncdent lght, so accordng to the frst sgn rule, s s postve. The mage pont P s on the same sde as the reflected lght, so accordng to the second sgn rule, the mage dstance s s also postve. The centre of curvature C s on the same sde as the reflected lght, so accordng to the thrd sgn rule, R, too, s postve.
SPH3UW Unt 7.3 Sphercal Concave Mrrors Page 3 of 1 The Lens Formula Dervaton (for those Math Guru Types) From the above dagram: and now by elmnatng between these two equatons we obtan. Now f s small, then the followng relatonshps can be used h h h,, s s R Now substtutng these nto, we obtan a general relatonshp among s, s, and R: 1 1 s s R Ths equaton does not contan the angle. Hence all rays from P that make suffcently small angles wth the axs ntersect at P after they are reflected. Such rays are nearly parallel to the axs and close to t and thus are called paraxal rays. Now remember that the equaton s only approxmately correct, n realty, unlke a plane mrror, the sphercal mrror does not reflect through a precse pont P but are rather smeared out. Ths property of a sphercal mrror s called sphercal aberraton. Focal Pont and Focal Length When the object P s very far away from the sphercal mrror s, the ncomng rays are parallel. Therefore the mage dstance s n ths case s gven by 1 1 s R thus R s The pont P s called the focal pont, F and the dstance s s called the focal length, f.
SPH3UW Unt 7.3 Sphercal Concave Mrrors Page 4 of 1 R f Now f the object s placed at the focal pont, F so the object dstance s dstance s gven by 1 or 1 0 s R s R s R s f and the mage The most common verson of 1 1 s s R s wrtten as: 1 1 1 s s f
SPH3UW Unt 7.3 Sphercal Concave Mrrors Page 5 of 1 Image Characterstcs for Concave Mrrors Ray Drawng Technques 3 1 C F r 1. Any lne drawn parallel to the prncpal axs wll reflect through the focus (F). 1. Any lne drawn through the focus wll reflect back parallel to the prncpal axs.. Any lne that s drawn to the ntersecton of the prncpal axs and mrror wll have the angle of ncdence equalng the angle of reflecton. 3. Any lne drawn through the radus of curvature (C) wll reflect back through the radus of curvature ( C). Prevously, ray dagrams were constructed n order to determne the general locaton, sze, orentaton, and type of mage formed by concave mrrors. Perhaps you notced that there s a defnte relatonshp between the mage characterstcs and the locaton where an object placed n front of a concave mrror. The purpose of ths porton of the lesson s to summarze these object-mage relatonshps - to practce the L O S T art of mage descrpton. We wsh to descrbe the characterstcs of the mage for any gven object locaton. The L of L O S T represents the relatve locaton. The O of L O S T represents the orentaton (ether uprght or nverted). The S of L O S T represents the relatve sze (ether magnfed, reduced or the same sze as the object). And the T of L O S T represents the type of mage (ether real or vrtual). The best means of summarzng ths relatonshp between object locaton and mage characterstcs s to dvde the possble object locatons nto fve general areas or ponts: Case 1: the object s located beyond the centre of curvature (C) Case : the object s located at the centre of curvature (C) Case 3: the object s located between the centre of curvature (C) and the focal pont (F) Case 4: the object s located at the focal pont (F) Case 5: the object s located n front of the focal pont (F)
SPH3UW Unt 7.3 Sphercal Concave Mrrors Page 6 of 1 Case 1: The object s located beyond C When the object s located at a locaton beyond the center of curvature, the mage wll always be located somewhere n between the center of curvature and the focal pont. Regardless of exactly where the object s located, the mage wll be located n the specfed regon. In ths case, the mage wll be an nverted mage. That s to say, f the object s rght-sde up, then the mage s upsde down. In ths case, the mage s reduced n sze; n other words, the mage dmensons are smaller than the object dmensons. If the object s a sx-foot tall person, then the mage s less than sx feet tall. Earler n Lesson, the term magnfcaton was ntroduced; the magnfcaton s the rato of the heght of the mage to the heght of the object. In ths case, the absolute value of the magnfcaton s less than 1. Fnally, the mage s a real mage. Lght rays actually converge at the mage locaton. If a sheet of paper was placed at the mage locaton, the actual replca of the object would appear projected upon the sheet of paper.
SPH3UW Unt 7.3 Sphercal Concave Mrrors Page 7 of 1 Case : The object s located at C When the object s located at the center of curvature, the mage wll also be located at the center of curvature. In ths case, the mage wll be nverted (.e., a rght-sde-up object results n an upsde-down mage). The mage dmensons are equal to the object dmensons. A sx-foot tall person would have an mage whch s sx feet tall; the absolute value of the magnfcaton s equal to 1. Fnally, the mage s a real mage. Lght rays actually converge at the mage locaton. As such, the mage of the object could be projected upon a sheet of paper. Case 3: The object s located between C and F When the object s located n front of the center of curvature, the mage wll be located beyond the center of curvature. Regardless of exactly where the object s located between C and F, the mage wll be located somewhere beyond the center of curvature. In ths case, the mage wll be nverted (.e., a rght-sde-up object results n an upsde-down mage). The mage dmensons are larger than the object dmensons. A sx-foot tall person would have an mage whch s larger than sx feet tall; the absolute value of the magnfcaton s greater than 1. Fnally, the mage s a real mage. Lght rays actually converge at the mage locaton. As such, the mage of the object could be projected upon a sheet of paper.
SPH3UW Unt 7.3 Sphercal Concave Mrrors Page 8 of 1 Case 4: The object s located at F When the object s located at the focal pont, no mage s formed. As dscussed earler n Lesson 3, lght rays from the same pont on the object wll reflect off the mrror and nether converge nor dverge. After reflectng, the lght rays are travelng parallel to each other and do not result n the formaton of an mage. Case 5: The object s located n front of F When the object s located at a locaton beyond the focal pont, the mage wll always be located somewhere on the opposte sde of the mrror. Regardless of exactly where n front of F the object s located, the mage wll always be located behnd the mrror. In ths case, the mage wll be an uprght mage. That s to say, f the object s rght-sde up, then the mage wll also be rght-sde up. In ths case, the mage s magnfed; n other words, the mage dmensons are greater than the object dmensons. A sx-foot tall person would have an mage whch s larger than sx feet tall; the magnfcaton s greater than 1. Fnally, the mage s a vrtual mage. Lght rays from the same pont on the object reflect off the mrror and dverge upon reflecton. For ths reason, the mage locaton can only be found by extendng the reflected rays backwards beyond the mrror. The pont of ther ntersecton s the vrtual mage locaton. It would appear to any observer as though lght from the object were dvergng from ths locaton. Any attempt to project such an mage upon a sheet of paper would fal snce lght does not actually pass through the mage locaton.
SPH3UW Unt 7.3 Sphercal Concave Mrrors Page 9 of 1 It mght be noted from the above descrptons that there s a relatonshp between the object dstance and object sze and the mage dstance and mage sze. Startng from a large value, as the object dstance decreases (.e., the object s moved closer to the mrror), the mage dstance ncreases; meanwhle, the mage heght ncreases. At the center of curvature, the object dstance equals the mage dstance and the object heght equals the mage heght. As the object dstance approaches one focal length, the mage dstance and mage heght approaches nfnty. Fnally, when the object dstance s equal to exactly one focal length, there s no mage. Then alterng the object dstance to values less than one focal length produces mages whch are uprght, vrtual and located on the opposte sde of the mrror. Fnally, f the object dstance approaches 0, the mage dstance approaches 0 and the mage heght ultmately becomes equal to the object heght. These patterns are depcted n the dagram below. Nne dfferent object locatons are drawn and labeled wth a number; the correspondng mage locatons are drawn n blue and labeled wth the dentcal number. Algebrac Calculatons Usng our algebrac formula 1 1 1, we can determne the mage locaton, s, when the f s s object s poston and the focal length of the mrror s gven. o Sgn conventon for concave mrror Postve Negatve f always s o In front of mrror s In front of mrror (Real mage) Behnd Mrror (vrtual mage)
SPH3UW Unt 7.3 Sphercal Concave Mrrors Page 10 of 1 Image Sze Suppose we have an object wth fnte sze, represented on the dagram wth the arrow PQ. The mage of P s located at P wth a heght represented wth arrow P Q. Geometry demonstrates that the lateral magnfcaton: y s s m or y s s 0 h s m h o s 0 Where h represents the heght and the ndexes and o represent the mage and object respectvely. If m s postve, the mage s erect n comparson to the object; f m s negatve; the mage s nverted relatve to the orgnal object. Example: A 10 cm tall candle s placed 6 cm n front of a concave mrror wth focal length f= cm. Determne the mage locaton and the mage sze. Soluton: Use our formula to determne the mage locaton. s f s o cm 6cm? 1 1 1 f s s o 1 1 1 s f s o 1 1 6cm cm 1 cm 6cm 1cm 3cm s 3cm Therefore the mage s 3cm n front of the mrror
SPH3UW Unt 7.3 Sphercal Concave Mrrors Page 11 of 1 Now for the mage sze. We wll use our magnfcaton formula. h m h o s o s m s o 3cm 6cm 1 s Therefore h h mh 0 1 10 cm 5cm Ths tells us that the mage s a real mage located 3cm from the reflectng sde of the mrror, and the mage s nverted (the negatve sgn n front of the h ), and the mage s 5cm tall.
SPH3UW Unt 7.3 Sphercal Concave Mrrors Page 1 of 1 Extra Notes and Comments