Physics 1502: Lecture 30 Today s Agenda
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1 Pyscs 1502: Lecture 30 Tday s Agenda Annuncements: Mdterm 2: Mnday Nv. 16 Hmewrk 08: due Frday Optcs Mrrrs Lenses Eye R- R & -R f 1
2 Te Mrrr Equatn We wll nw transfrm te gemetrc drawngs nt algebrac equatns: frm trangles, α α β γ elmnatng α, bject mage Nw we emply te small angle apprxmatns: R Pluggng tese back nt te abve equatn relatng te angles, we get: Defnng te fcal lengt f = R/2, Ts eqn s knwn as te mrrr eqn. Nte tat tere s n mentn f n ts equatn. Terefre, ts eqn wrks fr all, e we ave an mage! Magnfcatn We ave derved te mrrr eqn wc determnes te mage dstance n terms f te bject dstance and te fcal lengt: Wat abut te sze f te mage? Hw s related t?? Frm smlar trangles: R Nw, we can ntrduce a sgn cnventn. We can ndcate tat ts mage s nverted f we defne ts magnfcatn M as te negatve number gven by: 2
3 Mre Sgn Cnventns Cnsder an bject dstance s wc s less tan te fcal lengt: Ray Trace: Ray trug te center f te spere (lgt blue) s reflected stragt back. Ray parallel t axs (red) passes trug fcal pnt f. Tese rays dverge! e tese rays lk tey are cmng frm a pnt bend te mrrr. We call ts a vrtual mage, meanng tat n lgt frm te bject passes trug te mage pnt. Prf left t student: Ts stuatn s descrbed by te same mrrr equatns as lng as we take te cnventn tat mages bend te mrrr ave negatve mage dstances s. e: In ts case, < 0, wc leads t M > 0, ndcatng tat te mage s vrtual (<0) and nt nverted (M>0). R f Cncave-Planar-Cnvex Wat appens as we cange te curvature f te mrrr? Plane mrrr:» R = IMAGE: vrtual uprgt (nn-nverted) Cnvex mrrr:» R < 0 f IMAGE: vrtual uprgt (nn-nverted) 3
4 Lecture 30, ACT 1 In rder fr a real bject t create a real, nverted enlarged mage, a) we must use a cncave mrrr. b) we must use a cnvex mrrr. c) neter a cncave nr a cnvex mrrr can prduce ts mage. 4
5 5
6 Mrrr Lens Defntns Sme mprtant termnlgy we ntrduced last class, = dstance frm bject t mrrr (r lens) = dstance frm mrrr t mage pstve, pstve f n same sde f mrrr as. R = radus f curvature f spercal mrrr f = fcal lengt, = R/2 fr spercal mrrrs. Cncave, Cnvex, and Spercal mrrrs. M = magnfcatn, (sze f mage) / (sze f bject) negatve means nverted mage bject β R γ mage α α Lenses A lens s a pece f transparent materal saped suc tat parallel lgt rays are refracted twards a pnt, a fcus: Cnvergent Lens» lgt mvng frm ar nt glass wll mve tward te nrmal» lgt mvng frm glass back nt ar wll mve away frm te nrmal» real fcus Dvergent Lens» lgt mvng frm ar nt glass wll mve tward te nrmal» lgt mvng frm glass back nt ar wll mve away frm te nrmal» vrtual fcus 6
7 Cnvergng Lens Prncpal Rays F P.A. Image Object F 1) Rays parallel t prncpal axs pass trug fcal pnt. 2) Rays trug center f lens are nt refracted. 3) Rays trug F emerge parallel t prncpal axs. Image s: real, nverted and enlarged (n ts case). Assumptns: mncrmatc lgt ncdent n a tn lens. rays are all near te prncpal axs. ACT 2: Cnvergng Lens Wc way suld yu mve bject s mage s real and dmnsed? Dem (1) Clser t lens Object F (2) Furter frm lens (3) Cnvergng lens can t create real dmnsed mage. F P.A. 7
8 Te Lens Equatn We nw derve te lens equatn wc determnes te mage dstance n terms f te bject dstance and te fcal lengt. Cnvergent Lens: f Ray Trace: Ray trug te center f te lens (lgt blue) passes trug undeflected. Ray parallel t axs (wte) passes trug fcal pnt f. tw sets f smlar trangles: elmnatng /: magnfcatn: als same as mrrr eqn!! M < 0 fr nverted mage. same as mrrr eqn f we defne > 0 f > 0 Summary We ave derved, n te paraxal (and tn lens) apprxmatn, te same equatns fr mrrrs and lenses: wen te fllwng sgn cnventns are used: Varable f > 0 f < 0 > 0 < 0 > 0 < 0 Mrrr cncave cnvex real (frnt) vrtual (back) real (frnt) vrtual (back) Lens cnvergng dvergng real (frnt) vrtual (back) real (back) vrtual (frnt) 8
9 3 Cases fr Cnvergng Lenses Object Past 2F Image Inverted Reduced Real Ts culd be used n a camera. Bg bject n small flm Object Between F & 2F Image Inverted Enlarged Real Ts culd be used as a prjectr. Small slde n bg screen Image Object Insde F Uprgt Enlarged Vrtual Ts s a magnfyng glass Dvergng Lens Prncpal Rays F P.A. Object F Image 1) Rays parallel t prncpal axs pass trug fcal pnt. 2) Rays trug center f lens are nt refracted. 3) Rays tward F emerge parallel t prncpal axs. Image s vrtual, uprgt and reduced. 9
10 ACT 3: Dvergng Lenses Wc way suld yu mve bject s mage s real? Dem Object 1) Clser t lens 2) Furter frm lens 3) Dvergng lens can t create real mage. F F P.A. Lecture 30, ACT 4 A lens s used t mage an bject n a screen. Te rgt alf f te lens s cvered. Wat s te nature f te mage n te screen? (a) left alf f mage dsappears (b) rgt alf f mage dsappears (c) entre mage reduced n ntensty bject lens screen 10
11 Multple Lenses We determne te effect f a system f lenses by cnsderng te mage f ne lens t be te bject fr te next lens f = +1 f = -4 Fr te frst lens: 1 = +1.5, f 1 = +1 Fr te secnd lens: 2 = +1, f 2 = -4 Multple Lenses Objects f te secnd lens can be vrtual. Let s mve te secnd lens clser t te frst lens (n fact, t ts fcus): f = +1 f = -4 Fr te frst lens: 1 = +1.5, f 1 = +1 Fr te secnd lens: 2 = -2, f 2 = -4 Nte te negatve bject dstance fr te 2nd lens. 11
12 Multple Lenses If te tw lenses are tn, tey can be tucng.e. n te same pstn. We can treat as ne lens. f ttal =??? Fr te frst lens: = 1, 1 and f 1 Fr te secnd lens: 2 = - 1, 2 =, f 2 Addng, As lng as, 12
13 Te Lens Equatn Cnvergent Lens: f 13
14 Te Lensmaker s Frmula S far, we ave treated lenses n terms f ter fcal lengts. Hw d yu make a lens wt fcal lengt f? Start wt Snell s Law. Cnsder a plan-cnvex lens: Snell s Law at te curved surface: Assumng small angles, Te bend-angle β s just gven by: lgt ray R ar N β ar α Te bend-angle β als defnes te fcal lengt f: Te angle can be wrtten n terms f R, te radus f curvature f te lens : Puttng tese last equatns tgeter, Mre generally Lensmaker s Frmula Tw curved surfaces Tw arbtrary ndces f refractn Te cmplete generalzed case Nte: fr ne surface Planar, R > 0 f cnvex wen lgt ts t R < 0 f cncave wen lgt ts t 14
15 Cmpund Mcrscpe Objectve (f b < 1cm) L Eyepece (f eye ~5cm) 1 f b 1 f eye I 1 O 1 2 I 2 Magnfcatn: Refractng Telescpe Objectve (f b ~ 250cm) Eyepece (f eye ~5cm) Star f b 1 I 1 f eye ο ο 1 2 I 2 Angular Magnfcatn: 15
16 Te EYE ~f bjectve I 2 L I 1 ~f e eyepece Te Eye Wat des te eye cnsst f? Spere (balln) f water. - An aperture tat cntrls w muc lgt gets trug te Irs/pupl - Bulge at te frnt te crnea - A varable fcus lens bend te retna te lens - A screen tat s ked up t yur bran te retna Crnea Retna Irs Lens T bran 16
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