s s f h s s SPH3UW Unit 7.7 Concave Lens Page 1 of 7 Notes Properties of a Converging Lens

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1 SPH3UW Unt 7.7 Cncave Lens Page 1 f 7 Ntes Physcs Tl bx Thn Lens s an ptcal system wth tw refractng surfaces. The mst smplest thn lens cntan tw sphercal surfaces that are clse enugh tgether that we can neglect the dstance between them. Cnvergng Lens a lens that has the rays cnverge t a sngle pnt, such as a duble cnvex lens. Dvergng Lens a lens that has parallel rays dverge, thus gvng the lens a negatve value, such as a duble cncave lens. Object-Image Relatn - s s f h s s Lateral Magnfcatn - m h s s Prpertes f a Cnvergng Lens The prpertes f a duble cncave lens (called a dvergng lens r negatve lens) allw a beam f parallel rays that pass thrugh the lens t refract away s that the rays when prjected backward wuld cme f a pnt called the fcus, f whch s ne the same sde as the rgnal parallel rays. The same equatn that wrks wth the cnvergng lens s used wth a dvergng lens. s s f

2 SPH3UW Unt 7.7 Cncave Lens Page 2 f 7 Sgn cnventn The sgn cnventn fr lenses s smlar t that fr mrrrs. Agan, take the sde f the lens where the bject s t be the pstve sde. Because a lens transmts lght rather than reflectng t lke a mrrr des, the ther sde f the lens s the pstve sde fr mages. In ther wrds, f the mage s n the far sde f the lens as the bject, the mage dstance s pstve and the mage s real. If the mage and bject are n the same sde f the lens, the mage dstance s negatve and the mage s vrtual. Fr cnvergng mrrrs, the fcal length s pstve. Smlarly, a cnvergng lens always has a pstve f, and a dvergng lens has a negatve f. The sgns asscated wth magnfcatn als wrk the same way fr lenses and mrrrs. A pstve magnfcatn crrespnds t an uprght mage, whle a negatve magnfcatn crrespnds t an nverted mage. As usual, uprght and nverted are taken relatve t the rentatn f the bject. Nte that n certan cases nvlvng mre than ne lens the bject dstance can be negatve. Ths ccurs when the mage frm the frst lens les n the far sde f the secnd lens; that mage s the bject fr the secnd lens, and s called a vrtual bject. Real bject (n frnt f lens) Vrtual bject (behnd lens) Real mage (behnd lens) Vrtual mage (n frnt f lens) Centre f curvature (back f lens) Centre f curvature (n frnt f lens) Dvergng lens Cnvergng lens s s pstve s s negatve s' s pstve s' s negatve r 1, r 2 s pstve r 1, r 2 s negatve f s negatve f s pstve

3 SPH3UW Unt 7.7 Cncave Lens Page 3 f 7 Step-by-Step Methd fr Drawng Ray Dagrams The methd f drawng ray dagrams fr a duble cncave lens s descrbed belw. 1. Pck a pnt n the tp f the bject and draw three ncdent rays travelng twards the lens. Usng a straght edge, accurately draw ne ray s that t travels twards the fcal pnt n the ppste sde f the lens; ths ray wll strke the lens befre reachng the fcal pnt; stp the ray at the pnt f ncdence wth the lens. Draw the secnd ray such that t travels exactly parallel t the prncpal axs. Draw the thrd ray t the exact centre f the lens. Place arrwheads upn the rays t ndcate ther drectn f travel. 2. Once these ncdent rays strke the lens, refract them accrdng t the three rules f refractn fr duble cncave lenses. The ray that travels twards the fcal pnt wll refract thrugh the lens and travel parallel t the prncpal axs. Use a straght edge t accurately draw ts path. The ray whch traveled parallel t the prncpal axs n the way t the lens wll refract and travel n a drectn such that ts extensn passes thrugh the fcal pnt n the bject's sde f the lens. Algn a straght edge wth the pnt f ncdence and the fcal pnt, and draw the secnd refracted ray. The ray whch traveled t the exact center f the lens wll cntnue t travel n the same drectn. Place arrwheads upn the rays t ndcate ther drectn f travel. The three rays shuld be dvergng upn refractn. 3. Lcate and mark the mage f the tp f the bject. The mage pnt f the tp f the bject s the pnt where the three refracted rays ntersect. Snce the three refracted rays are dvergng, they must be extended behnd the lens n rder t ntersect. Usng a straght edge, extend each f the rays usng dashed lnes. Draw the extensns untl they ntersect. All three extensns shuld ntersect at the same lcatn. The pnt f ntersectn s the mage pnt f the tp f the bject. The three refracted rays wuld appear t dverge frm ths pnt. Ths s merely the pnt where all lght frm the tp f the bject wuld appear t dverge frm after refractng thrugh the duble cncave lens. Of curse, the rest f the bject has an mage as well and t can be fund by applyng the same three steps t anther chsen pnt. See nte belw.

4 SPH3UW Unt 7.7 Cncave Lens Page 4 f 7 4. Repeat the prcess fr the bttm f the bject. The gal f a ray dagram s t determne the lcatn, sze, rentatn, and type f mage whch s frmed by the duble cncave lens. Typcally, ths requres determnng where the mage f the upper and lwer extreme f the bject s lcated and then tracng the entre mage. After cmpletng the frst three steps, nly the mage lcatn f the tp extreme f the bject has been fund. Thus, the prcess must be repeated fr the pnt n the bttm f the bject. If the bttm f the bject les upn the prncpal axs (as t des n ths example), then the mage f ths pnt wll als le upn the prncpal axs and be the same dstance frm the lens as the mage f the tp f the bject. At ths pnt the cmplete mage can be flled n. Sme students have dffculty understandng hw the entre mage f an bject can be deduced nce a sngle pnt n the mage has been determned. If the bject s merely a vertcal bject (such as the arrw bject used n the example belw), then the prcess s easy. The mage s merely a vertcal lne. Ths s llustrated n the dagram belw. In thery, t wuld be necessary t pck each pnt n the bject and draw a separate ray dagram t determne the lcatn f the mage f that pnt. That wuld requre a lt f ray dagrams as llustrated n the dagram belw. A dvergng lens wll always prduce a vrtual mage (Remember an eye cannt dstngush the dfference between a vrtual mage r a real mage). As the bject mves clser and clser t the dvergng lens, the mage (whch starts at the fcus as a dt) als mves clser and clser t the lens (t als ncreases n sze as t appraches the lens).

5 SPH3UW Unt 7.7 Cncave Lens Page 5 f 7 Example A 4 cm bject s placed 30.0 cm frm a dvergng lens f fcal length 10.0 cm. Where s the mage f the bject prduced? What s the sze and rentatn s the mage? Slutn: f s s 10cm 30cm s s 10cm 30cm 4 cm 30 s 7.5cm Therefre the mage s a vrtual mage lcated n the same sde as the bject. The mage s 7.5 cm frm the lens. Nw fr the magnfcatn. s m s 7.5cm 30cm 0.25 The pstve value n the magnfcatn ndcates that the mage s n the same rentatn as the rgnal, but s ¼ the sze r 1 cm n heght.

6 SPH3UW Unt 7.7 Cncave Lens Page 6 f 7 Example A 4 cm bject s placed at the fcal pnt a dvergng lens f fcal length 10.0 cm. Where s the mage f the bject prduced? What s the sze and rentatn s the mage? Slutn: f s s 10cm 10cm s s 10cm 10cm 2 cm 10 s 5.0cm Therefre the mage s a vrtual mage lcated n the same sde as the bject. The mage s 5.0 cm frm the lens. Nw fr the magnfcatn. s m s 5.0cm 10cm 0.5 Ntce that f ths was a cnvergng lens, we wuld have an mage lcated at nfnty. The pstve value n the magnfcatn ndcates that the mage s n the same rentatn as the rgnal, but s 1/2 the sze r 2 cm n heght.

7 SPH3UW Unt 7.7 Cncave Lens Page 7 f 7 Extra Ntes and Cmments