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Ethelbert Roberts
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1 Otical Instrumnts 363 ANSWERS TO CONCEPTUAL QUESTIONS Th bjctiv lns th micrsc must rm a ral imag just insid th cal int th yic lns In rdr r this t ccur, th bjct must b lcatd just utsid th cal int th bjctiv lns Sinc th cal lngth th bjctiv lns is tyically quit shrt ( ~ cm), this mans that th micrsc can cus rrly nly n bjcts cls t th nd th barrl and will b unabl t cus n bjcts acrss th rm 4 Fr a lns t rat as a siml magniir, th bjct shuld b lcatd just insid th cal int th lns I th wr th lns is +00 ditrs, its cal lngth is P + 00 m 00 m m 5 0 Th bjct shuld b lacd slightly lss than 50 in rnt th lns 6 Th artur a camra is a cls arximatin t th iris th y Th rtina th y crrsnds t th ilm th camra, and a cls arximatin t th crna th y is th lns th camra 8 Yu want a ral imag rmd at th lcatin th ar T rm such an imag, th bjct distanc must b gratr than th cal lngth th lns 0 Undr lw ambint light cnditins, a htlash unit is usd t insur that light ntring th camra lns will dlivr suicint nrgy r a rr xsur t ach ara th ilm Thus, th mst imrtant critrin is th additinal nrgy r unit ara (rduct intnsity and th duratin th lash, assug this duratin is lss than th shuttr sd) rvidd by th lash unit Th angular magniicatin rducd by a siml magniir is m 5 cm Nt that this is rrtinal t th tical wr a lns, P, whr th cal lngth is xrssd in mtrs Thus, i th wr th lns is dubld, th angular magniicatin will als dubl PROBLEM SOLUTIONS 5 Th -numbr (r cal rati) a lns is dind t b th rati cal lngth th lns t its diamtr Thrr, th -numbr th givn lns is 8 cm -numbr D I a camra has a lns with cal lngth 55 mm and can rat at -numbrs that rang rm t, th artur diamtrs r th camra must rang rm t D D -numbr -numbr 55 mm 55 mm 5 mm 46 mm

2 364 Chatr 5 53 Th thin lns quatin, +, givs th imag distanc as q q ( 00 m )( 5 0 mm ) 50 mm 3 00 m m Frm th magnitud th latral magniicatin, M h h q, whr th hight th imag is h m 9 0 mm, th hight th bjct (th building) must b 00 m h h ( 9 0 mm) q 5 0 mm 77 m 54 Cnsidr rays cg rm sit dgs th bjct and assing undviatd thrugh th cntr th lns as shwn at th right Fr a vry distant bjct, th imag distanc quals th cal lngth th lns I th angular width th bjct is q, th ull imag width n th ilm is 0 h tan ( θ ) ( 55 0 mm ) tan 9 mm s th imag asily its within a 35 mm by 350 mm ara 55 Th xsur tim is bing rducd by a actr t t 56 s 3 s 8 Thus, t maintain crrct xsur, th intnsity th light raching th ilm shuld b incrasd by a actr 8 This is dn by incrasing th ara th artur by a actr 8, s in trms th diamtr, π D 4 8( π D 4) r D 8 D Th nw -numbr will b -numbr ( -numbr) 40 4 r 4 D 8 D 8 8

3 Otical Instrumnts (a) Th intnsity is a masur th rat at which nrgy is rcivd by th ilm r unit ara th imag, r I A imag Cnsidr an bjct with hrizntal and vrtical dimnsins hxand hy as shwn at th right I th vrtical dimnsin intrcts angl q, th vrtical dimnsin th imag is hy qθ, r hy q Similarly r th hrizntal dimnsin, hx q, and th ara th imag is A h imag xh y q Assug a vry distant bjct, q, s A imag and w cnclud that I Th intnsity th light raching th ilm is als rrtinal t th crss-sctinal ara th lns and hnc t th squar th diamtr that lns, r I D Cmbining this with ur arlir cnclusin givs D I ( D ) r I ( -numbr ) Th ttal light nrgy hitting th ilm is rrtinal t th rduct intnsity and xsur tim, It Thus, t maintain crrct xsur, this rduct must b kt cnstant, r It It, giving I t ( -numbr ) t I ( -numbr) t s 00 s 57 Sinc th xsur tim is unchangd, th intnsity th light raching th ilm must b dubld i th nrgy dlivrd is t b dubld Using th rsult Prblm 6 (art a), w btain I ( -numbr) -numbr I 6, r -numbr Thus, yu shuld us th 80 stting n th camra 58 Th imag must always b cusd n th ilm, s th imag distanc is th distanc btwn th lns and th ilm Frm th thin lns quatin, + q, th bjct distanc is q ( q ), and th rang bjct distancs this camra can wrk with is rm t q q ( 0 mm )( 75 mm ) 0 mm 75 mm mm 05 m q q ( 80 mm )( 75 mm ) 80 mm 75 mm mm 630 m

4 366 Chatr 5 59 Th crrctiv lns must rm an uright, virtual imag at th nar int th y (i, q 60 in this cas) r bjcts lcatd 5 in rnt th y ( +5 ) Frm th thin lns quatin, + q, th rquird cal lngth th crrctiv lns is + ( 5 )( 60 ) 5 60 q and th wr (in ditrs) this lns will b P + 33 ditrs in mtrs m cm 50 (a) Th rsn is arsightd, abl t s distant bjcts but unabl t cus n bjcts at th nrmal nar int r a human y (c) (d) With th crrctiv lns 0 in rnt th y, th bjct distanc r an bjct 0 in rnt th y is Th uright, virtual imag rmd by th crrctiv lns will srv as th bjct r th y, and this bjct must b 40 in rnt th y With th lns 0 in rnt th y, th magnitud th imag distanc r th lns will b q Th imag must b lcatd in rnt th crrctiv lns, s it is a virtual imag and th imag distanc is ngativ Thus, q 38 () Frm th thin lns quatin, + q, th rquird cal lngth th crrctiv lns is + ( 8 )( 38 ) 8 38 q + 34 cm () (g) Th wr th crrctiv lns is thn P + 9 ditrs in mtrs m With a cntact lns, th lns t y distanc wuld b zr, s w wuld hav 0, q 40, giving a rquird cal lngth + ( 0 )( 40 ) q 0 40 and a wr in ditrs P + 50 ditrs in mtrs m His lns must rm an uright, virtual imag a vry distant bjct ( ) at his ar int, 80 in rnt th y Thrr, th cal lngth is q 80 I this lns is t rm a virtual imag at his nar int (q 8 ), th bjct distanc must b q ( q 8 )( 80 ) cm

5 Otical Instrumnts (a) Whn th child clarly ss bjcts at hr ar int 5 cm, th lns-crna cmbinatin has assumd a cal lngth suitabl r rg th imag n th rtina q 00 cm Th thin lns quatin givs th tical wr undr ths cnditins as P ar ditrs in mtrs q 5 m m Whn th y is cusd q 00 cm n bjcts at hr nar int 0, th tical wr th lns-crna cmbinatin is P + nar in mtrs m + q m ditrs clarly, hr yglass lns must rm an I th child is t s vry distant bjcts rct virtual imag at th ar int hr y ( q 5 cm ) Th tical wr th rquird lns is P ditrs q 5 in mtrs m Sinc th wr, and hnc th cal lngth, this lns is ngativ, it is divrging 53 (a) Th lns shuld rm an uright, virtual imag at th ar int q 50 r vry distant bjcts Thrr, q 50, and th rquird wr is P 00 ditrs m I this lns is t rm an uright, virtual imag at th nar int th unaidd y q 3, th bjct distanc shuld b ( ) 7 6 q q cm 54 (a) Ys, a singl lns can crrct th atint s visin Th atint nds crrctiv actin in bth th nar visin (t allw clar viwing bjcts btwn 45 and th nrmal nar int 5 cm) and th distant visin (t allw clar viwing bjcts mr than 85 away) A singl lns slutin is r th atint t war a bical r rgrssiv lns Altrnatly, th atint must urchas tw airs glasss, n r rading, and n r distant visin T crrct th nar visin, th lns must rm an uright, virtual imag at th atint s nar int (q 45 ) whn a ral bjct is at th nrmal nar int ( +5 ) Th thin lns quatin givs th ndd cal lngth as + ( 5 )( 45 ) 5 45 q cm s th rquird wr in ditrs is P + 78 ditrs in mtrs m cntinud n nxt ag

6 368 Chatr 5 (c) T crrct th distant visin, th lns must rm an uright, virtual imag at th atint s ar int (q 85 ) r th mst distant bjcts ( ) Th thin lns quatin givs th ndd cal lngth as q 85, s th ndd wr is P 8 ditrs in mtrs m 55 Cnsidring th imag rmd by th crna as a virtual bjct r th imlantd lns, w hav ( cm ) 5 33 cm and q +80 cm Th thin lns quatin thn givs th cal lngth th imlantd lns as + q 533 cm 80 cm 533 cm + 80 cm +590 cm s th wr is P ditrs m 56 (a) Th ur rtin th lns shuld rm an uright, virtual imag vry distant bjcts ( ) at th ar int th y ( q 5 m ) Th thin lns quatin thn givs q 5 m, s th ndd wr is P 067 ditrs 5 m Th lwr art th lns shuld rm an uright, virtual imag at th nar int th y q 3 whn th bjct distanc is 5 cm Frm th thin lns quatin, + ( 5 cm )( 3 ) q 5 cm cm +5 m Thrr, th wr is P ditrs + 5 m 57 Th crrctiv lns shuld rm an uright, virtual imag at th wman s ar int (q 40 ) r a vry distant bjct ( ) Th thin lns quatin givs th rquird cal lngth as q m Sinc < 0, it is a divrging lns, and th rquird wr is P 50 ditrs in mtrs m 58 (a) 0 50 m 5 P 400 ditrs Th crrctiv lns rm virtual imags vry distant bjcts ( ) at q 5 Thus, th rsn must b vry narsightd, unabl t s bjcts clarly whn thy ar vr ( ) cm 70 cm in rnt th y (c) I cntact lnss ar t b wrn, th ar int th y will b 7 in rnt th lns, s th ndd cal lngth will b q 7, and th wr is P 370 ditrs in mtrs 0 70 m

7 Otical Instrumnts (a) Th siml magniir (a cnvrging lns) is t rm an uright, virtual imag lcatd 5 cm in rnt th lns q 5 cm Th thin lns quatin thn givs q q 5 cm 7 5 cm + 58 cm 5 cm 7 5 cm s th stam shuld b lacd 58 cm in rnt th lns Whn th imag is at th nar int th y, th angular magniicatin rducd by th siml magniir is m m + 5 cm 5 cm cm 50 (a) Th imum magniicatin a siml magniir is m + ( 5 cm) Thus, i m +60, th cal lngth th lns is 5 cm 5 cm 50 cm m 60 Whil using a siml magniir, th y is mst rlaxd i th lns rms th virtual imag at ininity (s aralll rays mrg rm th lns) rathr than at th nar int th y Undr ths cnditins, th magniicatin rducd is m 5 cm 5 cm cm 5 (a) Frm th thin lns quatin, + ( 35 )( 5 ) q cm With th imag at th nrmal nar int, th angular magniicatin is m m cm 5 (a) Whn th bjct is at th cal int th magniying lns, a virtual imag is rmd at ininity and aralll rays mrg rm th lns Undr ths cnditins, th y is mst rlaxd and th magniicatin rducd is m 5 cm 5 cm cm Whn th bjct is sitind s th magniir rms a virtual imag at th nar int th y (q 5 cm), imum magniicatin is rducd and this is m + 5 cm 5 cm cm (c) Frm th thin lns quatin, th bjct distanc ndd t yild th imum magniicatin cmutd in art abv is q q 5 cm 5 5 cm 5 4 cm

8 370 Chatr 5 53 (a) Frm th thin lns quatin, a ral invrtd imag is rmd at an imag distanc q ( 7 )( 39 ) cm s th latral magniicatin rducd by th lns is M h q 86 5 cm h 7 and th magnitud is M I h is th actual lngth th la, th small angl arximatin givs th angular width th la whn viwd by th unaidd y rm a distanc d 6 cm cm as h h θ 0 d 97 cm Th lngth th imag rmd by th lns is h M h h, and its angular width whn viwd rm a distanc d 6 cm q 39 5 cm is h θ d h 395 cm Th angular magniicatin achivd by viwing th imag instad viwing th la dirctly is θ h 395 cm 97 cm h 97 cm 395 cm θ (a) With th imag at th nrmal nar int ( q 5 cm ), th angular magniicatin is m + 5 cm + 5 cm cm Whn th y is rlaxd, aralll rays ntr th y and 5 cm 5 cm m cm 55 Th vrall magniicatin is m M m M 5 cm whr M is th latral magniicatin rducd by th bjctiv lns Thrr, th rquird cal lngth r th yic is M 5 cm 5 cm m 40 cm 56 Th arximat vrall magniicatin a cmund micrsc is givn by m ( L )( 5 ), whr L is th distanc btwn th bjctiv and yic lnss, whil and ar th cal lngths th bjctiv and yic lnss, rsctivly Thus, th dscribd micrsc shuld hav an arximat vrall magniicatin L 5 m cm

9 Otical Instrumnts Th magnitud th magniicatin a tlsc is m, whr and ar th cal lngths th bjctiv lmnt and th yic, rsctivly Thus, i m 45 and 40 cm, th cal lngth th bjctiv must b m ( 45)( 4 ) 8 Th vrall lngth th tlsc will thrr b L cm 84 m 58 It is sciid that th inal imag th micrsc rms th bld cll is 9 in rnt th y and that th diamtr this imag intrcts an angl θ 43 mrad Th diamtr this inal imag must thn b ( ) h rθ m rad m At this int, it is tmting t us Equatin (57) rm th txtbk r th vrall magniicatin a cmund micrsc, and cmut h h m as th siz th bld cll srving as th bjct r th micrsc Hwvr, th drivatin that quatin is basd n svral assumtins, n which is that th y is rlaxd and viwing a inal imag lcatd an ininit distanc in rnt th yic This is clarly nt tru in this cas, and th us Equatin (57) wuld intrduc cnsidrabl rrr Instad, w shall rturn t basics and us th thin lns quatin t ind th siz th riginal bjct Th imag rmd by th bjctiv lns is th bjct r th yic, and w labl th siz this imag as h Th latral magniicatin th bjctiv lns is M h h q and that th yic is M h h q Th vrall magniicatin rducd by th micrsc is M h h h h h h which givs th siz th riginal bjct as h h M Frm th thin lns quatin, th rquird bjct distanc r th yic is q q and th magniicatin rducd by th yic is M q Th imag distanc r th bjctiv lns is thn q L cm and th bjct distanc r this lns is q q 8 cm 6 cm 8 cm 6 cm Th magniicatin by th bjctiv lns is M q 8 cm 7 cm cm ( + ) and th vrall latral magniicatin is M MM Th siz th rd bld cll srving as th riginal bjct is h 45 0 h M 53 4 m m μm

10 37 Chatr 5 59 Sm th arximatins mad in th txtbk whil driving th vrall magniicatin a cmund micrsc ar nt valid in this cas Thrr, w start with th yic and wrk backwards t dtr th vrall magniicatin I th y is rlaxd, th yic imag is at ininity q, s th bjct distanc is 50 cm, and th angular magniicatin by th yic is 5 5 m cm Th imag distanc r th bjctiv lns is thn q L cm q 5 cm 0 and th bjct distanc is q 5 cm 0, and th vrall magni- Th magniicatin by th bjctiv lns is M icatin th micrsc is m Mm q 09 5 cm 09 cm cm (a) Fr a rracting tlsc, th vrall lngth is L + and th magniicatin rducd is m, whr and ar th cal lngths th bjctiv lmnt and th yic, rsctivly Thus, w may writ m t btain L m m m m Using th rsult art (a), th rquird chang in th lngth th tlsc will b ΔL m + m + m m 0 00 ( 00 m) cm cm r th tlsc must b shrtnd by mving th yic 0 rward tward th bjctiv lns 53 Th lngth th tlsc is L + 9 cm and th angular magniicatin is m 45 Thrr, 45 and L cm, giving 0 cm and 9 cm r 9

11 Otical Instrumnts Th mn may b cnsidrd an ininitly distant bjct viwd with this lns, s th imag distanc will b q 50 whn Cnsidring th rays that ass undviatd thrugh th cntr this lns as shwn in th sktch, bsrv that th angular widths th imag and th bjct ar qual Thus, i w is th linar width an bjct rg a 0 wid imag, thn w 0 cm 0 cm θ m 50 r w cm m 0 mi m 6 0 mi 533 (a) Frm th thin lns quatin, q, s th latral magniicatin by th bjctiv lns is M h h q ( ) Thrr, th imag siz will b h h M h h h I >>, thn and h (c) Sus th tlsc bsrvs th sac statin at th znith h Thn, ( 4 00 m )( 08 6 m ) h m 07 mm m 534 Us th largr cal lngth (lwst wr) lns as th bjctiv lmnt and th shrtr cal lngth (largst wr) lns r th y ic Th cal lngths ar m, and + 0 m + 0 ditrs ditrs (a) Th angular magniicatin (r magniying wr) th tlsc is thn m m + 0 m Th lngth th tlsc is 750 L m + 0 m m

12 374 Chatr Th lns r th lt y rms an uright, virtual imag at q L 50 whn th bjct distanc is L 5, s th thin lns quatin givs its cal lngth as L L L + q L L ( 5 )( 50 ) Similarly r th thr lns, q R 0 whn R 5, and R 33 3 cm (a) Using th lns r th lt y as th bjctiv, m L cm R Using th lns r th right y as th yic and, r imum magniicatin, rquiring that th inal imag b rmd at th nrmal nar int q 5 givs q q cm cm cm Th imum magniicatin by th yic is thn m cm and th imag distanc r th bjctiv is q L 0 43 cm 4 3 cm Th thin lns quatin thn givs th bjct distanc r th bjctiv as q q 43 5 cm cm Th magniicatin by th bjctiv is thn q ( 43 M cm ) + 40 cm and th vrall magniicatin is m Mm ( + )( + 75 ) 9 cm 536 Nt: W slv art br answring art (a) in this rblm Th bjctiv rms a ral, diishd, invrtd imag a vry distant bjct at q This imag is a virtual bjct r th yic at, giving + 0 q and q cntinud n nxt ag

13 Otical Instrumnts 375 (a) Paralll rays mrg rm th yic, s th y bsrvs a virtual imag (c) Th angular magniicatin is m 300, giving 300 Als, th lngth th tlsc is L cm, giving cm and cm 537 I just rslvd, th angular saratin is λ m θ θ D m Thus, th altitud is rad d 00 m 5 h m 49 km 6 θ 03 0 rad 538 Fr a narrw slit, Rayligh s critrin givs λ m 3 θ mrad 3 a m λ 539 Th limit rslutin in air is θ 060 μ D In il, th limiting angl rslutin will b λ λ il ( nil ) λ θ il D D D air rad n il r θ il θ air 060 μrad n 5 il 040 μrad 540 (a) Th wavlngth th light within th y is λn λ n Thus, th limiting angl rslutin r light assing thrugh th uil (a circular artur with diamtr D 00 mm) is λ λ m n θ 3 D nd ( 33 ) 00 0 m 9 0 Frm s rθ, th distanc rm th y that tw ints saratd by a distanc s 00 cm will intrct this imum angl rslutin is s 00 cm 3 r m 4 θ 9 0 rad 54 Th angular saratin th hadlights whn viwd rm a distanc r 0 0 km is s 00 m 4 θ 00 0 rad 3 r 00 0 m I th hadlights ar t b just rslvd, this saratin must qual th limiting angl rslutin r th circular artur, θ λ D, s th diamtr th artur is λ λ m D 4 θ θ 00 0 rad m 540 mm 4 rad

14 376 Chatr 5 54 Diractin ccurs whn wavs ass thrugh an artur, causing th intnsity t g thrugh ima and ima as n gs rm th cntr th bam utward as illustratd in th igur at th right Th angular saratin th irst imum rm th cntral imum is a cnstant dtrd by th dimnsin th artur, th wavlngth th wav, and th sha th artur Fr a circular artur, this angular saratin is givn by θ λ D, whr D is th diamtr th artur Th ull angular width th cntral imum is thn α θ 44λ D Th latral width th cntral imum, d, incrass as th distanc r rm th artur incrass Whn a bam lasr light having wavlngth λ 63 8 nm diracts thrugh a circular ning diamtr D 0 0, w stimat th diamtr th bam at distanc r 300 km ast th ning as qual t th diamtr th cntral imum in th diractin attrn at this lcatin This givs d r r 44λ α D ( m) m 3 m m λ 543 I just rslvd, th angular saratin th bjcts is θ θ D m and s rθ ( 80 0 km) 500 m 98 km λ 544 I just rslvd, th angular saratin th bjcts is θ θ D and s rθ ( 00 0 m) 035 m m 038 m 38cm cm Th grating sacing is d m, and th highst rdr nm light that can b bsrvd is m 6 d sin m 78 rdrs λ m Th ttal numbr slits is N ( ) slits cm , and th rslving wr th grating in th scnd rdr is R availabl Nm Th rslving wr rquird t sarat th givn sctral lins is R ndd λ nm Δλ 0003 nm Ths lins cannt b saratd with this grating

15 Otical Instrumnts 377 λ 546 Th rslving wr a diractin grating is R Nm Δλ (a) Th numbr lins th grating must hav t rslv th H α lin in th irst rdr is N R m λ Δλ 656 nm 08 nm In th scnd rdr ( m ), N R lins 656 nm 08 nm lins 547 A ring shit ccurs whn th mirrr mvs distanc λ 4 Thus, i th mirrr mvs distanc ΔL 0 80 mm, th numbr ring shits bsrvd is N shits ΔL λ ΔL m λ m ring shits 548 (a) Whn th cntral st in th intrrmtr attrn gs thrugh a ull cycl rm bright t dark and back t bright, tw ring shits hav ccurrd and th mvabl mirrr has mvd a distanc ( λ 4) λ Thus, i N cycls 700 such cycls ar bsrvd as th mirrr mvs distanc d 0 38 mm, it must b tru that λ d Ncycls r λ d N cycls and th wavlngth th light illuating th intrrmtr is 3 ( m) λ m 449 nm which is in th blu rgin th visibl sctrum Rd light has a lngr wavlngth than blu light, s wr wavlngths wuld cvr th givn dislacmnt, hnc N cycls wuld b smallr 549 A ring shit ccurs whn th mirrr mvs distanc λ 4 Thus, th distanc mvd (lngth th bactrium) as 30 shits ccur is ΔL N shits λ m m 50 4 μm 550 A ring shit ccurs whn th mirrr mvs distanc λ 4 Thus, th distanc th mirrr mvs as 50 ring shits ar cuntd is ΔL N shits λ m m 39 6 μm

16 378 Chatr 5 55 Whn th tical ath lngth that light must travl as it gs dwn n arm a Michlsn s intrrmtr changs by n wavlngth, ur ring shits will ccur (n shit r vry quartr-wavlngth chang in ath lngth) Th numbr wavlngths (in a vacuum) that it in a distanc qual t a thicknss t is N vac t λ Th numbr wavlngths that it in this thicknss whil travling thrugh th transarnt matrial is Nn t λn t ( λ n) nt λ Thus, th chang in th numbr wavlngths that it in th ath dwn this arm th intrrmtr is t ΔN Nn Nvac ( n) λ and th numbr ring shits that will ccur as th sht is insrtd will b t # ring shits 4( ΔN) 4( n) 4( 40) m λ m A ring shit will ccur ach tim th ctiv lngth th tub changs by a quartr a wavlngth (that is, r ach additinal wavlngth ittd int th lngth th tub, 4 ring shits ccur) I L is th lngth th tub, th numbr ring shits bsrvd as th tub is illd with gas is N Hnc, n shits gas L L L L L 4 4 n n gas 4 λ λ λ λ λ n gas L N m + shits λ m ( 60 ) (a) Fr a rracting tlsc, th magniicatin is m, whr and ar th cal lngths th bjctiv lns and th yic, rsctivly Thus, whn th Yrks tlsc uss an yic with 50 cm, th magniicatin is m 0 0 m m Standard astrnmical tlscs rm invrtd imags Thus, th bsrvr Martian lar cas ar usid dwn 554 Whn viwd rm a distanc 50 mtrs, th angular lngth a mus (assumd t hav an actual lngth ) is s 00 m θ radians r 50 m Thus, th limiting angl rslutin th y th hawk must b θ θ rad

17 Otical Instrumnts Th rslving wr th grating is R λ Δ λ Nm Thus, th ttal numbr lins ndd n th grating t rslv th wavlngths in rdr m is (a) N R m m λ ( Δλ ) Fr th sdium dublt in th irst rdr, nm 0 59 nm N nm nm In th third rdr, w nd N (a) Sinc this y can alrady cus n bjcts lcatd at th nar int a nrmal y (5 cm), n crrctin is ndd r nar bjcts T crrct th distant visin, a crrctiv lns (lcatd rm th y) shuld rm virtual imags vry distant bjcts at 3 cm in rnt th lns (r at th ar int th y) Thus, w must rquir that q 3 cm whn This givs P ditrs q 03 m A crrctiv lns in cntact with th crna shuld rm virtual imags vry distant bjcts at th ar int th y Thrr, w rquir that q 5 cm whn, giving P ditrs q 05 m Whn th cntact lns 5 cm is in lac, th bjct distanc which yilds a P virtual imag at th nar int th y (that is, q 6 cm) is givn by ( ) q q 6 cm 5 cm 6 cm 5 cm 44 cm 557 (a) Th lns shuld rm an uright, virtual imag at th nar int th y q 75 whn th bjct distanc is 5 Th thin lns quatin thn givs + ( 5 )( 75 ) 5 75 q 37 5 cm m s th ndd wr is P + 67 ditrs m I th bjct distanc must b 6 t sitin th imag at q 75, th actual cal lngth is + ( 6 )( 75 ) q 6 75 and P + 5 ditrs m m Th rrr in th wr is ΔP 67 5 ditrs 0 6 ditrs t lw

18 380 Chatr (a) I q 00cm whn 00 m 0, th thin lns quatin givs th cal lngth as + ( 0 )( 0 ) q 9 6 cm Th -numbr a lns artur is th cal lngth th lns dividd by th diamtr th artur Thus, th smallst -numbr ccurs with th largst diamtr th artur Fr th tyical yball cusd n bjcts 00 m away, this is 96 cm ( -numbr) D (c) Th largst -numbr th tyical yball cusd n a 00-m-distanc bjct is 96 cm ( -numbr) D (a) Th imlantd lns shuld giv an imag distanc q 4 mm r distant bjcts Th thin lns quatin thn givs th cal lngth as q 4 mm, s th wr th imlantd lns shuld b P imlant ditrs m Whn th bjct distanc is 33, th crrctiv lns shuld rduc aralll rays ( q ) Thn th imlantd lns will cus th inal imag n th rtina Frm th thin lns quatin, th rquird cal lngth is 33, and th wr this lns shuld b 560 W us n n n + q P crrctiv ditrs m n R R q n n n, with and q qual t th crna t rtina distanc Thn, 00 cm cm 5 07 mm 56 Whn a cnvrging lns rms a ral imag a vry distant bjct, th imag distanc quals th cal lngth th lns Thus, i th scut startd a ir by cusing sunlight n kindling 50 rm th lns, q 500 cm (a) Whn th lns is usd as a siml magniir, imum magniicatin is rducd whn th uright, virtual imag is rmd at th nar int th y (q 5 cm in this cas) Th bjct distanc rquird t rm an imag at this lcatin is q q 5 cm 5 5 cm 5 cm 5 40 q and th latral magniicatin rducd is M 5 cm cm 40 cntinud n nxt ag

19 Otical Instrumnts 38 Whn th bjct is viwd dirctly whil sitind at th nar int th y, its angular siz is θ 0 h 5 cm Whn th bjct is viwd by th rlaxd y whil using th lns as a siml magniir (with th bjct at th cal int s aralll rays ntr th y), th angular siz th uright, virtual imag is θ h Thus, th angular magniicatin gaind by using th lns is h m θ h 5 cm θ 5 cm cm 5 56 Th angular magniicatin is m θθ, whr θ is th angl subtndd by th inal imag, and θ is th angl subtndd by th bjct as shwn in th igur Whn th tlsc is adjustd r imum ystrain, th rays ntring th y ar aralll Thus, th bjctiv lns must rm its imag at th cal int th yic Frm triangl ABC, θ tanθ h q and rm triangl DEF, θ tanθ h Th angular θ h q magniicatin is thn m θ h q Frm th thin lns quatin, th imag distanc th bjctiv lns in this cas is q ( cm )( cm ) 4 cm 30 0 With an yic cal lngth 00 cm, th angular magniicatin r this tlsc is m q 4 cm 0 0 7

Magnifiers. 5.1 Optical Instruments - Polarization. Angular size

Magnifiers. 5.1 Optical Instruments - Polarization. Angular size 5.1 Otical Instrumnts - Plarizatin Otical Instrumnts Siml magniir Cmund micrsc Tlsc Wav tics Plarizatin Magniirs W magniy t imag a small bjct by bringing it cls t ur y. But w cannt bring it clsr tan t