Ch 35 Images Eunl Won Department of Physcs Korea Unversty Fundamentals of Physcs by Eunl Won, Korea Unversty
We wll cover... Plane mrrors Thn Lenses Three proofs Sphercal mrrors Sphercal refrtng Surfes Optcal Instruments (mcroscope,telescope) We are begnnng to learn geometrcal optcs Fundamentals of Physcs by Eunl Won, Korea Unversty 2
Mrrors Plane mrror: a flat reflectng surfe O: a pont source of lght, called object I : mage of the object O Object dstances p (>0) Image dstances (<0) = -p Extended objects: represented by an uprght arrow Vrtual mage I has same orentaton and heght Fundamentals of Physcs by Eunl Won, Korea Unversty 3
Sphercal Mrrors One could make sphercal mrrors by bendng plane mrrors Concave mrror C : center of curvature n front of mrror feld of vew: smaller than the case of plane mrror heght of I : greater than heght of O : > p Convex mrror C : center of curvature behnd mrror feld of vew: larger than the case of plane mrror heght of I : smaller than heght of O : < p Fundamentals of Physcs by Eunl Won, Korea Unversty 4
Focal Ponts of Sphercal Mrrors Focal pont: when parallel rays reh a mrror, they are reflected to a common pont F Focal length f: dstance from c (mrror) to F Concave mrror : real focal pont (f>0) Convex mrror : vrtual focal pont (f<0) f = /2 r (sphercal mrror) * r>0 for concave and r<0 for convex mrrors Fundamentals of Physcs by Eunl Won, Korea Unversty 5
Images from Sphercal Mrrors O nsde F: observer sees a vrtual mage of O vrtual mage: form on opposte sde of the object O on F: mage s ambguous and mperceptble h: heght of the object h : heght of the mage m: lateral magnfcaton m = h h * m > 0 f O and F has same orentaton m < 0 f O and F has opposte orentaton O outsde F: mage s nverted real mages: on the sde where the object s * For convex or plane mrrors only vrtual mage can be formed Later we wll prove followngs for sphercal mrrors p + = f m = p Fundamentals of Physcs by Eunl Won, Korea Unversty 6
Locatng Images (mrror) Drawng Rays:. A ray parallel to the central axs reflects through F 2. A ray passes F reflects parallel to the central axs 3. A ray passes C returns along tself 4. A ray reflects at c s reflected symmetrcally about the central axs The mage of pont s at the ntersecton of the rays of -4 From ray 4 n (b), trangle abc and dec are smlar: de ab = cd ca Left hand sde: -m (negatve magnfcaton) Rght hand sde: cd: ca: p m = p (lateral magnfcaton) Fundamentals of Physcs by Eunl Won, Korea Unversty 7
Sphercal Refrtng Surfes Our concern here s whether the lght rays form a real or vrtual mage Medum wth the greater ndex of refrton s shaded Object O s always n the medum wth n real mage: located opposte sde of O vrtual mage: located same sde of O Conventon: convex refrtng surfe : r>0 concave refrtng surfe : r<0 (note: ths s reverse of the sgn conventon we have for mrrors) We wll prove the followng later: Fundamentals of Physcs by Eunl Won, Korea Unversty 8 n p + n 2 = n 2 n r
Thn Lenses A lens s a transparent object wth two refrtng surfes whose central axes concde Conversng lens Dvergng lens Thn lens: thckness of lens << p,, r, r 2 (rad of curvature of two surfes of the lens) We wll also prove that: p + = ( ) and = (n ) f f r r 2 Fundamentals of Physcs by Eunl Won, Korea Unversty 9
Images from Thn Lenses Object O outsde F : a real nverted mage I Object O nsde F : a vrtual, same orentaton mage I Regardless of the object dstance: a vrtual mage I Real mages : form on the sde of a lens that s opposte the object Vrtual mages : form on the sde where the object s Fundamentals of Physcs by Eunl Won, Korea Unversty 0
Locatng Images (Lens). A ray parallel to the central axs of the lens : pass through focal pont F 2 2. A ray passes through F: emerge from lens parallel to the central axs 3. A ray drected center of lens: no change n ts drecton Fundamentals of Physcs by Eunl Won, Korea Unversty
Optcal Instruments: Smple Magnfer Smple Magnfyng Lens : Human eye can focus an object on the retna (at the rear of the eye) to a certan mnmum dstance (P n : near pont, ~25 cm for 20 years old person) One can restore the clarty by lookng at O through a convergng lens The angular magnfcaton: m θ = θ If we assume O s at the focal pont: θ h 25 cm, θ = h f Fundamentals of Physcs by Eunl Won, Korea Unversty 2 θ m θ 25 cm f (smple magnfer)
Optcal Instruments: Compound Mcroscope O to be vewed s pled just outsde F (p ~ f ob ) The separaton s arranged to have the enlarged, nverted real I just nsde the F (of the eyepece) Tube length s >> f ob, ~ s The lateral magnfcaton: m = p = s f ob The eyepece ts as a smple magnfyng lens: observer sees a fnal (vrtual, nverted) I The overall magnfcaton: M = mm θ = s f ob 25 cm f ey (mcroscope) Fundamentals of Physcs by Eunl Won, Korea Unversty 3
Optcal Instruments: Refrtng Telescope Telescopes are desgned to vew large objects, such as galaxes, stars, and planets, at large dstances (parallel rays are ncomng) 2nd focal pont F 2 concdes wth st focal pont F The angular magnfcaton: m θ = θ ey θ ob θ ob = h and θ ey = h f ob f ey m θ = f ob f ey (telescope) Fundamentals of Physcs by Eunl Won, Korea Unversty 4
Three proofs: sphercal mrror formula Concave sphercal mrror The sphercal mrror formula: p + = f We wrte three angles n radan measure: α co = p, Then the above equaton becomes An exteror angle of a trangle = sum of two opposte nteror angles β = α + θ and γ = α + 2θ α + γ = 2β β = cc = r, p + = 2 r = f γ ci = Fundamentals of Physcs by Eunl Won, Korea Unversty 5
Three proofs: Refrtng Surfe The refrtng surfe formula: n p + n 2 = n 2 n r From the law of refrton: n 2 sn θ 2 = n sn θ n θ n 2 θ 2 f α << From two trangles COa and ICa: θ = α + β and β = θ 2 + γ n (α + β) = n 2 (β γ) n α + n 2 γ = (n 2 n )β n p + n 2 = (n 2 n ) α Fundamentals of Physcs by Eunl Won, Korea Unversty 6 r p, β = r, n p + n 2 γ = n 2 n r
Three proofs: the thn-lens formulas We start wth (a): O s pled near the left surfe and makes mage I In (b): Left surfe makes vrtual mage I so we can use: n p + n 2 = n 2 n r p n = n r n =, n 2 = n, and =- (mage dstance s negatve) In (c): I becomes O by an observer at a and note that p = + L Fundamentals of Physcs by Eunl Won, Korea Unversty 7
Three proofs: the thn-lens formulas p = + L gves n and p + n 2 n + L + = n r = n 2 n r Thn lens approxmaton allows us to set L = 0, so Combnng two equatons (one from prevous slde): p + n + = n r Changng notatons as: (p p, ) we eventually get p + (n = n, n 2 = ) n + = n r p n = n r n + = n r = (n ) ( ) r r Fundamentals of Physcs by Eunl Won, Korea Unversty 8
Summary Real and Vrtual Images Sphercal Mrror: p + = f Sphercal Refrtng Surfe: n p + n 2 = n 2 n r Thn lens: p + = (n ) ( ) r r Fundamentals of Physcs by Eunl Won, Korea Unversty 9