Principles of Physics III

Transcription

1 Priniples of Physis III J. M. Veal, Ph. D. version Contents Mehanial Waves 3. Basis Speed Wave equation String wave s speed String wave s power Superposition Standing waves Homework Exerises Sound 3 2. Propagation speed Intensity Interferene Beats Doppler effet Homework Exerises Heat 4 3. Basis Thermal expansion Kilograms & moles Latent heat Transfer Homework Exerises Kineti Theory 4 4. Basis The van der Waals equation Pressure as ollisions Translational kineti energy Mean free path Moleular speeds Homework Exerises Exam First Law of Thermodynamis 5 5. Work Internal energy Proesses Speifi heats Ratio of speifi heats Ideal adiabats Homework Exerises Seond Law of Thermodynamis 5 6. Basis Internal-ombustion engines Refrigerators Two statements Carnot yle Entropy Introdution to statistial mehanis Homework Exerises Eletromagneti Waves 6 7. Basis Transmission Amplitude ratio Speed of light Energy transfer Radiation pressure Polarization

2 J. M. Veal, Priniples of Physis III Homework Exerises Preliminary Optis 6 8. Refletion & refration Chromati dispersion Total internal refletion Polarized refletions Homework Exerises Images 7 9. Basis Plane mirror Spherial mirrors Spherial lenses Thin lenses Homework Exerises Exam Interferene 8 0. Basis Fringes Intensity Thin films Mihelson s interferometer Homework Exerises Veloities Doppler effet Momentum and fore Energies Homework Exerises Photons 0 3. Basis Photoeletri effet Momentum Compton shift Probability waves Homework Exerises Matter Waves 0 4. Basis Unertainty priniple Wave funtions Shrödinger s equation Free partile Homework Exerises Exam Final Exam A All Formulae Diffration 9. Single-slit minima Single-slit intensity Cirular aperture Double slit Grating X-ray Homework Exerises Speial Relativity 9 2. Two postulates Spaetime Simultaneity Time dilation Length ontration Lorentz transformation

3 J. M. Veal, Priniples of Physis III 3 Mehanial Waves 2 Sound. Basis.2 Speed v λν.3 Wave equation Given a wave ypx, tq Y sin pkx ωtq having propagation speed v ω{k, derive the wave equation:.4 String wave s speed B 2 ypx, tq Bx 2 v 2 B 2 ypx, tq Bt 2. If a strethed string has tension T and linear mass density µ, show that a string wave s propagation speed is given by.5 String wave s power v a T {µ. Given de dk du and a string of linear mass density µ, show that a string wave desribed by ypx, tq Y sin pkx ωtq will have the following power:.6 Superposition.7 Standing waves.8 Homework Exerises P µvω 2 Y 2 os 2 pkx ωtq. View The Mehanial Universe and Beyond, 8: Waves. Read your text, hapter 5: Mehanial Waves. 2. Propagation speed Given the definition of the bulk modulus as B p{p v{vq, show that the propagation speed of a sound wave in a fluid of density ρ is desribed by d B v ρ. 2.2 Intensity Given that the osillation of a slie of air in a sound wave is desribed by spx, tq S sin pkx ωtq, and assuming that x 9 Ky x 9 Uy, show that the intensity of a sound wave is expressed by 2.3 Interferene 2.4 Beats I 2 ρvω2 S 2. Given two overlapping waves of different wavelengths, show that the beat frequeny is equal to f b f f Doppler effet Consider a soure and a detetor of sound waves. If they move, they move with speeds v s and v d, respetively. If the soure emits a wave of frequeny f, show that the detetor will reeive a frequeny given by where the signs are hosen suh that. 2.6 Homework Exerises f f v v d v v s, 9r 0 Ñ f f and 9r 0 Ñ f f View The Mehanial Universe and Beyond, 0. Fundamental Fores. Read your text, hapter 6: Sound and Hearing.

4 J. M. Veal, Priniples of Physis III 4 3 Heat 3. Basis 3.2 Thermal expansion Consider the thermal expansion of large area of material that is very thin. Given that the linear expansion is given by L αl 0 T, show that the area expansion is given by A 2αA 0 T. Consider the thermal expansion of a lattie struture. Given that the linear expansion is given by L αl 0 T, show that the volume expansion is given by V 3αV 0 T. 3.3 Kilograms & moles dq m dt C dq n dt Given the speifi heat p{mqdq{dt and the heat apaity C p{nqdq{dt, show that the ratio is equal to the molar mass of a substane. 3.4 Latent heat 3.5 Transfer 3.6 Homework Exerises View The Mehanial Universe and Beyond, 45: Temperature and Gas Laws. Read your text, hapter 7: Temperature and Heat. 4 Kineti Theory 4. Basis pv NkT 4.2 The van der Waals equation pp a η 2 qpη b q kt Given the van der Waals equation, show that the units work out as they should. Given the van der Waals equation, show that for low density it redues to the ideal gas law: pv NkT. Show that the van der Waals equation an be onverted to the alternate form, pp an 2 {V 2 qpv nbq nrt, where p is the gas pressure, a is a onstant depending on attrative intermoleular fores (units: J m 3 /mol 2 ), n is the number of moles of gas, V is the volume of the gas, b is roughly the volume per mole of gas, R is the gas onstant, and T is the gas temperature. 4.3 Pressure as ollisions Given the number density of gas partiles as η, the mass of a typial gas partile as m, and the typial gas partile s veloity omponent perpendiular to a surfae as v x, show that the pressure of the gas on that surfae is given by p ηmv 2 x. 4.4 Translational kineti energy Given that a gas partile s speed an be represented in terms of its omponents as v 2 v 2 x v 2 y v 2 z, show that the average translational kineti energy of a gas partile an be given by v rms b 3kT m 4.5 Mean free path xk tr y 3 2 kt. If a typial gas partile has volume 4πr 3 {3 and moves a distane xvydt in time dt, and the number density of the gas is η, show that the average distane between ollisions is given by xλy p4π? 2r 2 ηq.

5 J. M. Veal, Priniples of Physis III Moleular speeds m 3{2 P pvq 4π v 2 e mv2 {2kT 2πkT 5.5 Ratio of speifi heats Given that a moleule s average tranlational kineti energy is xk tr y 3kT {2, show that the ratio of speifi heats is given by v P 2kT m, xvy 4.7 Homework Exerises 8kT πm, v rms 3kT m View The Mehanial Universe and Beyond, 46: Engine of Nature. Read your text, hapter 8: Thermal Properties of Matter. γ p { V C p {C V 5.6 Ideal adiabats T V γ onst. pv γ onst. p V γ Exam Exam overs material up to here. 5 First Law of Thermodynamis 5. Work W» Vf V i pdv 5.2 Internal energy I Q W 5.3 Proesses 5.4 Speifi heats By omparing isohori heating from T to T 2 with isobari heating from T to T 2, show that the speifi heats are related: p V k{m. Given that an ideal gas behaves aording to I fpt q, show that pv γ α, where γ is the ratio of speifi heats and α is a onstant. (Hint: As an intermediate step, you will show that T V γ β, where β is a onstant.) 5.7 Homework Exerises View The Mehanial Universe and Beyond, 47: Entropy. Read your text, hapter 9: The First Law of Thermodynamis. 6 Seond Law of Thermodynamis 6. Basis 6.2 Internal-ombustion engines Consider the Otto yle for an internal-ombustion engine. a) Sketh, label, and explain the assoiated pv diagram. b) Use the definition of engine effiieny, the definition of speifi heat at onstant volume, and a property of adiabats to show that the effiieny of an Otto engine is given by e r γ, where r is the ompression ratio and γ is the ratio of speifi heats. Consider the Diesel yle for an internal-ombustion engine. Sketh, label, and explain the assoiated pv diagram.

6 J. M. Veal, Priniples of Physis III Refrigerators 6.4 Two statements 6.5 Carnot yle Consider a Carnot yle. a) By what means does this yle attain the maximum possible effiieny for an engine? b) Sketh, label, and explain the assoiated pv diagram. ) Use the definition of engine effiieny, an isothermal version of the first law of thermodynamis for an ideal gas, and a property of adiabats to show that the effiieny of a Carnot engine is given by 6.6 Entropy S» f i dq T e T T h. 6.7 Introdution to statistial mehanis S k ln w 6.8 Homework Exerises View The Mehanial Universe and Beyond, 48: Low Temperatures. Read your text, hapter 20: The Seond Law of Thermodynamis. 7 Eletromagneti Waves 7. Basis 7.2 Transmission 7.3 Amplitude ratio ¾ Beginning with Faraday s law, E d s Φ 9 B, show that the ratio of the strengths of the eletri and magneti fields in an eletromagneti wave is given by E m B m. 7.4 Speed of light ¾ 9 Beginning with Maxwell s law, B d s µ 0 ε 0 Φ E, show that the speed of the wave is given by 7.5 Energy transfer S µ 0 E B I µ 0 E 2 rms u E u B 7.6 Radiation pressure p r I 7.7 Polarization 7.8 Homework Exerises pµ 0 ε 0 q {2. View The Mehanial Universe and Beyond, 39: Maxwell s Equations. Read your text, hapter 32: Eletromagneti Waves, and hapter 33: The Nature and Propagation of Light (setion 5). 8 Preliminary Optis 8. Refletion & refration n sin θ n 2 sin θ 2 n {v

7 J. M. Veal, Priniples of Physis III Chromati dispersion 9.6 Homework Exerises 8.3 Total internal refletion θ sin n 2 n 8.4 Polarized refletions θ B tan n 2 n 8.5 Homework Exerises View The Mehanial Universe and Beyond, 40: Optis. Read your text, hapter 33: The Nature and Propagation of Light (setions -4, 6-7). 9 Images 9. Basis 9.2 Plane mirror 9.3 Spherial mirrors m i p Derive the spherial mirror formula: 9.4 Spherial lenses Derive the refrating surfae formula: 9.5 Thin lenses n p p Derive the lens maker s equation: n 2 i i f. n 2 n. r f pn qp r r 2 q. Read your text, hapter 34: Geometri Optis (setions -4).. (0 points) a) Fill in Table. Eah row refers to a different objet-mirror ombination. Distane are in entimeters. For any number laking a sign, the sign must be found. b) Sketh eah ombination in Table and draw the number of rays speified in the table. Table : Mirrors type f r i p m real? inverted? rays a) onave b) +0 + no 2 ) d) e) f) g) onvex h) yes 3 2. (0 points) a) Fill in Table 2. Eah row refers to a different objet-lens ombination. Distane are in entimeters. For any number laking a sign, the sign must be found. The medium in whih the objet resides has index of refration n, the lens has index of refration n 2. b) Sketh eah ombination in Table 2 and draw the number of rays speified in the table. 3. (0 points) a) Fill in Table 3 wherever possible. Eah row refers to a different objetlens (thin) ombination. Distane are in entimeters. For any number laking a sign, the sign must be found. For type, use C or D for onverging or diverging.

8 J. M. Veal, Priniples of Physis III 8 Table 2: Spherial Refrating Surfaes 9.7 Exam 2 n n 2 p i r inverted? rays a) ,2 b) ,2 ) , d) e) f) ,2 g) h) Table 3: Thin Lenses type f r r 2 i p n m real? inverted? a) C b) ) 0 +5 d) 0 +5 e) f) g) h) +0.5 no i) b) Sketh eah ombination in Table 3 and draw three rays for eah diagram. 4. (5 points) a) A onverging lens with a foal length of +20 m is loated 0 m to the left of a diverging lens having foal length of -5 m. If an objet is loated 40 m to the left of the onverging lens, loate and desribe ompletely the final image formed by the diverging lens. b) Create an aurate ray diagram, drawn to sale. 0 Interferene 0. Basis 0.2 Fringes Exam 2 overs material between Exam and here. d sin θ mλ; m 0,, 2,... d sin θ pm 2qλ; m 0,, 2,... Consider Young s two-slit interferene experiment, where the distane to the sreen is muh larger than the distane between the slits. Begin with a sketh of the experiment, label the sketh, and dedue the the position angles of the onstrutive and destrutive interferene maxima and minima. θ pmq sin, pmλ{dq. 0.3 Intensity θ d pmq sin rpm 2 qλ{ds - m 0,, 2,... Consider Young s two-slit interferene experiment, where the distane to the sreen is muh larger than the distane between the slits. Show that the intensity of the fringe pattern may be desribed by 0.4 Thin films πd I 4I 0 os 2 λ sin θ. Consider interferene by refletion from a thin film of water on a glass surfae. Derive the following expressions for the film thikness in both the onstrutive and destrutive interferene ases., L mλ{2n. L d pm 2 qλ{2n - m 0,, 2,...

9 J. M. Veal, Priniples of Physis III Mihelson s interferometer 0.6 Homework Exerises View The Mehanial Universe and Beyond, 4: The Mihelson-Morley Experiment. Read your text, hapter 35: Interferene. Diffration. Single-slit minima Consider a single-slit diffration pattern for wavelength λ. If the slit width, a, is muh smaller than the sreen distane, show that the angular loations of the minima are desribed by the following equation..2 Single-slit intensity a sin θ mλ; m, 2, 3,... Consider a single-slit diffration pattern for wavelength λ. If the slit width, a, is muh smaller than the sreen distane, show that the intensity pattern is given by sin p πa λ sin θq 2 Ipθq I max πa λ sin θ..3 Cirular aperture.4 Double slit Consider a double-slit diffration pattern for wavelength λ. If the slit width, a, is muh smaller than the sreen distane, show that the intensity pattern is given by sin p Ipθq I max os 2 p πd πa λ sin θq λ sin θq 2 πa λ sin θ..5 Grating.6 X-ray.7 Homework Exerises View The Mehanial Universe and Beyond, 27: Beyond the Mehanial Universe. Read your text, hapter 36: Diffration. 2 Speial Relativity 2. Two postulates 2.2 Spaetime 2.3 Simultaneity 2.4 Time dilation β v{; γ p β 2 q {2 Consider a light lok. Show that if t 0 is the proper time between two spaetime events and t is the time between these same two events as measured in an inertial frame moving at speed v relative to the propertime frame, then t γ t Length ontration Consider a light lok. Show that if l 0 is the proper length of an objet and l is the length of that objet as measured in an inertial frame moving at speed v relative to the proper-length frame, then 2.6 Lorentz transformation l l 0 {γ. Consider two inertial frames, S moving at speed v relative to S. Derive the following Lorentz transformation. x γpx vtq; y y; z z; t γpt xv{ 2 q x γpx vt q; y y ; z z ; t γpt x v{ 2 q X β L β αx α ; X β L β α X α 2.7 Veloities u x u x v u x v{ 2 ; u x u x v u xv{ 2

10 J. M. Veal, Priniples of Physis III Doppler effet Consider the Doppler effet for light. If f 0 is the frequeny of light as measured in the frame of the light soure and f is the frequeny of the same light as measured in an inertial frame moving at speed v relative to the soure s frame, show that 9r 0 Ñ f f 0 ; 9r 0 Ñ f f Momentum and fore p γm v f f 0 v v. 3 Photons 3. Basis E hf 3.2 Photoeletri effet K e E ph Φ 3.3 Momentum p h{λ 3.4 Compton shift F γ 3 ma; m γm 2.0 Energies F γma Beginning with the work-kineti energy theorem of lassial mehanis, show that the kineti energy of a mass m moving at speed v relative to the observer is given by K pγ qm 2. E m 2 Given that the momentum of a partile is p γmv and the total energy of a partile is E γm 2, show that the energy as a funtion of momentum is given by E 2 ppq 2 pm 2 q Homework Exerises View The Mehanial Universe and Beyond, 42: The Lorentz Transformation, 43: Veloity and Time, and 44: Mass, Momentum, and Energy. Read your text, hapter 37: Relativity. Consider Compton s experiment sattering an x-ray from an eletron. Given relativisti versions of onservation of momentum and onservation of energy, show that the sattered x-ray s wavelength is greater than its inident wavelength by an amount equal to 3.5 Probability waves 3.6 Homework Exerises λ h p os φq. m View The Mehanial Universe and Beyond, 50. Partiles and Waves. Read your text, hapter 38: Photons: Light Waves Behaving as Partiles. 4 Matter Waves 4. Basis λ h{p 4.2 Unertainty priniple x p x h

11 J. M. Veal, Priniples of Physis III 4.3 Wave funtions Ψpx, y, z, tq ψpx, y, zqe iet{ h» 8 ψ ψ dv Shrödinger s equation h2 d 2 ψpxq 2m dx 2 Upxqψpxq Eψpxq 4.5 Free partile a) Consider a free partile and show that the solution to the onedimensional, time-independent Shrödinger equation is given by ψpxq e ikx 2 e ikx. b) Modify this solution to inlude the time-dependent part and let 2 ψ 0. Show that Ψpx, tq 2 2ψ 2 0p os 2kxq. 4.6 Homework Exerises Read your text, hapter 40: Quantum Mehanis (setion ). 4.7 Exam 3 Exam 3 overs material between Exam 2 and here. 4.8 Final Exam The final exam is umulative up to this point. A All Formulae

12 J. M. Veal, Priniples of Physis III 2 v P Mehanial Waves B 2 ypx, tq Bx 2 v λν v 2 B 2 ypx, tq Bt 2 v a T {µ P µvω 2 Y 2 sin 2 pkx ωtq Sound v a B{ρ I 2 ρvω2 S 2 f b f f 2 f f v v d v v s 9r 0 Ñ f f; 9r 0 Ñ f f pp Heat V 3αV 0 T dq m dt C dq n dt P pt h T qka{l Kineti Theory pv NkT a η 2 qpη b q kt xk tr y 3 2 kt 3kT v rms m xλy p4π? 2r 2 ηq m 3{2 P pvq 4π v 2 e mv2 {2kT 2πkT 2kT 8kT m, xvy πm, v rms 3kT m st Law of Thermodynamis W» Vf V i pdv I Q W p V k{m p V 5 3 γ p { V C p {C V T V γ onst. pv γ onst. 2 nd Law of Thermodynamis e r γ e T C S» f i T H dq T S k ln w Eletromagneti Waves E m B m pµ 0 ε 0 q {2 S µ 0 E B I µ 0 E 2 rms u E u B p r I f Preliminary Optis n sin θ n 2 sin θ 2 n p n {v θ sin n 2 n θ B tan n 2 n Images m i p p i f n 2 i n 2 n r pn q r r 2 Interferene d sin θ mλ; m 0,, 2,... d sin θ pm 2qλ; m 0,, 2,... I 4I 0 os 2 πd λ sin θ L mλ ; m 0,, 2,... 2n L pm 2 qλ ; m 0,, 2,... 2n Diffration a sin θ mλ; m, 2, 3,... sin p πa λ sin θq Ipθq I max πa λ sin θ 2 sin p πa Ipθq I max os 2 p πd λ sin θq λ sin θq πa λ sin θ 2

13 J. M. Veal, Priniples of Physis III 3 Speial Relativity β v{; γ p β 2 q {2 t γ t 0 l l 0 {γ x γpx vtq; y y; z z; t γpt xv{ 2 q x γpx vt q; y y ; z z ; t γpt x v{ 2 q X β L β αx α ; X β L β X α α u x u x v u x v{ 2 ; u x u x v u xv{ 2 f f 0 v v 9r 0 Ñ f f 0 ; 9r 0 Ñ f f 0 p γm v F γ 3 ma; m γm F γma K pγ qm 2 E m 2 E 2 ppq 2 pm 2 q 2 λ Photons E hf K e E ph Φ p h{λ h p os φq m Matter Waves λ h{p x p x h Ψpx, y, z, tq ψpx, y, zqe iet{ h» 8 8 ψ ψ dv h2 d 2 ψpxq 2m dx 2 Upxqψpxq Eψpxq Matter Waves II h 2 E n 8mL 2 n 2, n, 2, 3,... E E high E low hf E E high E low nπ ψ n pxq A sin L x, n, 2, 3,... ψ 2 npxq A 2 sin 2 nπ L x, n, 2, 3,... E nx,n y E nx,n y,n z» 8 8 ψ 2 npxqdx h2 n 2 x 8m L 2 x h2 n 2 x 8m L 2 x n 2 y L 2 y n 2 y L 2 y n 2 z L 2 z