Principles of Physics II


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1 Principles of Physics II J. M. Veal, Ph. D. version Contents 1 Fluid Mechanics Fluid pressure Buoyancy Fluid flow Bernoulli s equation Homework Exercises Electric Force & Field 3.1 What is charge? Coulomb s law Vector fields & field lines Electric field Point charge Dipole, ring, & disk Dipole: torque & energy Homework Exercises Gauss Law Gaussian surface Flux & electric flux One of Maxwell s equations Coulomb s law Conductors Math: volume integrals Symmetry: cylinder, plane, & sphere Homework Exercises Electric Potential Force, field, energy, potential From a field Point charge Four negative signs Dipole & disk Find the field Homework Exercises Capacitance Ratio Capacitors: plate, cylinder, & sphere In parallel & in series Energy Homework Exercises Exam 1a Dielectrics Polar & nonpolar Gauss law Homework Exercises Current Definition Density & drift Homework Exercises Resistance Definition Ohm s law & drift Emf & current Four rules Resistance: in series & in parallel Homework Exercises
2 J. M. Veal, Principles of Physics II 9 Circuits Math: firstorder linear differential equations: nonhomogeneous and homogeneous Capacitance: charging and discharging Homework Exercises Magnetic Force & Field Definition Hall effect Circulating charge On current: force & torque Dipole moment Dipole: torque & energy Homework Exercises Ampere s Law BiotSavart law Wires: one & two One of Maxwell s equations Fat wires: inside & out Solenoid & toroid Loop s axis Homework Exercises Exam Faraday s Law One of Maxwell s equations Lenz s law Induction Homework Exercises Magnets One of Maxwell s equations Spin & orbital moments Loop model Three types Homework Exercises Electromagnetic Oscillations Simple harmonic motion Capacitorinductor system Math: secondorder linear nonhomogeneous differential equations Damping: the LRC circuit Resonance Homework Exercises Alternating Current Three simple circuits Impedance Power Transformer Homework Exercises Maxwell s Equations Speed of light Maxwell s law Displacement current Beauty and light Homework Exercises Exam Final Exam A All Formulae Inductance Definition Solenoid & toroid Selfinduction Inductors: current rise & decay Energy Mutual induction Homework Exercises
3 J. M. Veal, Principles of Physics II 3 1 Fluid Mechanics 1.1 Fluid pressure dp dy ρg 1. Buoyancy 1.3 Fluid flow ρ 1 A 1 v 1 ρ A v 1.4 Bernoulli s equation As discussed in class, with a sketch to define the quantities involved, derive Bernoulli s equation: p 1 ρgy Homework Exercises 1 ρv 1 p ρgy Read your text, chapter 1: Fluid Mechanics. Electric Force & Field.1 What is charge?. Coulomb s law F 1 q 1q 4πε 0 r ˆr.3 Vector fields & field lines.4 Electric field.5 Point charge E q 4πε 0 r ˆr 1 ρv..6 Dipole, ring, & disk Consider an electric dipole with charges q and q on the x axis at locations x a and x a, respectively. a) Show that the electric field on the y axis is given by Ep0, yq kqapy a q 3{ î. b) Show that the electric field on the x axis is given by $ & 4kqaxpx a q î, x a Epx, 0q kqpx % a qpx a q î, a x a 4kqaxpx a q î, a x. Consider a ring of uniform charge density and total charge Q, located at y z a, x 0. Show that the electric field on the x axis is given by Epx, 0, 0q xq 4πε 0 px a q 3{. Consider a disk of uniform charge density σ, and radius R, located in the xy plane and centered at the origin. Show that the electric field on the z axis is given by Ep0, 0, zq σ z 1?. ε 0 z R.7 Dipole: torque & energy τ p E U p E.8 Homework Exercises Read your text, chapter 1: Electric Charge and Electric Field. 3 Gauss Law 3.1 Gaussian surface 3. Flux & electric flux Φ E E da
4 J. M. Veal, Principles of Physics II One of Maxwell s equations E da q enc ε Coulomb s law Use Gauss law to derive the electric field of a point charge: 3.5 Conductors E σ ε Math: volume integrals E q 4πε 0 r. 3.7 Symmetry: cylinder, plane, & sphere Use Gauss law to derive the electric field of a line of charge: E λ πε 0 r. Use Gauss law to derive the electric field of a sheet of charge: E σ ε 0. Use Gauss law to derive the electric field of a sphere of charge: 3.8 Homework Exercises E qr 4πε 0 R 3. Read your text, chapter : Gauss s Law. 4 Electric Potential 4.1 Force, field, energy, potential 4. From a field V f V i f i E d s 4.3 Point charge Given a point charge q, show that its electric potential varies with distance r as follows: V q 4πε 0 r. 4.4 Four negative signs 4.5 Dipole & disk Consider an electric dipole with charges q and q, and charge separation d. Let θ be the angle measured from the axis running from the negative to the positive charge, and let r be the distance from the center of the dipole. If r d, show that the electric potential is given by V qd cos θ 4πε 0 r. Consider a disk of radius R and uniform charge density σ. Let z be the distance from the disk along its axis. Show that the electric potential is given by 4.6 Find the field E s BV Bs V 4.7 Homework Exercises σ ε 0 a z R z. Read your text, chapter 3: Electric Potential. 5 Capacitance 5.1 Ratio q V C
5 J. M. Veal, Principles of Physics II 5 5. Capacitors: plate, cylinder, & sphere 5.6 Exam 1a Consider a capacitor that consists of two parallel plates of area A and separation distance d. Show that the capacitance is given by C ε 0A d. Consider a capacitor that consists of two coaxial cylinders of radii a and b (a b). Show that the capacitance is given by L C πε 0 ln b{a. Consider a capacitor that consists of two concentric spherical shells of radii a and b (a b). Show that the capacitance is given by 5.3 In parallel & in series C eq C eq 5.4 Energy U q C U 1 CV u E 1 ε 0E ņ C j ņ 1 C j Homework Exercises ab C 4πε 0 b a. Read your text, chapter 4: Capacitance and Dielectrics (sections 13). 6 Dielectrics 6.1 Polar & nonpolar 6. Gauss law Exam 1 covers material up to here. Show Gauss Law for dielectrics is E d A q enc κε 0. E da q end ε 6.3 Homework Exercises Read your text, chapter 4: Capacitance and Dielectrics (sections 46). 7 Current 7.1 Definition i dq 7. Density & drift i J da v d i nea J pneq v d
6 J. M. Veal, Principles of Physics II Homework Exercises Read your text, chapter 5: Current, Resitance, and Electromotive Force (section 1). 8 Resistance 8.1 Definition R V i R ρ L A 8. Ohm s law & drift i9v ρ m e nτ 8.3 Emf & current E dw dq i E R 8.4 Four rules 8.5 Resistance: in series & in parallel R eq R eq ņ R j ņ 1 R j Homework Exercises Read your text, chapter 5: Current, Resitance, and Electromotive Force (sections  6). 9 Circuits 9.1 Math: firstorder linear differential equations: nonhomogeneous and homogeneous Show that the solution to an ordinary firstorder linear homogeneous differential equation is given by y ce ³ P pxqdx. Show that the solution to an ordinary firstorder linear nonhomogeneous differential equation is given by y e ³ P pxqdx P pxqdx e³ fpxqdx. 9. Capacitance: charging and discharging q CE p1 e t{rc q Consider a charging capacitor. Begin by applying the loop rule to an RCcircuit with the switch closed in the charging position. Show that the charging current is given by the following formula. E i e t{rc R q q 0 e t{rc Consider a discharging capacitor. Begin by applying the loop rule to an RCcircuit with the switch closed in the discharging position. Show that the discharging current is given by the following formula. 9.3 Homework Exercises i q 0 RC e t{rc Read your text, chapter 6: DirectCurrent Circuits. 10 Magnetic Force & Field 10.1 Definition F B q v B
7 J. M. Veal, Principles of Physics II Hall effect 11. Wires: one & two As discussed in class, show that the number density of charge carriers in a conductor is given by n ibd eav. Include a discussion of the Hall effect Circulating charge 10.4 On current: force & torque F B i L B As discussed in class, show that the torque on a loop of current is given by 10.5 Dipole moment µ NiA 10.6 Dipole: torque & energy τ µ B U µ B 10.7 Homework Exercises τ NiAB sin θ. Read your text, chapter 7: Magnetic Field and Magnetic Forces. 11 Ampere s Law 11.1 BiotSavart law db µ 0 id s r 4π r 3 You have a long straight wire with current i running through it. Use the BiotSavart law to show that the magnetic field a distance R from the wire is given by B µ 0i πr. F ab µ 0i a i b L πd 11.3 One of Maxwell s equations B d s µ 0 i enc 11.4 Fat wires: inside & out B µ 0ir πr B µ 0iN πr 11.5 Solenoid & toroid You have an ideal solenoid with current i and turn density n. Show that the ideal internal magnetic field is given by 11.6 Loop s axis B µ 0 in. As discussed in class, show that a current loop s axial magnetic field is given by 11.7 Homework Exercises Bpzq 1 µ 0iR pz R q 3{. Read your text, chapter 8: Sources of Magnetic Field (sections 17).
8 J. M. Veal, Principles of Physics II Exam Exam covers material between Exam 1 and here. 1 Faraday s Law 1.1 One of Maxwell s equations Φ B B da E d s Φ 9 B E ind 9 Φ B 1. Lenz s law 1.3 Induction Consider a rectangular current loop in the xy plane of wih L parallel to the y direction. There is a magnetic field B Bˆk in the region x 0. The loop is forced to move with constant velocity v vî. As discussed in class and beginning with E 9 Φ B, show that the rate at which the applied force does work is given by 1.4 Homework Exercises P pblvq {R. Read your text, chapter 9: Electromagnetic Induction (sections 16). 13 Inductance 13.1 Definition L NΦ B i 13. Solenoid & toroid L l µ 0n A 13.3 Selfinduction E L L di 13.4 Inductors: current rise & decay As discussed in lecture, show that the current rise in an RL circuit is given by i E R p1 e Rt{L q. As discussed in lecture, show that the current decay in an RL circuit is given by i i 0 e Rt{L Energy U B 1 Li As discussed in lecture, analyze an inductor to show that the energy density of a magnetic field is given by 13.6 Mutual induction E 1 M di E M di Homework Exercises u B B µ 0. Read your text, chapter 30: Inductance (sections 14). 14 Magnets 14.1 One of Maxwell s equations B da 0
9 J. M. Veal, Principles of Physics II Spin & orbital moments 14.3 Loop model µ orb e m L orb 14.4 Three types Compare the following two statements. i) A material placed in an external magnetic field B ext develops a magnetic dipole moment directed opposite B ext. If the field is nonuniform, the material is repelled from a region of greater magnetic field toward a region of lesser field. ii) A material placed in an external magnetic field B ext develops a magnetic dipole moment in the direction of B ext. If the field is nonuniform, the material is attracted toward a region of greater magnetic field from a region of lesser field. Name the two materials and explain these phenomena. (Hint: You may wish to use the loop model for electron orbits.) 14.5 Homework Exercises Read your text, chapter 8: Sources of Magnetic Field (section 8) Math: secondorder linear nonhomogeneous differential equations 15.4 Damping: the LRC circuit d qptq Qe Rt{L 1 R cos pω 1 t φq, ω 1 LC L As discussed in class, show that an LRC circuit s charge as a function of time is given by d qptq Qe Rt{L cos 1 R t LC L You may take the following step as given Resonance φ. xptq c 1 cos ωt c sin ωt Ñ xptq x max cos pωt φq 15.6 Homework Exercises Read your text, chapter 30: Inductance (sections 56). 15 Electromagnetic Oscillations 15.1 Simple harmonic motion Consider simple harmonic motion as discussed in class for a blockspring system. Given U 1 kx, K 1 mv, and E mec 0, show that the position as a function of time is given by xptq x m cos pωt 15. Capacitorinductor system t qptq Q cos? φ LC φq. 16 Alternating Current E E m sin ω d t 16.1 Three simple circuits i R V R R sin ω i C V C sin ω d t X C π i L V L sin ω d t π X L
10 J. M. Veal, Principles of Physics II Impedance d Z R ω d L 1 ω d C Consider an RLC circuit with alternating current. As discussed in class, use phasor diagrams, algebra, and the definition of impedance to show that 16.3 Power E m IZ. As discussed in class, show that the average power associated with alternating current is given by 17.4 Beauty and light E da q enc, E d s Φ ε 9 B 0 B da 9 0, B d s µ 0 ε 0 Φ E 17.5 Homework Exercises µ 0 i enc Read your text, chapter 3: Electromagnetic Waves (section 1) Exam Transformer xp y 1 IE m cos φ. Exam 3 covers material between Exam and here. V p {N p V s {N s 16.5 Homework Exercises Read your text, chapter 31: Alternating Current and chapter 9: Electromagnetic Induction (section 7) Final Exam The final exam is cumulative up to this point. 17 Maxwell s Equations 17.1 Speed of light c pµ 0 ε 0 q 1{ A All Formulae 17. Maxwell s law 9 B d s µ 0 ε 0 Φ E 17.3 Displacement current 9 i d ε 0 Φ E B d s µ 0 pi enc i d,enc q
11 J. M. Veal, Principles of Physics II 11 Fluid Mechanics dp dy ρg ρ 1 A 1 v 1 ρ A v 1 p 1 ρgy 1 ρv 1 p ρgy Electric Force & Field F 1 q 1q 4πε 0 r ˆr E q 4πε 0 r ˆr Epyq kqapy a q 3{ Epxq 4kqaxpx a q xq Epxq 4πε 0 px a q 3{ σ z Epzq 1? ε 0 z R τ p E U p E Gauss Law Φ E E da E da q enc E ε 0 q 4πε 0 r E σ ε 0 1 ρv Electric Potential V f V i V f i q 4πε 0 r E d s qd cos θ V 4πε 0 r V σ a z R ε z 0 C eq E s BV Bs Capacitance q V C C ε 0A d C πε 0L ln pb{aq C 4πε 0ab b a C eq ņ ņ C j C 1 j U q C U 1 CV u E 1 ε 0E 1 R eq Current i dq i J da v d i nea J pneq v d Resistance R V i R ρ L A i9v ρ m e nτ E dw dq R eq i E R ņ ņ R j R 1 j Circuits 1 y ce ³ P pxqdx y e ³ P pxqdx e ³ P pxqdx f pxqdx E λ πε 0 r E σ ε 0 E qr 4πε 0 R 3 Dielectrics E d A q enc κε 0 E d A q enc ε q q 0 e t{rc i q 0 RC e t{rc q CE p1 e t{rc q i E R e t{rc
12 J. M. Veal, Principles of Physics II 1 Magnetic Force & Field F B q v B Φ B B da E d s Φ 9 B Alternating Current E E m sin ω d t n idb eav F B i L B τ tot NiAB sin θ µ NiA τ µ B U µ B Ampere s Law E ind 9 Φ B P pblvq {R Inductance L NΦ B i L l µ 0n A E L L di i E R p1 e Rt{L q i i 0 e Rt{L Z i R V R R sin ω i V C C sin ω d t X C π i V L L sin ω d t π X L d R ω d L 1 ω d C E IZ db µ 0 id s r 4π r 3 B µ 0i πr F ab µ 0i a i b L πd B d s µ 0 i enc B µ 0ir πr B µ 0 in B µ 0iN πr Bpzq 1 µ 0iR pz R q 3{ Faraday s Law U B 1 Li u B B µ 0 E 1 M di E M di 1 Magnets B da 0 µ orb e m L orb Electromagnetic Oscillations t qptq Q cos? φ LC d qptq Qe Rt{L cos pω 1 t φq, ω 1 1 LC R L xp y 1 IE m cos φ V p {N p V s {N s Maxwell s Equations c pµ 0 ε 0 q 1{ B d s µ 0 ε 0 9 Φ E i d ε 0 9 Φ E B d s µ 0 pi enc E d A q enc ε 0, B d A 0, i d,enc q E d s 9 Φ B B d s µ 0 ε 0 9 Φ E µ 0 i enc
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