# 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: first-order 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 Biot-Savart 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 Capacitor-inductor system Math: second-order 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 Self-induction 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 1-3). 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 4-6). 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: first-order linear differential equations: nonhomogeneous and homogeneous Show that the solution to an ordinary first-order linear homogeneous differential equation is given by y ce ³ P pxqdx. Show that the solution to an ordinary first-order 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 RC-circuit 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: Direct-Current 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 Biot-Savart law db µ 0 id s r 4π r 3 You have a long straight wire with current i running through it. Use the Biot-Savart 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 1-7).

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 1-6). 13 Inductance 13.1 Definition L NΦ B i 13. Solenoid & toroid L l µ 0n A 13.3 Self-induction 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 1-4). 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: second-order 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 5-6). 15 Electromagnetic Oscillations 15.1 Simple harmonic motion Consider simple harmonic motion as discussed in class for a block-spring 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. Capacitor-inductor 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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