Superposition of Forces r r F 12 qq 1 2 eˆ r 2 4 r 0 We find the total force by adding the vector sum of the individual forces. Work Problem 21-18 (III) Two charges, Q and 4Q are a distance apart. These two charges are free to move but do not because there is a third charge nearby. What must be the magnitude of the third charge and its placement in order for the first two to be in equilibrium? 21-18. Electric Field of a Point Charge E F q0 = kq r 2 eˆ r E k q r2 k 1 4 0 Relation between F and E If w e put a charge q1 in an electric field E , then the charge q1 feels a force of value F1 q1 E * * * T his is the really useful part. Don’t confuse this charge q1 with the test charge q0 or the original charges q that produced E. The test charge q0 was used to find the electric field. This is a real charge q1 placed in the electric field. Electric Field Lines for a Point Charge Electric Field Lines for Systems of Charges We call this a dipole. It is a dipole field. The Electric Field of a Charged Plate E 2 0 A Parallel-Plate Capacitor The electric field near a conducting surface must be perpendicular to the surface when in equilibrium. If w e place conductor in electric field, the E lines m ust be to surface. If not, charges w ould m ove. E m ust be zero inside . Conductor placed around a charge +Q C onductor 21-86. An electron moves in a circle of radius r around a very long uniformly charged wire in a vacuum chamber, as shown in the figure. The charge density on the wire is λ = 0.14 μC/m. (a) What is the electric field at the electron (magnitude and direction in terms of r and λ? (b) What is the speed of the electron? F = qE = m a a= qE m We can work all kinds of problems with charged particles moving in electric fields. Electron entering charged parallel plates Electric flux: F E = E ×A = E A co s q Electric flux through an area is proportional to the total number of field lines crossing the area. Flux through a closed surface: 0 for closed surface that contains no ch arge E positive negative The net number of field lines through the surface is proportional to the charge enclosed, and also to the flux, giving Gauss’s law: E E dA Q encl o This can be used to find the electric field in situations with a high degree of symmetry. Electric field of charged sheet E (2 A ) Q 0 A 0 w here is charge/area on the sheet. E2A E A 2 0 0 Electric Potential V V U q0 W *** U nit: J/C = volt, V q0 definition!! Electric potential, or potential, is one of the most useful concepts in electromagnetism. This is a biggie!! Electrostatic Potential Energy and Potential Difference The electrostatic force is conservative – potential energy can be defined. Change in electric potential energy is negative of work done by electric force: U b - U a = - W = - qEd Copyright © 2009 Pearson Education, Inc. The Potentials of Charge Distributions If the electric field is known: ò DV = - rb E ds ra For one point charge: V = q 4p e0 r For many point charges: V = 1 4p e0 å i qi ri The Potentials of Charge Distributions If the electric field is known: ò DV = - rb E ds ra For differential charge: dV = dq 4p e0 r For many point charges: V = 1 4p e0 For a continuous charge distribution: V = å 1 4p e0 i ò qi ri dq r Equipotential Surfaces V = q 4p e0 r An equipotential is a line or surface over which the potential is constant. Electric field lines are perpendicular to equipotentials. The surface of a conductor is an equipotential. Copyright © 2009 Pearson Education, Inc. Equipotential Surfaces Another case showing electric field lines are perpendicular to equipotentials. The surface of a conductor is an equipotential. We can also see that equipotentials are perpendicular to electric fields from the equation DV = Copyright © 2009 Pearson Education, Inc. òE d Equipotential Surfaces Equipotential surfaces are always perpendicular to field lines; they are always closed surfaces (unlike field lines, which begin and end on charges). Electric field and equipotentials for electric dipole. Copyright © 2009 Pearson Education, Inc. 23-74. Four point charges are located at the corners of a square that is 8.0 cm on a side. The charges, going in rotation around the square, are Q, 2Q, -3Q and 2Q, where Q = 3.1 μC. What is the total electric potential energy stored in the system, relative to U = 0 at infinite separation? When a capacitor is connected to a battery, the charge on its plates is proportional to the voltage: Q = CV The quantity C is called the capacitance. C = Q V Copyright © 2009 Pearson Education, Inc. Parallel plate capacitor V Ed So Q 0 d 0A Q d A 0 and d V C Q V 0A d The capacitance value depends only on geometry! Capacitors in Parallel Capacitors in parallel have the same voltage across each one. The equivalent capacitor is one that stores the same charge when connected to the same battery: C eq = C 1 + C + C 3 Copyright © 2009 Pearson Education, Inc. (parallel) Capacitors in Series Capacitors in series have the same charge. In this case, the equivalent capacitor has the same charge across the total voltage drop. Note that the formula is for the inverse of the capacitance and not the capacitance itself! V V V V Q Q Q Q 1 2 3 C C C eq C 1 Q Q 1 1 1 C eq C C C 1 2 3 1 1 1 1 C eq C C C 1 Copyright © 2009 Pearson Education, Inc. 2 3 2 3 Effect of a Dielectric on the Electric Field of a Capacitor E E0 E E 0 / and V V 0 / w here is the dielectric constant A dielectric is an insulator, and is characterized by a dielectric constant k . Capacitance of a parallel-plate capacitor filled with dielectric: C 0 A d C C0 for parallel-plate capacitor Using the dielectric constant, we define the permittivity: 0 Copyright © 2009 Pearson Education, Inc. Energy in electric field The energy U in a capacitor is 2 A U 1CV 2 1 0 E d 2 2 d U 1 E 2 Ad 2 0 The volume is Ad, and the energy density u is 1 E 2 Ad u energy 2 0 1 E2 2 0 volum e Ad Defibrillator Potential energy of a charged capacitor: All three expressions are equivalent! U 1 2 QV 1 2 CV 2 Q 2 2C A complete circuit is one where current can flow all the way around. Note that the schematic drawing doesn’t look much like the physical circuit! Open circuit Copyright © 2009 Pearson Education, Inc. Direction of Current and Electron Flow V IR O hm 's law V in this case is the em f of battery R is the resistance of bulb Resistors are color coded to indicate the value of their resistance. 22 10 10% 6 Resistivity The resistance of a wire is directly proportional to its length and inversely proportional to its cross-sectional area: R= r A The constant ρ, the resistivity, is characteristic of the material. Copyright © 2009 Pearson Education, Inc. Energy and Power P ow er P U t dU dt ( dQ )V IV dt The unit of electrical power is watt W (J/s). If we use Ohm’s law with this equation, we have P IV I ( IR ) I R 2 P IV V R V V 2 R AC Voltage and Current for a Resistor Circuit V V 0 sin 2 ft V 0 sin t V0 I sin t I 0 sin t R R I 0 I m ax peak current (m axim um current) V Note that I and V are in phase!! 25-36. (II) A 120-V hair dryer has two settings: 850 W and 1250 W. (a) At which setting do you expect the resistance to be higher? After making a guess, determine the resistance at (b) the lower setting; and (c) the higher setting. 25-43. (II) How many 75-W lightbulbs, connected to 120 V as in Fig. 25–20, can be used without blowing a 15-A fuse? Resistors in Series V1 V 2 V 3 Current is not used up in each resistor. Same current I passes through each resistor in series. IR eq IR1 IR 2 IR 3 I ( R1 R 2 R 3 ) IR eq R eq R1 R 2 R 3 A parallel connection splits the current; the voltage across each resistor is the same: I I1 I 2 I 3 V R eq 1 R eq 1 1 1 V R1 R 2 R 3 R R R 2 3 1 V V V 1 1 1 R R R 2 3 1 Copyright © 2009 Pearson Education, Inc. Analyzing a Complex Circuit of Resistors 1 ' R eq ' R eq 1 R R/2 1 R 2 R R eq R R R / 2 R eq 2.5 R Kirchhoff’s Junction Rule In + Out - The sum of currents meeting at a junction must be zero. I1 – I2 – I3 = 0 or I1 = I2 + I3 Kirchhoff’s Loop Rule The sum of potential differences around any closed circuit loop is zero. Our rules: 1) When going from – to + across an emf the V is +. (+ to -, it is -). 2) When going across resistor in direction of assumed I, the V is -. (Opposite, it is +). Measuring the Current in a Circuit Am m eter We want ammeter to have very low resistance so it will not affect circuit. Ammeters go in series. Measuring the Voltage in a Circuit V oltm eter 1 R eq 1 R 1 R voltm eter We want voltmeter to have very large resistance so it will not affect circuit. Voltmeters go in parallel across what is being measured. An ohmmeter measures resistance; it requires a battery to provide a current. These circuits are much more complicated. Rsh is a shunt resistor to change scales. Rser is a resistor to adjust galvanometer scale zero. Copyright © 2009 Pearson Education, Inc. Magnetic Field Lines for a Bar Magnet B S N Imagine using a test pole N; place it at any point and see where the force is. Just like we do for electric fields. We actually use small compasses to do this. The Magnetic Force Right-Hand Rule FB q v B Units of magnetic field: teslas kg 1T = 1 C ×s The Lorentz force is the sum of the electric and magnetic forces acting on the same object: æ çç çç çç çè F = q E + v´ B ö ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ø The Earth’s magnetic field is similar to that of a bar magnet. Note that the Earth’s “North Pole” is really a south magnetic pole, as the north ends of magnets are attracted to it. Copyright © 2009 Pearson Education, Inc. Operating Principle of a Mass Spectrometer r mv q B Several applications Magnetic Force on a Current-Carrying Wire F qvB dF I d B F Id B F I LB d dq F IL B sin The Magnetic-Field Right-Hand Rule Put thumb along direction of current, and fingers curl in direction of B. r B ~ I r discuss Magnetic Forces on a Current Loop F 0 I I F = Ih B Forces cause a torque F 0 Ftotal = 0 I F I LB An electric motor uses the torque on a current loop in a magnetic field to turn magnetic energy into kinetic energy. Copyright © 2009 Pearson Education, Inc. 27-23. (II) A 6.0-MeV (kinetic energy) proton enters a 0.20-T field, in a plane perpendicular to the field. What is the radius of its path? See Section 23–8. F 2 I 2 B1 We have to look closely at fields and forces to see how the forces occur. Ampère’s Law We have seen that 0I B 2 r 0 I B ds B ds B (2 r ) 2 r 2 r 0 I B ds I enclosed 0 To find the field inside, we use Ampère’s law along the path indicated in the figure. B ds I enclosed 0 T he dot product of B and ds is zero everyw h ere except from cd B d ds B ds B 0 N I c B 0 I N 0 nI N # turns, n turns/length B 0 nI Induced Current Produced by a Moving Magnet v v ind We conclude that it is the change in magnetic flux that causes induced current. F = B A B This is called Faraday’s Law of Induction after Michael Faraday. Induced em f N is num ber of turns N dB dt Lenz’s Law The induced current will always be in the direction to oppose the change that produced it. In d u ced em f Û In d u ced cu rren t Applying Lenz’s Law to a Magnet Moving Toward and Away From a Current Loop Induced current v v An Electrical Generator Falling w ater, Current is induced Produces AC power steam Magnetic flux changes! A Simple Electric Motor/Generator Inductance B LI m agnetic flux depends on current dB dt L dI dt L is called inductance (actually self ind uctance here). The inductance L is a proportionality constant that depends on the geometry of the circuit There will be a magnetic flux in Loop 1 due to current I1 flowing in Loop 1 and due to current I2 flowing in Loop 2. B (1) L I M 1 1 I 12 2 S im ilarly, B (2) L I M 2 2 I 21 1 Now it is clearer why we call L self inductance and M mutual inductance. For exam ple, tw o nearby coils A rea A Solenoid Self-Induction B 0 nI B N B A 0 nN IA 0 A n B L I , so L 0 An 2 I 2 O nly depends on geom etry. General energy density uB 1 B uE 1 2 general result 2 0 2 0E 2 1B 2 u uB uE 0E 2 0 2 30-34. (II) A 425-pF capacitor is charged to 135 V and then quickly connected to a 175-mH inductor. Determine (a) the frequency of oscillation, (b) the peak value of the current, and (c) the maximum energy stored in the magnetic field of the inductor. RL Circuit V 0 I Think about what will happen when the switch is closed. V 0 L dI IR 0 dt Current as a Function of Time in an RL Circuit L/R I V0 R 1 e t / V0 R 1 e tR / L Oscillations in LC Circuits Start with charged capacitor. Close switch. It will discharge through inductor, and then recharge in opposite sense. If no resistance, will continue indefinitely. The charge and current are both sinusoidal, but with different phases. Q Q 0 cos( t ) I I 0 sin( t ) Copyright © 2009 Pearson Education, Inc. LC Oscillations with Resistance (LRC Circuit) Any real (nonsuperconducting) circuit will have resistance. Copyright © 2009 Pearson Education, Inc. Damped Oscillations in RLC Circuits Charge equation: Solution: where and w '= 0 w hen R2 = 4L C This is a step-up transformer – the emf in the secondary coil is larger than the emf in the primary: VP VS Copyright © 2009 Pearson Education, Inc. NP NS Lots of applications for transformers, the bug zapper. Power distribution Transformers work only if the current is changing; this is one reason why electricity is transmitted as ac. Single elements with AC Source Resistors, capacitors, and inductors have different phase relationships between current and voltage when placed in an ac circuit. The current through a resistor is in phase with the voltage. I = I cos w t 0 Resistive element Inductive Element The voltage across the inductor is determined by I = I cos w t 0 V = L dI = - w L I sin w t 0 dt or 0 V L I cos t 90 0 V V cos 0 0 t 90 Therefore, the current through an inductor lags the voltage by 90°. Copyright © 2009 Pearson Education, Inc. I = I cos w t 0 t1 t2 Inductive Circuit The voltage across the inductor is related to the current through it: V m ax V LI X L I 0 0 0 The quantity XL is called the inductive reactance, and has units of ohms: X L º w L = 2p f L For very low frequencies the inductive reactance is small. That is because for direct currents (zero frequency), an inductor has little or no effect. Direct current passes right through an inductor. Copyright © 2009 Pearson Education, Inc. Capacitive Circuit The voltage across the capacitor is given by Q 1 0 V I cos t 90 0 C C V cos 0 0 t 90 Therefore, in a capacitor, the current leads the voltage by 90°. Copyright © 2009 Pearson Education, Inc. t1 t2 Capacitive Circuit The voltage across the capacitor is related to the current through it: 1 0 0 V I cos t 90 V cos t 90 0 C 0 1 I X V I 0 0 C 0 C The quantity XC is called the capacitive reactance, and (just like the inductive reactance) has units of ohms: XC º Copyright © 2009 Pearson Education, Inc. 1 = 1 w C 2p f C Effects of frequency on capacitive reactance Note that when the frequency increases to large values that XC becomes very small. The current then becomes very large. 1 XC Why? C The frequency is so high that the capacitor doesn’t have time to fully charge. It almost acts as a short circuit. At low frequencies, it acts as an open circuit. Either capacitors or inductors can be used to make either AC or DC filters: AC & DC input LRC Series AC Circuit Analyzing the LRC series AC circuit is complicated, as the voltages are not in phase – this means we cannot simply add them. Furthermore, the reactances depend on the frequency. Copyright © 2009 Pearson Education, Inc. Some Applications Diodes and Rectifiers • A diode conducts electricity in one direction only • Can use diodes and combinations of diodes to make half- and full-wave rectifiers Half-wave Rectifier Some Applications Full-wave Rectifier

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