Conductors in EConductors in E--fields: dynamic conditionsfields: dynamic conditions••If the EIf the E--field is maintained, field is maintained, then the dynamics persist, then the dynamics persist, i.e. charge continues to flow i.e. charge continues to flow indefinitely.indefinitely.••This is no longer strictly the This is no longer strictly the domain of electrostatics.domain of electrostatics.••Note the direction of flow of Note the direction of flow of the charge carriers the charge carriers (electrons).(electrons).iElectrical current:dqidt=SI unit: 1 ampere (A) = 1 coulomb per second (C/s)Conductors in EConductors in E--fields: dynamic conditionsfields: dynamic conditions••If the EIf the E--field is maintained, field is maintained, then the dynamics persist, then the dynamics persist, i.e. charge continues to flow i.e. charge continues to flow indefinitely.indefinitely.••This is no longer strictly the This is no longer strictly the domain of electrostatics.domain of electrostatics.••Note the direction of flow of Note the direction of flow of the charge carriers the charge carriers (electrons).(electrons).oridijAdA=id=⋅∫jAGGCurrent density:Current density and drift speedCurrent density and drift speed∆L()()11/ddenA LiqjenvAAt ALv∆∆== = =∆∆n = # electrons per unit volumedne=−jvGGStarting pointfor Ohm’s LawOhmic materials (Ohm’s law)Ohmic materials (Ohm’s law)••We will see in a minute We will see in a minute that vthat vddis proportional is proportional to E. Thus, j to E. Thus, j ∝∝E, i.e.E, i.e.σ=jEGGOhm’s Lawρ=EjGGσis the electrical conductivityρis the electrical resistivity1σρ=SI unit for resistivity is ohm⋅ meter: 1 ohm = 1 volt/ampereSI unit for conductivity is siemens per meter:1 siemens = 1 ampere/volt = (1 ohm)-1Ohmic materials (Ohm’s law)Ohmic materials (Ohm’s law)••We will see in a minute We will see in a minute that vthat vddis proportional is proportional to E. Thus, j to E. Thus, j ∝∝E, i.e.E, i.e.σ=jEGGOhm’s LawandiVjEAL∆==oriV L LVi i iRAL A Aρσσ∆⇒= ∆= ==LRAρ=SI unit for resistance is ohm (Ω)Resistance:CapacitorsCapacitors•The transfer of charge from one terminal of the capacitor to the other creates the electric field.•Where there is a field, there must be a potential gradient, i.e. there has to be a potential difference between the terminals.qCV=∆•qrepresents the magnitude of the excess charge on either plate. Another way of thinking of it is the charge that was transferred between the plates.SI unit of capacitance: 1 farad (F) = 1 coulomb/volt(after Michael Faraday)•This leads to the definition of capacitance C:Capacitances more often have units of picofarad (pF) and microfarad (µF)Capacitors connected in parallelCapacitors connected in parallel∆V+q2-q2-q1+q1+(q1+q2)-(q1+q2)∆V11qCV=∆22qCV=∆12 eqqq CV+= ∆()12 eqCC VCV+∆= ∆12eqCCC=+Capacitors connected in seriesCapacitors connected in series12111eqCCC=+11neq nCC=∑In fact:Energy stored in electric fieldsEnergy stored in electric fieldsdqdU = ∆V × dqqdU dqC=()2221201122QqQUdU dq Q CVCCC== =≡××=∆∫∫212ouEε=Energy densityA dielectric in an electric fieldA dielectric in an electric field0'=+EE EGGGE = E0− E' E' opposes E0A dielectric in an electric fieldA dielectric in an electric fieldLinear materials: E' ∝ E, ⇒ E0= (1 + χe)E0011(1)1eeeeEEEκχχκ⇒= = =++χeis the electric susceptibility (dimensionless)κeis the dielectric constant (dimensionless)ε = κeεοis the permittivityDielectrics in capacitorsDielectrics in capacitorsoencencoe encencdqq'dqdqεεκε⋅= −⋅=⋅=∫∫∫EAEAEAGGGGGGvvveC' Cκ=DC CircuitsDC CircuitsKirchoff’s first law:At any junction in an electrical circuit, the total current entering the junction must equal the total current leaving the junction.Electromotive force (emf)Electromotive force (emf)•Source of electrical energy in a circuit.•Represents the potential energy provided to each coulomb of charge that passes through the device.•IT IS NOT A FORCE!!!•Most often, emf is provided by a battery (a chemical cell).•The emf is the same as the potential difference between the negative and positive terminals of a battery WHEN NO CURRENT FLOWS.•In general, when a current flows, the potential difference at the terminals of a battery is lower than the emf.•An emf can also store energy.E = dW/dq SI unit: joule/CoulombCircuit analysisCircuit analysisKirchoff’s second law:The algebraic sum of all differences in potential around a complete circuit loop must be zero.Energy transfer in electric circuitsEnergy transfer in electric circuits•A battery does work by providing each coulomb of charge that leaves its positive terminal 1 joule of energy. •If charge flows at a rate of 1 coulomb per second, then the battery does work at a rate of 1 joule per second, i.e.joule coulomb joulePower= × = =wattcoulomb second secondP = Ei = dW/dt•In a resistor, energy is lost in an amount iRper coulomb.⇒ Pcharge= i∆V = i(−iR) = −i2RPheat= i2RRC circuits (charging a capacitor)RC circuits (charging a capacitor)Kirchoff’s 2nd law:0qiRCε −−=dq qRdt Cε =+()//1tRC tRCqC e i eRεε−−= − =RC circuits (discharging a capacitor)RC circuits (discharging a capacitor)Kirchoff’s 2nd law:0ε =0dq qRdt C+=///00tRC tRC tRCqqqe i e eRC Rε−−−==− = −Kirchoff’s 2nd law:0qiRC−− =q0= εCThe The Magnetic Magnetic Force on a Moving ChargeForce on a Moving ChargeWe define the magnetic field:,max,sinBBFBorFqvBqvφ==In fact:Bq=×FvBGGGnewton newton1tesla=1 =1coulomb×meter/second ampere×meterThe Lorentz ForceThe Lorentz Force()q=+×FEvBGG GGThe Lorentz ForceThe Lorentz Force•The velocity filter:EvB=(undeflected trajectories in crossed E and B fields)•Cyclotron motion:2BrvFma qvBmr= ⇒ =mv prqB qB==2qBfmωπ==2222122qBRKmvm==•Orbit radius:•Orbit frequency:•Orbit energy:momentum (p) filtermass detectionThe classical Hall effectThe classical Hall effect••Lorentz force likes to deflect jLorentz force likes to deflect jxx••However, EHowever, E--field is set up which balances Lorentz forcefield is set up which balances Lorentz force••Balance occurs when EBalance occurs when Eyy= v= vxxBBzz= = VVyy//llyy••But jBut jxx= = nevnevxx(or (or iixx= = nevnevxxAAxx))⇒⇒RRxyxy= V= Vyy//iixx= R= RHHBBzz××((llyy/A/Axx)),,where where RRHH= 1/= 1/neneWhere lWhere lyyis
View Full Document
Unlocking...