Presentation on theme: "PHY 042: Electricity and Magnetism"— Presentation transcript:
1PHY 042: Electricity and Magnetism ElectrodynamicsProf. Hugo Beauchemin
2IntroductionThe strategy adopted to reveal the structure of E&M was to:Start from a concrete experimental setup to infer uniformities about nature and quantify them with laws;Introduce new concepts allowing the extension of these laws to a wider range of experiments and phenomena;Use powerful mathematical tools to express the laws in terms of the new concepts with differential equations, completing the extension.We will continue to adopt this strategy to develop E&M further…So far:We described all electric phenomena with static chargesWe described all magnetic phenomena with steady currentsTo generalize E&M phenomena further, we need to understand:The impact of using steady currents on E-field phenomenaThe impact of non-steady currents on both electricity and magnetismWill lead to Maxwell equations, electromagnetism and…light!
3Ohm’s lawThe connection between electric fields and steady currents begin again with an experimental setup:A device to set charges in motion, i.e. to generate an E-field:Voltaic pile generates a potential differenceA source of free charges that can move in the E-field:Metallic wire allows for steady currentsOhm measured the relationship between current (moving charges) and potential (electric field) and found an empirical law:R depends on the nature of the material and its geometryUses newly made galvanometer to measure currents when varying VWe want a connection between a steady current and the E-field:Charges move in response to an E-field
4An hint for a generalization The previous equation is a modern version of the Ohm’s law giving a relationship between currents and electric field in materialDescribe the effect of an E-field on free charges (currents)Expected from Newton’s law (an electric force set the charges in motion)External mechanical force were needed to maintain equilibriumWe are not yet studying moving charges as a source (effect of J on E)Assuming steady currents (J) and uniform conductivity (s):andLaplace equation can be used to also describe the electric potential in homogenous material when the E-field produces a steady currentNo unbalanced charge inside the conductorWe are limited to steady current in simple geometries but it shows that: Generalization: The E-field of electrostatics can also describesome situations inside conductors where it produces a steadycurrent!
5Inconsistencies???Can we extend the generalization further? Studying Ohm’s law more carefully reveals some potential inconsistencies:Is this in contradiction with Newton’s second law?No: it means that another “force” balances the electric forceObjects in free fall eventually reach a constant velocity…Something was hidden in Ohm’s law: A friction force opposite to the motion of the charge leads to a constant velocity (in average):Another potential inconsistency: the E-field must be null in metalsA: This is true only at equilibrium, but the system must first reachsuch equilibrium (with E=0) and to do this it must dispatchcharges accordingly. This takes some time. To generalize further: need to study systems out of equilibriumand, at equilibrium:
6Ensuring equilibriumOhm’s law implies that a friction insures constant velocity for the charge carrier, but doesn’t stop the motion there is something elseQ1: How is it possible that the current is the same in each segment of a circuit, i.e. that the current is constant (I=V/R)?Q2: Why currents circulate so quickly in a circuit while the drift velocity of charge carrier is smallA1: The E-field distributes charges in the conductor in such away to reach equilibrium and this guarantees steady currentIf current varies, it means that for a given segment Iin≠IoutCharge will accumulate, generating an E-field removing the charge accumulation and restoring Iin=Iout. The current is steady, in averageA2: Intermediate charges relay the effect of E to each others atthe speed of light, but charge reaction might be slow in average
7But there is more…From Q1 and Q2: we discussed how an E-field, the same as the one described in electrostatics, propagates in the circuit, moves charges around and reach an equilibrium state in the most efficient wayThis describes the situation in the wires, but this is not the full circuitWhat is the origin of this E-field satisfying electrostatic field equations?Why doesn’t the friction kill the current?There is another electric force, beside the electrostatics field in the wire, which is involved in a Ohm’s circuit:The force of the source of potential fs, contained in the batteryDoesn’t need to be a battery (chemical reaction) to provide the source of energy, could be piezoelectric, thermocouple, etc.To make sense of the empirical Ohm’s law, there must be:a dissipative force;a source force, confined to a small part of the circuit, doing work on the system to compensate dissipation;E-force of electrostatics propagating the effect of the source elsewhere
8Electromotive forceOn a close circuit, as considered in Ohm’s law, two components yield a force on the charge carriers: the source and the E-fieldThe (conservative) electrostatic field doesn’t contribute to the potential over the close circuit (no net work)But this f performs a work per unit charge on the system (EMF) which compensates the energy dissipation of the resistance:fs is a source of non-conservative electric fieldThe battery is analogous to a pump that does work on a fluidThis non-conservative force is localized in the batteryIt is responsible for the potential difference giving the current
9Motional EMF (I)Studying EMF is the best prospect for a complete generalization of electrostatics: it is a non-conservative fieldAn EMF converts some form of energy into an electric potentialLet’s consider devices that convert mechanical works into electric potential using… a magnetic field: generatorsPull the conducting bar at speed vFree charge in the bar are subject to a magnetic force along the barMoving charges = current EMFThe EMF is localized in the bar
10Motional EMF (II)Q1: How could the magnetic force generate an EMF if it does nowork on a system (normal to the direction of motion)Q2: How could the magnetic field keep the current for more thanan infinitesimal fraction of the time after the start of motion?F is always normal to the direction of motionThe expression of the EMF in terms of B is partly misleading:the work converted by the EMF to potential, doesn’t come fromthe constant magnetic field, but from the force pulling the circuitThe magnetic field is transmitting the force to produce the currentThe magnetic force generate a resistance that must be balanced by the external force to keep the current constant✔
11Varying magnetic fluxes A motional EMF results from a dynamics, a set of forces which alter the system over time.The external force is doing a work, which, in effect, alters the B field that generates the force setting the charges in motion EMFWe can express the EMF as a variation of the magnetic flux through the system: Flux ruleA this point, this equation is bounded to the empirical context by which it has been obtained:mechanical force giveing rise to a magnetic force producing a currentStill describes systems with electrostatics and magnetostatics fieldNet work due to external mechanical agentBut everything is in place for electrodynamics!!!
12Electromagnetic induction (I) Once again, the generalization of electrostatics to electrodynamics proceeds from a set of experiments leading to quantified laws:Experiment 1:Experiment 2:This is just the motional EMF described above, understandable in terms of statics fieldLorentz force at workMove the magnet rather than the circuitNO mechanical force is applied on circuitOnly relative motion matters as this makes B-field varyingvIBvIB
13Electromagnetic induction (II) Experiment 3:These experiences add to what is already established, and can lead to laws using the math structures already elaboratedThey generalize the conditions of applicability of previous experimentsStill have conditions of applicability themselves (e.g. classical physics), but don’t lead to new limitations not already in Coulomb or Ampere lawsThis is completely new!Stationary systemNo charges set to motionA current is obtainedAn E-field must have set the charges in motion Electromagnetic inductionIB(t)
14Faraday’s lawThe results of these experiments establish the connection between non-statics electric fields and variations of magnetic fields:Generalization of Flux ruleDoesn’t explain ALL EMF, but allowed to produce our generalizationConnection between EMF and circulation of electric fieldConsidering TOTAL E-field, with both cons. and non-cons. componentsElectric and magnetic fields are coupled like two aspects of the same thingSpecial relativity will finalize thisAll the laws and experiments of electrostatics can be described in electrodynamics by assuming B statics (dB/dt = 0)orElectrodynamics!
15Out-of-equilibrium (I) To illustrate how magnets generate EMF:consider a magnet of constant magnetization along its axismoving at constant velocity v through a loop wireBut the field is not uniform at the edgesThe flux will start from 0, reach a max ofm0Mpa2 and get back to 0 as the magnetapproaches, passes through and movesaway from the circuitThe induced EMF is the derivative of thisCurrent densitiesaLThe field inside a magnet is equivalent to B-field in a solenoid: The electric induction can be graphed as
16Out-of-equilibrium (II) Faraday’s law and EM induction have always been there in all earlier experiments, but have been neglected as side effectsNeglected geometrical complications on edges of distributionsAlways supposed long time after equilibrium gets achievedLong after the circuit get plugged, B-field sets to nominal value, etc.A lot of interesting physics is included in the out-of-equilibrium conditions that were neglectedObservations and experimentations are guided by theoryThe electrostatics field was already very general in the sense that it is the same field as the one studied by Ohm and FaradayTwo sources two undistinguishable components of E:E (cons.) generated by charges:E (non-cons.) generated by varying B-field:In absence of charges, dynamics of E is analogous to that of B
17Lenz’s law How do we know in which direction does the current flow? With motional EMF, we can use the right-hand rule as in magnetostaticsWhile it works in the case of an electromagnetic induction …If dF > 0, I is clockwiseIf dF < 0, I is counterclockwise… it doesn’t make much sense in the situation of the third Faraday’s experiment because:There are no moving free charges in a perpendicular B fieldThere are no cross products between two quantitiesLenz’s law is another rule to determine the direction of the current:Nature opposes to a change of flux so generates a current tocancel the change of fluxSee slide 15
18Preview of relativity (I) When considering a general circuit C moving in a varying magnetic field B we can express the variation of the B-field flux through C as:This flux variation generates an EMF which gives Faraday’s law:Prime on E’ means that the E-field is seen from the point of view of a charge on C, i.e. of an observer in the moving frame C. For such an observer, the charge is not moving in the B-field and the only possible source of motion for the charge is the E’-field in the circuitFor on observer in the lab, the force on the charge carrier comes from both an E-field in C and the B-field (Lorentz force):The force must be the same in all inertial framevs
19Preview of relativity (II) The electric field seen by the stationary observer is the one produced by the varying B field, exactly as prescribed by Faraday’s law:Two important conclusions from this derivation:Faraday’s law is valid for any state of motion of a system. We can always apply it to cases where a charge is stationary, and change the referential to the state of motion to the system keeping the same law Generalizes laws of electrostatics + Faraday to any state of motion We can redefine potential in a more general wayThe definition of E and B and the interpretation of phenomena depend on the state of motion of observers, but laws of physics stay the same special relativity
20InductanceFaraday’s law says that a varying B-field generates an EMF, and Biot-Savart’s law says that a current in a circuit produces a magnetic field. Putting these effects together:A varying current in a circuit C generates an EMF in any circuit C’in the vicinity of C, including C itselfThe EMF generated by the varying current is characterized by the the concept of inductanceAuto-reactive effect of electromagnetic induction, similar to the mutual adjustment of P/M and E/BThe inductance is a property of a system as a wholeThis is what is handled in actual experiment and techno applicationsComplementary to diff. equations that adopt a local point of viewHave beneficial and detrimental practical effects on circuits
21An analogyWe already encountered a property describing systems of conductors as a whole, which modifies the EMF of a circuit: the capacitanceBy analogy, let’s try to find a linear relationship between a varying current and the generated EMF, for which the proportionality constant depends only on the geometry of the whole systemStrategy to achieve this:Consider the flux of a B-field (the area is a geometric factor)Take the setup of the empirical Ampere’s experiment, but with I only in CUse Biot-Savart’s law to express the B-field in terms of the current IThe factor multiplying I would be only geometricalApply Faraday’s law (differentiate I with respect to t)
22Neumann formulaApplying the above strategy to the empirical Ampere’s law setup, to get the B-field flux through C2, allows us to meet our objectives:M21 depends only on the geometry of the systemApplying this to the EMF generated in a circuit, by varying currents in a set of circuits, yields:This is in complete analogy to the potential on a conductor generated by charges on a set of conductorsMij is called the mutual inductance of the circuit j on circuit iNewmann formula reveals that:The mutual inductance only depends on the size, shape and relative positions of the circuitsInductance is unchanged if we inverse the roles of circuits: M12=M21
23Self-inductance (I)A variation of current in a circuit C generates an EMF on C itselfSelf-inductance: L = M11Measured in unit of Henry (H)Positive definite quantityThis is the magnetic equivalent of the capacitance on a circuitThis definition of L is favored over a definition from F = LIIt offers a prescription of how it can be measuredThe sign informs on the physical effects:The self-inductance acts as a back EMF, a “protest” of a system againstchanges of its stateInductance plays a similar role in current as mass does in mechanics:It is an inertia of the system that opposes to changesHowever allows to solve problems with Neumann formula
24Self-inductance (II)Inductance can be added to a circuit by mean of an inductorA solenoid connected to the circuit is an inductorAssume L of such solenoid to be much bigger than the L of the wireStarting from no current in a circuit, the inductance slows down the build-up of the current by generating a back EMFIt is part of the EMF that establishes the voltage across the resistorUseful in many applications:E.g.: Can be used to delay and shape currents (electronics) and can be used to filter signal frequencies (radio)
25Energy of a B field (I)The B-field does no work on a charge, but it acts on a system, exerts a force, and energy must be provided to charges to create itThe B-field is expected to store energyWe can’t use the same approach as the one used in electrostatics to estimate this energy, but we will use Faraday’s lawStrategy: use the fact that the E-field, produced as B builds up, generates a force, which can do a work that can be used to get UBWe can estimate the work done by a battery against the back EMF in order to reach a steady current and the maximal B-field in the inductor;This work is reversible because it will be released from the inductor when the circuit will be switched off;We cannot use the energy given by the battery because, during the build-up, part of the energy supplied by the battery also gets dissipated as heat
26Energy of a B field (II)Procedure to estimate the work done by the source to be stored in the magnetic field of the inductor:At the beginning (t=0), there is no current in the circuit (i=0);i is the current at time t, and I is the max current reached at t=T;When i increases, B changes and the source must supply an EMF equal and opposite to the back EMF to maintain the current i;During a time dt, dq=idt charges will pass through the EMF, which will exert a work on them;Assuming stable geometrical configuration (L(t)=L)
27Where is the energy stored? This work, needed to yield the B-field in the inductor, can be expressed in terms of the vector potential and the current:This can be generalized to any current densityThe energy seems to be stored in the current density distribution!But… using Ampere’s law, we can express the current density in terms of the magnetic fieldThe energy is stored in the magnetic fieldThe situation is completely analogous to what we saw for UEor