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PHY 042: Electricity and Magnetism Electrodynamics Prof. Hugo Beauchemin 1

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Introduction The strategy adopted to reveal the structure of E&M was to: A. Start from a concrete experimental setup to infer uniformities about nature and quantify them with laws; B. Introduce new concepts allowing the extension of these laws to a wider range of experiments and phenomena; C. 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 charges We described all magnetic phenomena with steady currents To generalize E&M phenomena further, we need to understand: 1) The impact of using steady currents on E-field phenomena 2) The impact of non-steady currents on both electricity and magnetism Will lead to Maxwell equations, electromagnetism and…light! 2

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Ohm’s law The 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 difference A source of free charges that can move in the E-field: Metallic wire allows for steady currents Ohm 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 geometry Uses newly made galvanometer to measure currents when varying V We want a connection between a steady current and the E-field: 3 Charges move in response to an E-field

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An 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 material Describe 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 equilibrium We are not yet studying moving charges as a source (effect of J on E) Assuming steady currents (J) and uniform conductivity ( ): and Laplace equation can be used to also describe the electric potential in homogenous material when the E-field produces a steady current No unbalanced charge inside the conductor We are limited to steady current in simple geometries but it shows that: Generalization: The E-field of electrostatics can also describe some situations inside conductors where it produces a steady current! 4

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Inconsistencies??? 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 force Objects 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 metals A: This is true only at equilibrium, but the system must first reach such equilibrium (with E=0) and to do this it must dispatch charges accordingly. This takes some time. To generalize further: need to study systems out of equilibrium 5 and, at equilibrium :

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Ensuring equilibrium Ohm’s law implies that a friction insures constant velocity for the charge carrier, but doesn’t stop the motion there is something else Q1: 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 small A1: The E-field distributes charges in the conductor in such a way to reach equilibrium and this guarantees steady current If current varies, it means that for a given segment I in ≠I out Charge will accumulate, generating an E-field removing the charge accumulation and restoring I in =I out. The current is steady, in average A2: Intermediate charges relay the effect of E to each others at the speed of light, but charge reaction might be slow in average 6

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But 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 way This describes the situation in the wires, but this is not the full circuit What 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 f s, contained in the battery Doesn’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: 1. a dissipative force; 2. a source force, confined to a small part of the circuit, doing work on the system to compensate dissipation; 3. E-force of electrostatics propagating the effect of the source elsewhere 7

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Electromotive force On a close circuit, as considered in Ohm’s law, two components yield a force on the charge carriers: the source and the E-field The (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: f s is a source of non-conservative electric field The battery is analogous to a pump that does work on a fluid This non-conservative force is localized in the battery It is responsible for the potential difference giving the current 8

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Motional EMF (I) Studying EMF is the best prospect for a complete generalization of electrostatics: it is a non-conservative field An EMF converts some form of energy into an electric potential Let’s consider devices that convert mechanical works into electric potential using… a magnetic field: generators 9 Pull the conducting bar at speed v Free charge in the bar are subject to a magnetic force along the bar Moving charges = current EMF The EMF is localized in the bar

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Motional EMF (II) Q1: How could the magnetic force generate an EMF if it does no work on a system (normal to the direction of motion) Q2: How could the magnetic field keep the current for more than an infinitesimal fraction of the time after the start of motion? F is always normal to the direction of motion The expression of the EMF in terms of B is partly misleading: the work converted by the EMF to potential, doesn’t come from the constant magnetic field, but from the force pulling the circuit The magnetic field is transmitting the force to produce the current The magnetic force generate a resistance that must be balanced by the external force to keep the current constant 10 ✔

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Varying 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 EMF We can express the EMF as a variation of the magnetic flux through the system: Flux rule A 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 current Still describes systems with electrostatics and magnetostatics field Net work due to external mechanical agent But everything is in place for electrodynamics!!! 11

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Electromagnetic induction (I) Once again, the generalization of electrostatics to electrodynamics proceeds from a set of experiments leading to quantified laws: Experiment 1: Experiment 2: 12 I B v I B v This is just the motional EMF described above, understandable in terms of statics field Lorentz force at work Move the magnet rather than the circuit NO mechanical force is applied on circuit Only relative motion matters as this makes B-field varying

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Experiment 3: These experiences add to what is already established, and can lead to laws using the math structures already elaborated They generalize the conditions of applicability of previous experiments Still have conditions of applicability themselves (e.g. classical physics), but don’t lead to new limitations not already in Coulomb or Ampere laws 13 Electromagnetic induction (II) I B(t) This is completely new ! Stationary system No charges set to motion A current is obtained An E-field must have set the charges in motion Electromagnetic induction

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Faraday’s law The results of these experiments establish the connection between non-statics electric fields and variations of magnetic fields: Generalization of Flux rule Doesn’t explain ALL EMF, but allowed to produce our generalization Connection between EMF and circulation of electric field Considering TOTAL E-field, with both cons. and non-cons. components Electric and magnetic fields are coupled like two aspects of the same thing Special relativity will finalize this All the laws and experiments of electrostatics can be described in electrodynamics by assuming B statics (dB/dt = 0) 14 or Electrodynamics!

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Out-of-equilibrium (I) To illustrate how magnets generate EMF: consider a magnet of constant magnetization along its axis moving at constant velocity v through a loop wire But the field is not uniform at the edges The flux will start from 0, reach a max of 0 M a 2 and get back to 0 as the magnet approaches, passes through and moves away from the circuit The induced EMF is the derivative of this 15 The field inside a magnet is equivalent to B-field in a solenoid: L a The electric induction can be graphed as Current densities

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Faraday’s law and EM induction have always been there in all earlier experiments, but have been neglected as side effects Neglected geometrical complications on edges of distributions Always supposed long time after equilibrium gets achieved Long 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 neglected Observations and experimentations are guided by theory The electrostatics field was already very general in the sense that it is the same field as the one studied by Ohm and Faraday Two 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 16 Out-of-equilibrium (II)

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Lenz’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 magnetostatics While it works in the case of an electromagnetic induction … If d > 0, I is clockwise If d < 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 field There are no cross products between two quantities Lenz’s law is another rule to determine the direction of the current: Nature opposes to a change of flux so generates a current to cancel the change of flux 17 See slide 15

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Preview 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 circuit For 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 frame 18 vs

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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: 1. 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 way 1. The 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 19 Preview of relativity (II)

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Inductance Faraday’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 itself The EMF generated by the varying current is characterized by the the concept of inductance Auto-reactive effect of electromagnetic induction, similar to the mutual adjustment of P/M and E/B The inductance is a property of a system as a whole This is what is handled in actual experiment and techno applications Complementary to diff. equations that adopt a local point of vi ew Have beneficial and detrimental practical effects on circuits 20

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An analogy We already encountered a property describing systems of conductors as a whole, which modifies the EMF of a circuit: the capacitance By 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 system Strategy to achieve this: 1. 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 C 2. Use Biot-Savart’s law to express the B-field in terms of the current I The factor multiplying I would be only geometrical 3. Apply Faraday’s law (differentiate I with respect to t) 21

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Neumann formula Applying the above strategy to the empirical Ampere’s law setup, to get the B-field flux through C 2, allows us to meet our objectives: M 21 depends only on the geometry of the system Applying 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 conductors M ij is called the mutual inductance of the circuit j on circuit i Newmann formula reveals that: The mutual inductance only depends on the size, shape and relative positions of the circuits Inductance is unchanged if we inverse the roles of circuits: M 12 =M 21 22

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Self-inductance (I) A variation of current in a circuit C generates an EMF on C itself Self-inductance: L = M 11 Measured in unit of Henry (H) Positive definite quantity This is the magnetic equivalent of the capacitance on a circuit This definition of L is favored over a definition from = LI It offers a prescription of how it can be measured The sign informs on the physical effects: The self-inductance acts as a back EMF, a “protest” of a system against changes of its state Inductance plays a similar role in current as mass does in mechanics: It is an inertia of the system that opposes to changes However allows to solve problems with Neumann formula 23

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Inductance can be added to a circuit by mean of an inductor A solenoid connected to the circuit is an inductor Assume L of such solenoid to be much bigger than the L of the wire Starting from no current in a circuit, the inductance slows down the build-up of the current by generating a back EMF It is part of the EMF that establishes the voltage across the resistor Useful in many applications: E.g.: Can be used to delay and shape currents (electronics) and can be used to filter signal frequencies (radio) 24 Self-inductance (II)

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Energy 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 it The B-field is expected to store energy We can’t use the same approach as the one used in electrostatics to estimate this energy, but we will use Faraday’s law Strategy : 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 U B We 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 25

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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) 26 Energy of a B field (II)

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Where 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 density The 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 field The energy is stored in the magnetic field The situation is completely analogous to what we saw for U E 27 or

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