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

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Presentation on theme: "PHY 042: Electricity and Magnetism Electrodynamics Prof. Hugo Beauchemin 1."— Presentation transcript:

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

2 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

3 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

4 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

5 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 :

6 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

7 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

8 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

9 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

10 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 ✔

11 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

12 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

13  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

14 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!

15 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

16  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)

17 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

18 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

19  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)

20 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

21 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

22 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

23 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

24  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)

25 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

26  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)

27 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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