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PHY 042: Electricity and Magnetism Conductors Prof. Hugo Beauchemin 1 Introduction We have seen most of the powerful concepts needed to understand electrostatics We now turn to concrete applications, solving the equations we saw for various types of system encountered in experiments and technological devices We start with conductors. Why??? They are used in most of the experiments and devices probing or using electrostatics Why conductors are so used in E&M? They supply free charges The potential on their surface can easily be controlled ⇒ Provide controllable boundary conditions in many setups 2 What is a conductor? In a solid, the wave functions of electrons in atoms form bands Valence: electrons bounded to an atom Conduction: electrons delocalized and free to move to other atoms Conductors: No energy supply is needed to move electrons in the conduction band 3 Properties of conductors in E fields I There are five general electrostatic properties of conductors: 1) E=0 inside a conductor in electrostatic conditions If E≠0, an E-forced would be exerted on the free charges of the metal currents Q: Could it be that electrostatics simply doesn’t apply to conductors??? A: No. The solution comes from a phenomena known before Coulomb: electrostatics induction 4 Steps showing how induction can be used to charge a conductor Properties of conductors in E fields II The charges induced by the external field in the conductor are, after an equilibrium is reached, responsible for E=0 inside that conductor Q: Do the induced charges cancel E outside the conductor too? A: Not necessary, they generate a surface charge density and boundary conditions tell us that Eout ≠ Ein Q: Is Ein= 0 inside a charged conductor if Eext= 0? 5 Properties of conductors in E fields III 2) r = 0 inside a conductor Charges migrate to the surface to cancel Ein Can we prove it formally? 3) Any NET charge resides on the surface If r = 0 but Q ≠ 0, then we must have s ≠ 0 Since this is a consequence of Gauss’ law (see 2), measuring the net charge Qin inside the conductor is a really precise test of 1/r2 Concrete way to produce surface charge distribution empirically 4) The full conductor is an equipotential 6 Properties of conductors in E fields IV 5) The E-field is perpendicular to the surface just outside the conductor If ETang≠0, charges on the surface will move until equilibrium is reached Also, if the surface is an equipotential (from 4), then, since gradient of V, it is perpendicular to surface Also, can be proved from boundary conditions on general ground Get more “spherical” as r increases is the 7 Challenges Q1: If an external E-field acts on a surface of charge density s, this surface will experience a force density N/m2, but which E will it be? E is discontinuous at the surface so will E be Eabove? Ebelow? Sometime one, sometime the other? An intermediate value? Q2: If a charge +q is brought close to a neutral conductor, will it be attracted by it? Will the system have a back-reaction? Cavity Q3: If charges are put in a cavity inside a neutral conductor, will an observer outside the conductor know if there is a charge in the cavity? Can he know what is the shape of the cavity? Q4: What will be the effect of an external field inside an empty cavity? 8 Answers to the challenges I A1: It will be the average of the two electric field: If we take a tiny flat patch on the conductor, the force on that patch will be due to all E-field contributions except from the patch itself However, the discontinuity in the E-field just above and below the patch is due to the charge density distributed on the patch itself We must take the average of Eabove and Ebelow to eliminate the patch contribution to E, which should not affect the force on the patch A2: The field outside is non-null and will produce an induced charge of opposite sign on the conductor. The charge q and the induced charge –q will attract each other Of course, we must maintain the system at equilibrium 9 Answers to the challenges II A3: A charge distribution will be induced on the inner and outer surfaces of conductor. The outer one will communicate the presence of the charge q in the cavity to the outside world. The E-field seen by the observer is due to the induced charge distribution on the outer surface of the conductor. The information about the shape of the cavity is thus completely lost. The charge in the cavity is isolated from the outside world A4: If we place an apparatus inside a cavity in a conductor, it will suffer no effect of any external charge or electric field ⇒ Faraday Cage (shielding) 10 Capacitors I One of the oldest and most frequent use case of conductors in electrostatics is to: Store electric energy (in a static electric field) CAPACITORS are the devices that can do this Capacitors consist in a system of two conductors containing equal and opposite charges. The energy is stored in the E-field between them. This is analogous to the capacity of a tank in containing water. The maximum quantity of electrostatic energy that can be stored in a capacitor is determined by: The capacitance (C) It is a geometrical factor that just depends on the shape of the surfaces forming the capacitor, independent of the charge it accumulates. The actual energy stored depends on the charge accumulated if not max 11 Capacitors II We can find a relationship between the potential Vi at the surface of a conductor i due to the E-field of a set of N-1 charged conductors, and the charge Qj contained on each of these N-1 conductors: This is a matrix relationship P is the geometrical-only factor where: This math relationship tells us how we can actually measure Pij: Apply a known variation of Qj and measure with a potentiometer how Vj varies 12 Capacitors III The “amount” of electric field stored between two conductors of a capacitor is given by the potential difference between them Using the general relationship V = PQ applied to a system of two conductors with Q1 = -Q2 The geometrical factor P exactly corresponds to the capacitance: Units: [C] = C/V = Farad (F) Use typically pF=10-12 F By doing a work to transfer charges from a conductor to the other, we store energy in the electric field of the device 13