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 5  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 E out ≠ E in Q: Is E in = 0 inside a charged conductor if E ext = 0?

  = 0 inside a conductor  Charges migrate to the surface to cancel E in  Can we prove it formally? 3) Any NET charge resides on the surface  If  = 0 but Q ≠ 0, then we must have  ≠ 0  Since this is a consequence of Gauss’ law (see 2), measuring the net charge Q in inside the conductor is a really precise test of 1/r 2  Concrete way to produce surface charge distribution empirically 4) The full conductor is an equipotential 6 Properties of conductors in E fields III

5) The E-field is perpendicular to the surface just outside the conductor 7 Properties of conductors in E fields IV  If E Tang ≠0, charges on the surface will move until equilibrium is reached  Also, if the surface is an equipotential (from 4), then, since is the gradient of V, it is perpendicular to surface  Also, can be proved from boundary conditions on general ground Get more “spherical” as  increases

Challenges 8 Q1 : If an external E -field acts on a surface of charge density  this surface will experience a force density N/m 2, but which E will it be? E is discontinuous at the surface so will E be E abov e ? E belo w ? 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? 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? Cavity

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 E above and E below 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 V i at the surface of a conductor i due to the E-field of a set of N-1 charged conductors, and the charge Q j 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 P ij : Apply a known variation of Q j and measure with a potentiometer how V j 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 Q 1 = -Q 2  The geometrical factor P exactly corresponds to the capacitance:  Units: [C] = C/V = Farad (F)  Use typically pF= 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