Chapter 27: Current & Resistance

Transcription

1 Chapter 27: Current & Resistance When motion takes place within a conucting path that forms a close loop, the path is calle an electric circuit. Funamentally, electric circuits are a means for transferring energy from one place to another. As charge particles move within a circuit, PE is transferre from a source to a evice in which that energy is either store or converte to another form: soun (stereos), heat (toasters), or light (light bulbs). From a technological stanpoint, electric circuits are useful because they allow energy to be transferre without any moving parts. To unerstan the behavior of currents in circuits, we ll escribe the properties of conuctors (resistivity, length, area an temperature). That is, why short, fat, an col copper wires are better conuctors than long, skinny an hot steel wires. Reminer: Electric Potential Difference or Voltage sets up the electrical lanscape that charges will follow. CURRENTS IN CONDUCTING MATERIALS Electric Current is any motion of charge from one region to another just as water current is the flow of water molecules. We ll iscuss currents in conucting materials in this section. The vast majority of technological applications of charges in motion involve currents of this kin. There are two situations to consier: (1) E = 0 an (2) E V insie = 0 is only for perfect conuctors In electrostatic situations, V insie = constant or V insie = 0 everywhere within a conuctor. If there is no potential ifference, then there is NO net current insie a conuctor. However, this oes not mean that all charges within a conuctor are at rest. In fact, there is lots of motion insie of orinary metals (such as Cu or Al), it is just ranomly moving electrons in all irections (very similar to gas molecules) at very high spees ( 10 6 m/s 2 million mph). Because it is ranom, there is NO net irection of charge flow an therefore, NO net current insie a conuctor. 2. V insie 0 for a real conuctor Most conuctors are not perfect an therefore, have a small E-fiel insie the wires/conuctors. Suppose V insie 0 is establishe insie a conuctor. A free charge will slie own the potential incline. If the free charge ha no interactions whatso ever, the free charge woul gain to enormous KE or spees ( 10 6 m/s). However, in a conuctor the free charge sees massive, nearly stationary ions (i.e. vibrating ions) in a wire, which are in the way. Analogy: electrons are like little bb s an the ions are like softballs. This results in charges unergoing a massive number of collisions that has a ranomizing effect on the motion of the charges. The effect of V insie on the charge gives it a very slow net motion or DRIFT in the irection from high to low potential. So, there are two parts to the motion of a free charge in a conuctor: (1) ranom natural motion cause by collisions & (2) rift cause by V insie. When we speak of a moving charge insie a conuctor we escribe it in terms of the DRIFT VELOCITY v. 27.1

2 DEMOs (i) incline with no pegs vs (ii) pinchecho machine to show iffuse motion 2D computer simulation of electron flow in a gas. Is it art or science? The electrons enter at top left an sprea out over the electrical lanscape where the tracks vary from ark (few electrons) to bright (many electrons). Applet EJSS Drift Velocity Moel In terms of work an energy, the work one by the electric fiel on the electrons is converte into K, which transfers it to the copper ions an increase the amount of heat (or temperature) in the wire. In summary: most of the work one goes into heating the conuctor, not in making the charge move faster an faster. Heating the conuctor has some useful applications electric toasters, but in many situations it is an unwante an unavoiable byprouct. Different materials have ifferent moving charge particles: Metals negative charges Ionize gases, ionic solutions, an semiconuctors positive or negative charges In a segment of a carrying-current wire, a moving positive charge moves in the irection of high to low potential (with the E-fiel) whereas a moving negative charge moves from low to high potential (against the E-fiel). In either case, the same rift spee is the same. Definition The current i is efine to be in the irection in which there is a positive charge flow. Caution: Although we refer to the irection of a current, current is NOT efine as a vector quantity. In a current-carrying wire, the current is always along the length of the wire, regarless of whether the wire is straight or curve. No single vector coul escribe motion along a curve path, which is why current is not a vector! We escribe current as though it consiste entirely of positive charge flow, even in cases in which we know that the actual current is ue to electrons: Conventional current flow of positive charge Again: the irection of the conventional current is NOT necessarily the same as the irection in which charge particles are actually moving! However, the minus sign has NO EFFECT in analyzing electric circuits. 27.2

3 Explain Galvani, Volta an Ben Franklin s role in this matter. Definition The current is the net charge q that flows through the crosssectional area A in a time t: q Current = i = t Units: [i] = [q/t] = C/s = 1 Ampere 1 A Microscopic Current an Drift Velocity To unerstan the current insie a wire, I first nee two efinitions to express current in terms of rift velocity: 1. Assume all charges have the same rift velocity v. 2. The concentration n of charges e per unit volume is Concentration of number of charges # n = = 3 particles in a volume volume m To calculate the current, we first nee to etermine the total charge Q passing through a cross sectional area. A current-carrying wire contains n charges per volume, each with charge q an moving with average spee v. In a time t, all the charges will move a istance x = v t as shown. A cylinrical portion of the wire of length x an cross-sectional area A contains the total charge of ( ) ( ) Q = charge per carrier number of carriers number of charges = volume = q n x A = nqv A t ( charge per carrier) ( volume create by moving charges) ( ) x= v t All the charges ( Q) will pass through the right en of the cyliner uring a time interval t, an the resulting current is Q i = = nev A microscopic current when v = constant t It is customary to efine instea, at the microscopic level, the current ensity: Current i 2 J Current Density = J = = nev Units: [J] = [i]/[a] = A/m Area A constant v Either we have constant or varying current ensities: Example 26.1 nev A, J = J(V), v v = constant an inepenent of voltage constant an epens on voltage An 18-gauge copper wire (the size usually use by lamp cors) has a iameter of 1.02 mm. This wire carries a constant current of 1.67 A to a 200-watt lamp. The ensity of free electrons is electrons/m 3. Fin magnitues of the (a) current ensity an (b) rift velocity. SOLUTION a. The current ensity is 27.3

4 i 1.67A 6 3 J = = = A/m = J A π( ) 2 A= 1π 4 b. Using the current ensity to solve for the rift velocity, solving J A/m 4 v for v = = J = nev = m/s = 0.15 mm/s = v ne ( m )( C) At this spee an electron woul require 1 hr, 50 min to travel the length of a wire 1 m long. The spee of ranom motion (10 6 m/s) is times faster. Picture wise, the electrons are bouncing aroun frantically, with a very slow an sluggish rift! If the electrons move this slowly, you may woner why the light comes on right away when you turn on the switch. Let use an analogy to explain it. Interpretation: consier what happens when a jump start is given to a ba battery. If the jumper cables are 2m-long, the charges coming out of one car woul take 3 hrs, 40 min to reach the other car. If you are of the unerstaning that charges flow out of the goo battery an charge up the ba battery, then your wait shoul be a consierable to await the arrive of the electrons. Picture wise, when the jump cables are attache to the car with the ba battery, the ba battery an most importantly, the car s starter is on the potential incline. As soon as the cables are attache, ALL of the charges will move simultaneously an the charges alreay in the starter will slie from high to low potential. Once the car s engine starts, then the ba car s alternator begins the process of charging the ba battery. Other examples that are similar are (i) electric shock an (ii) light bulbs Interesting point Charges in a conuctor have very high average ranom velocities (2 million mph). One woul think that the electrons woul leave the surface at this spee. The answer is no since there is an attractive electrical force between the ions an the electrons. By contrast, the rift spee ue to the electric fiel is very slow: 10 4 m/s = 0.1 mm/s! The electrons are literally trying to cross a mosh pit. Question: how can the current be moving so slowly? Answer: it is the vast number of collisions that the electron experiences. For the wire: when you throw the switch an electric fiel is set up in the wire with a spee approaching the spee of light, an electrons start to move all along the wire at very nearly the same time. The time that it takes an iniviual electron to get from the switch to the light bulb isn t really relevant. Electrons at the en immeiately start coming out. When you buy electrical energy, we on t buy electrons; there are plenty of those in the wire we alreay have we buy aitional amounts of electron motion. The power company pushes aroun our electrons an sens us a bill for how much work they i in the process. RESISTIVITY/CONDUCTIVITY The current ensity J in a conuctor epens on the E-fiel an on the properties of the material. In general, this epenence can be quite complex. But for such materials like metals at fixe temperatures, there is a simple linear relationship: E E J = constant J That is, the E-fiel prouces a certain J-value where the proportionality constant is the electrical resistivity ρ of a material. This is known as microscopic form of Ohm s Law: 27.4

5 E(causes the current) V/m V J = or E = ρ J Units: [ ρ] = [E]/[J] = = 2 m Ω m ρ(scales the current) A/m A On the other han, the Drue s moel of microscopic conuctivity is a powerful moel of a material s ability to transfer charge. Electrical conuctivity σσ, which is the reciprocal of resistivity is conuctivity σ= Units: [ σ ] = = ρ [ ρ] Ω m where E σ= Drue's moel J Physical meaning: A goo conuctor, like copper, has a large σ an a small ρ, while a goo electrical insulator, like glass, has an extremely small σ an a huge ρ. The table below is of electrical conuctivities an resistivities of various materials. The range of conuctivities an resistivities foun in orinary materials varies by many orers of magnitue. We can then write the following states: perfect conuctor ρ = 0 or σ= perfect insulator ρ = or σ= 0 The electrical resistivities of insulators to metals or electrical conuctivities of conuctors to insulators are enormous factors: 27.5

6 14 ρinsulators ρ 16 σ woo conuctors ρconuctors 10 ρ σ copper insulators This means that in general, insulators are more resistive than conuctors. For example, the ratio of resistivities of woo to copper implies that electrons travel through a piece of copper with ease, they can harly move at all through pieces of woo since woo is more resistive than copper by a factor of State similarly but ifferent, the ratio of conuctivities states exactly the same thing, conuctors are better at conucting electrons than insulators are. Electrical vs. Thermal Conuctors Conuctivity is the irect electrical analog of thermal conuctivity: goo conuctors of electricity are also goo conuctors of heat (metals) whereas poor electrical conuctors are also poor heat conuctors (insulators). The reason for this is in a metal the free electrons that carry charge in electrical conuctors also provie the principal mechanism for heat conuction. Therefore, we expect a correlation between electrical an thermal conuctivity. However, there is a huge ifference: heat conuction is just the collision of particles moving through a meium. Question: Why can t we just heat up a wire to prouce a current (or move charge)? It s because of the σ-values! There is an enormous ifference between σ electrical vs. σ thermal. Suppose we have an electrical wire with insulation aroun it. Why oesn t current flow through the insulation? One has to compare the ratio of electrical an thermal conuctivities: σ metal 22 σ metal 3 10 vs. 10 σinsulator σ electrical insulator thermal The electrical ratio ( ) show an enormous ifference metal an insulator conuctivities, so electrically it is easy to confine electric currents to well-efine paths in copper wires in circuits since insulators greatly iminish current flow. On the other han, the thermal ratio ( 10 3 ) shows just the opposite, that there is a slightly avantage for heat current to move through the copper wire than moving through an insulator. Therefore, it is impossible to confine heat currents to the same extent as electrical currents. A material that obeys Ohm s law reasonably well is calle an ohmic (or linear) conuctor, E ρ= = constant Ohmic J When Ohm s law is obeye, ρ is constant an inepenent of the magnitue of the E- fiel, so E is irectly proportional to J. However, if there are materials that show substantial epartures from this, these materials are known as Non-ohmic (nonlinear) conuctors (where J has some complicate epenence on E): ρ= E constant Nonohmic J For example, a ioe has an exponential epenence on the voltage: Applet EJSS Drift Velocity Moel 2/3 J A V + constant MACROSCOPIC OHM S LAW As I have been saying for some time now, the E-fiel pictures is complicate an want the potential picture instea. The relationship between resistivity an resistance an E- fiel an potential relationships are ρl E V ρ R = an J = i = A ρ R 27.6

7 It is straight forwar to move from the microscopic picture to the macroscopic picture of Ohm s law. This can be observe from RA E = ρ J = J EL = V = Ri R=ρL/A L In summary, L V = Ri where R =ρ A Applet EJSS Drift Velocity Moel Electrical resistance has four characteristics: area, length, material, an temperature ρ(t)l R = A 1. Material Depenence: Conuctors move charge whereas insulators o not conuctors lower resistance insulators higher resistance 2. Area Depenence: with water, as the iameter of the pipe increases, so oes the amount of water that can flow through it. With electricity, conucting wires take the place of the pipe. As the cross-sectional area of the wire increases, so oes the amount of electric current (number of electrons) that can flow through it. Examples: Electrical fires Electrical fires are usually cause by having too much current in a smaller wire. Why the ifference in size between the outlet plug for a home clothes rier verses the outlet plug for a lamp? DEMO Compare a power supply wire vs. lamp lea 3. Length Depenence: if the length of a wire is increase, oes the number of collisions increase? Yes! longer length smaller current higher resistance more collisions shorter length higher current lower resistance less collisions Interesting point: Cray-2 Example: Cray-2. No wire was longer than 6 inches. Why? The Cray-2, installe in 1985 at LLNL, has 240,000 computer chips packe into the C-shape cabinet. A liqui coolant must wash continuously over the circuits to issipate heat that woul otherwise melt the machine. 4. Temperature Depenence: If one heats up a metal an its temperature increases, the resistance goes up. As the copper ions acquire KE from the heat energy, these 27.7

8 once stationary ions are now moving. It is now more likely that the electrons will collie more frequent with the copper ions, which results in a higher resistance an reuce current flow. higher temperature higher resistance more collisions lower temperature less collisions lower resistance Examples: lightbulbs change brightness as they heat up; rear car winows an resistors. The resistivity of a metallic conuctor nearly always increases with increasing temperature. Physically, as the temperature increases, the ions of the conuctor vibrate with greater amplitue, making it more likely that a moving electron will collie with an ion. We fin that resistivity is linear in temperature for small temperature regions accoring to ρ=ρ ρ ρ0 α (T T 0) α= = reference resistivity at T = 20 C thermal coefficient of resistivity Example 27.2 A wire 4.00 m long an 6.00 mm in iameter has a resistance of 15.0 mω. A potential ifference of 23.0 V is applie between the ens. (a) What is the current in the wire? (b) What is the magnitue of the current ensity? (c) Calculate the resistivity of the wire material an ientify the material. Solution a. Using the efinition of Ohm s law: V 23V 3 i = = = A = i 3 R Ω b. The current ensity is 3 i J = = = A/m = J 3 2 A π (3 10 m) c. The resistivity is RA (15 10 Ω) π ( m) 8 ρ= = = Ω m =ρ L 4.0m The material is platinum. Application of Ohm s Law Electric Shock Nerves are on the orer of 1μm, which is about the size of a spier web (1-4 μm). Because nerves are so thin, they cannot hanle very much current. Suppose that you are intereste in trying to fin out how much electric shock you can get out of a car battery (which has 12-volts). If you touch the two terminals of the car battery, the current path is then from your right han to your left han, that inclues your heart in the circuit. How much electric shock you will receive will epen upon your electrical resistance. There are three cases. 27.8

9 1. Dry fingers/han: 500,000Ω Your skin of your hans gives you the highest electrical resistance of the human boy. So when you grab the two battery terminals, Ohm s law preicts that 2. Wet fingers/han: 1,000Ω If you were able to bypass the skin of your fingers/hans, your resistance ramatically ecreases because the current path is most mae of water. So Ohm s law tells us that 3. Salty Wet fingers: 100Ω If you a salt to water, water becomes a better conuctor, which means that it lowers the resistance. Since your organs/bloo are really a salty water solution, when you bypass your skin Ohm s law tells us again that 12 volts I = = ma 500,000Ω too low of a resistance to be felt 12 volts I = = 12 ma 1,000Ω very painful shock but not enough to kill you 12 volts I = = 120 ma 100Ω A current this large is enough to stop the heart Current it takes with DMM that kills a sailor True Story (Darwin Awar Winner 1999) A US Navy safety publication escribes injuries incurre while oing on'ts. One page escribe the fate of a sailor playing with a DMM in an unauthorize manner. He was curious about the resistance level of the human boy. He ha a Simpson 260 DMM, a small unit powere by a 9- volt battery. That may not seem powerful enough to be angerous but it can be ealy in the wrong hans. The sailor took a probe in each han to measure his boily resistance from thumb to thumb. But the probes ha sharp tips, an in his excitement, he presse his thumbs har enough against the probes to break the skin. Once the salty conucting flui known as bloo was available, the current from the DMM travelle right across the sailor's heart, isrupting the electrical regulation of his heartbeat. He ie before he coul recor his resistance. The lesson? The Navy issues very few objects which are esigne to be stuck into the human boy. Microscopic View of Electric Circuits Questions: are charges use up in a circuit? How is it possible to create an maintain a nonzero E-fiel insie a wire? What is the role of a battery in a circuit? Review of previous chapter 1. Nonequilibrium Systems When an E-fiel is applie to a conuctor, the loose charges move until the conuctor prouces an inuce fiel that cancels out the external fiel, so that E insie = E ext E in = 0. We say that the system reaches equilibrium very quickly ( s). That is, equilibrium means that no current is flowing or that the rift velocity is zero (v ). Equilibrium E = 0 v = 0 (no net current) insie In an electric circuit, the system oes not reach equilibrium. Despite the motion of charges, the net fiel insie the conuctor oes not go to zero, an charge flow continues for a long perio. Nonequilibrium E 0 v = constant (net current) insie So steay state in a circuit means that charges are moving, but their rift velocity are constant (v = constant) at any location an no excess charge buil up anywhere. 2. Current in Different Parts of a Circuit What happens to the charges that flow through the circuit? Do they get use up? Is there a ifferent amount of current in ifferent parts of a (single loop) circuit? How woul you expect the amount of electron current at location A to compare to the electron current at location B? Most people chose one of the following possibilities: 27.9

10 There shoul be no current at all at B, because all of the electron current coming from the negative of the battery is use up in the bulb. The current at B shoul be less than the current at A, because some of the current is use up to make the bulb give off light an heat. The current shoul be the same at A an at B. Let look at these from a theoretical viewpoint. i. Can current be use up in the bulb? Current in a metal wire is simply the flow of electrons past a point. Can electrons be use up in the bulb? Electrons alone cannot be estroye, because this woul violate the funamental principle of conservation of charge. If negative particles were estroye, the universe woul become more an more positive! ii. Coul electrons just accumulate in the bulb, so that the current at B woul be less than at A? If electrons accumulate in the bulb, the bulb woul become negatively charge. The negative charge of the bulb woul become large enough to repel incoming electrons an stop the current. Since in the steay state current keeps flowing, this can t be happening. iii. We have rule out (by contraiction) that the current cannot be use up in a light bulb an conclue that the current shoul be the same at A an at B. Using a DMM, we woul fin that inee the current flowing into the light bulb is the same as the current flowing out of the light bulb. If electrons on t get use up in the bulb, what makes the light that we see? In other wors, what is use up in the light bulb? You can t get something for nothing! Rub your hans together as har as you can an as fast as you can for several secons. What change o you observe in your hans? Your hans get hot, but they are certainly not use up! So, what oes get use up to warm up your hans or to make a light bulb hot enough to emit visible light (blackboy raiation)? In both cases energy is being use up (or transforme) into thermal energy. In orer to move your hans, you must use up some of the store chemical energy in your boy. Similarly, forcing electrons through the filament heats the metal with a kin of friction, an this requires some of the chemical energy store in the battery to be use up. However, this process occurs without estroying the electrons that rub against each other. 3. What is pushing electrons through the wire? If the conuction electrons in a metal i not interact at all with the lattice of positively charge atomic ions, then once a current got starte, it woul flow forever. However, the moving electrons o interact an lose energy to the lattice, increasing the thermal motion of the atoms. We etect this by the wire getting hot. Unless an E-fiel is present to increase the momentum of the conuction electrons repeately after collisions, their energy will quickly be issipate (via atomic friction) an the current flow will stop. DEMO slie a block across the table an ask why oes it not keep moving? As we alreay know, there can be no excess charge insie a conuctor so that the wire is electrically neutral. Why? Because electrons repel each other an charges move to the surface as far as possible from each other. So is it the electrons that are pushing each other in wires that make them move? NO! What we mean by a neutral wire is that the number ensity of conuction electrons insie a metal wire must equal the number ensity of positive atomic ions. If we look at the forces on a conuction electron in a wire, there are two of them: electron repulsion an the attractive ion force, which cancel each other out. Therefore, electrons in a metal o not interact with each other an behave as an ieal gas. So electrons cannot continually push each other through the wire like peas pushe through a tube from one en of the tube. We conclue that it must be other charges somewhere outsie the wire that make an E-fiel throughout the wire that continually rives the electron current

11 How big oes an E-fiel have to be to push electrons through a wire? For copper, E copper 1 V/m. 4. Current an E-fiel To explain the role of the E-fiel in a wire, I introuce KCL, which is not a funamental law but a consequence of conservation of charge an efinition of steay state. Kirchhoff s Current Law (KCL): the current entering a noe in a circuit is equal to the current leaving that noe. Earlier we confirme that the current was the same everywhere in a circuit. However, we know that the current i epens on the area of a wire (i = neav ). Suppose we have a circuit in which a wire leas to another, thinner wire of the same material. KCL tells us that the current coming out of the thick wire must be the same as the current through the thin wire. We conclue that A i = constant = ne v = nea This makes complete since in a water hose, if you reuce the area of the water flow, the spee of the flui increases. Immeiately, this leas to an important conclusion: if the rift spee increases in the thin wire, then the thinner wire must have a larger E-fiel than the thick wire: thin wire higher E-fiel Ethin wire > Ethick wire thick wire lower E-fiel If the area of the wire oes not change, then the E-fiel oes not change an must be the same everywhere in the wire. In other wors, even if I twist or loop a wire, the E-fiel will be the same an uniform across the whole wire. thick thin v constant A E = constant 5. What charges make the E-fiel in the wires? As we have alreay conclue, there must be an E-fiel insie the bulb s filament, forcing electrons through it. Since E-fiels are prouce by charges, there must be excess charges somewhere to prouce the E-fiel insie a wire in a circuit. There are two important questions to answer: 1. Where might these charges be? 2. What istribution of charges coul prouce a pattern of E-fiel following the wire? We have alreay conclue that there are no excess charges in the interior of the bulb filament an wires because they are conuctors. The next reasonable assumption is that the charges that make the E-fiel are in an on the battery, since that s the active element in the circuit. Perhaps there are + charges on the + en of the battery, an charges on the en? One way to emonstrate the answer is that the battery will have a ipole fiel (in the absence of the wires), an we know that E ipole 1/r 3. Suppose I have a battery-bulb circuit with leas that are 10 cm from the battery. If I move the bulb until it is 1 cm closer, then a rough estimate is that the E-fiel shoul increase by a factor of stronger E-fiel an increase in current Eipole = r (r/10) r brighter bulb DEMO battery-bulb with long leas The bulb shoul be enormously brighter, yet this is not what happens. A secon argument is to rive the current through the bulb filament; the E-fiel must have a component parallel to the filament. If the charges responsible for making the E-fiel were solely in an on the battery, rotating a bulb shoul make a big ifference in the brightness (or even make the bulb go out), but this oesn t happen. We conclue that charges in an on the battery cannot be the only contributors to the E-fiel in the bulb 27.11

12 filament. There must be other charges somewhere else (an these charges can t be insie the wires or filament). This is eeply puzzling. DEMO Van e Graaff To simplify our analysis for the search for the location of the charges responsible for proucing the E-fiel in the filament, I introuce a mechanical battery (van e Graaff), which is easier to unerstan than a chemical battery, yet behaves in a circuit in a way that is very similar. The battery has a conveyor belt riven by a motor that pulls electrons out of one plate, making it positive, an pushes them onto another plate, making it negative. This action replenishes electrons that leave the negative plate an move through the wire with rift spee v, an it removes electrons that enter the positive plate after traveling though the wire. As long as the motor is able to maintain the charge separation across the two plates, we can have a steay-state current running in the wire. Very important question: what is the fiel ue to the battery in a wire that has turns? At the locations marke with an x (1 5), raw the E-fiel vectors ue solely to the charges on the metal plates. (Ignore the wire for the moment. Next, raw with a ifferent color the irection of the rift velocity at these points, assuming that the rift is only ue to the charges on the plates. Goo grief! The figure shows that we ve got the electron current running upstream at location 4. That can t be right in the steay state. We conclue that in steay state there must be some other charges somewhere that contribute to the net E-fiel in such a way that the E-fiel points upstream everywhere. The answer is that in aition to the fiel of the parallel plates, there is also charge builup on the surface of the wire that helps establish steay state current flow. 6. Charge builup on the surface of the wire Key point: if v = constant then E = constant everywhere. We ve alreay iscusse that the conuction electrons insie the metal on t interact with each other, so some other charges must contribute to the E-fiel that is responsible for pushing the electrons through the wire. Let s focus on the part of the wire that reas left section of the wire. Electrons are flowing into this section from both ens an will gain a negative charge an of course, these negative charges will buil up on the surface of the wire (that is, some of it peeks through the surface) while the section labele right ben will have excess positive ions that peek out at that surface. If we raw the E-fiel at location 4, note that there is a reuce fiel by the contribution of the negative charges that have pile up on the left ben section. As long as there is a net fiel to the right, electrons will continue to pile up at the left ben until there is so much charge on the bens that the net fiel at point 4 points to the left. The same thing will also happen at points 3 an 5. Only then will the electron current into a ben equal the electron current out of a ben, with no further change in the amount of charge on the surface of the ben. We see examples of feeback in both the left ben an the right ben of the circuit. If the intial fiel is such that it rives current that is ifferent in amount or irection from the steaystate current, surface charge automatically buils up in such a way as to alter the current 27.12

13 to be more like the steay-state current. Moreover, once the steay state is establishe, there is negative feeback : any eviation away from the steay state will prouce a change in the surface charge that tens to restore the steay-state conitions. In other wors, steay state is the lowest possible energy for this system. The istribution of excess surface charge in a circuit can be quite complicate, especially if there are bens an twists in the wires. So if we apply this to a simple circuit with the approximate surface charge (but gives you a sense of what is happening), we get that Near the poles of the battery, there is a high surface charge ensity but as I get further from the poles, the surface charge ensity ecreases (as inicate with less charge shown) where there is a smooth transition from one surface charge ensity to another. The surface charge is istribute along a ring of charge, an the E-fiel points from the higher to the lower surface charge region (this is usually calle a potential graient or graient). The key point is that it is the variation of the surface charge ensity along the wire that prouces the E-fiel that is constant along the wire, an therefore, a constant current (not rift spee). The surface charges an the E-fiel they make are essential to riving an guiing the current through the interiors of the wires. Two points on surface charge an E-fiels. i. Surface charges cannot inuce or polarize an object because the net charge of a wire is zero (equal amounts of negative an positives). ii. The present amount of surface charges on the wires an the E-fiel they prouce is very small. Facts: the amount of surface charge near the negative en of a 3Vbattery on the wire is about 10 6 e/cm. Compare this to the lab with the balloon, which was of the orer of e. The E-fiel in this 3V-battery is about 5V/m. 6 q 10 e E 1 5 V/m surface I foun online that it is possible to run a circuit with a 10,000 V power supply an emonstrate how the surface charges can repel charge objects. 7. Connecting a Circuit: the initial transient When you complete a circuit by making the final connection, feeback forces a rapi rearrangement of surface charges leaing to the steay state. This perio of ajustment before establishing the steay state is calle the initial transient. For a circuit with a gap, the circuit is in equilibrium so E = 0 everywhere insie the wires, so the approximate surface charge is shown below (we ignore the exact etails of the istribution because it is too complicate). wire 27.13

14 When a plug is brought close to an outlet, ielectric breakown can occur because E = V/ x. These charges are at the ens of the wires. If we look closely at the gap, the E-fiel that is ue to all the (i) surface charges E surface are in re while (ii) E-fiels ue to the other charges (that is, on the ens of the battery an along the wires) E other (in blue) must be equal an opposite, so that they cancel each other out. When the connection is mae to close the gap, the charges on the facing ens of the wires neutralize each other, leaving a tube of surface charge on the outsie of the wire. At this instant this surface charge istribution has a big, unstable iscontinuity in it. If we were to raw the fiels at the connection point, we see that there is a nonzero E-fiel at the connection, whereas in steay state, there is not. There is a ilution of the original surface charges where the left (right) sie of the gap is less positive (negative). If we zoom out an look at the whole circuit, as soon as the gap is close, the E-fiel on the left circuit (the circle area) travels outwars at approximately the spee of light, so the surface charges an the E-fiel at istance locations haven t been affecte yet (they haven t receive the information that the gap has been close). So the E = 0 is true everywhere in the circuit except at the gap. It is the spee of light that etermines the minimum time require to establish the steay state (1 foot = 30 cm per ns). Measurements inicate that it takes a few nanosecons for the rearrangement of the surface charges will exten all the way aroun the circuit. There is a ifference between the propagation of the E-fiel verses the rift velocity. Single resistor in a circuit For a single resistor raw the (i) surface charges, an plot the (ii) potential graient, an (iii) E-fiel

15 How oes current know how to ivie between two ifferent parallel resistors? Consier a circuit with one parallel branches is open. During the initial transient just after connecting the battery, electrons flowe into both branches, but the surface of the ea-en branch became so negatively charge that no more electrons coul enter. That is, the graient was higher in the ea-en branch than in the current-carrying wire. vs