Chapter 6Current and ResistanceCurrent and ResistanceElectric CurrentCurrent DensityOhm’s LawElectrical Energy and PowerSummarySolved ProblemsResistivity of a CableCharge at a JunctionDrift VelocityResistance of a Truncated ConeResistance of a Hollow CylinderConceptual QuestionsAdditional ProblemsCurrent and Current DensityPower Loss and Ohm’s LawResistance of a ConeCurrent Density and Drift SpeedCurrent SheetResistance and ResistivityPower, Current, and VoltageCharge Accumulation at the InterfaceChapter 6 Current and Resistance 6.1 Electric Current........................................................................................................ 1 6.1.1 Current Density................................................................................................. 1 6.2 Ohm’s Law .............................................................................................................. 3 6.3 Electrical Energy and Power.................................................................................... 6 6.4 Summary.................................................................................................................. 7 6.5 Solved Problems ...................................................................................................... 8 6.5.1 Resistivity of a Cable........................................................................................ 8 6.5.2 Charge at a Junction.......................................................................................... 9 6.5.3 Drift Velocity.................................................................................................. 10 6.5.4 Resistance of a Truncated Cone...................................................................... 11 6.5.5 Resistance of a Hollow Cylinder .................................................................... 12 6.6 Conceptual Questions ............................................................................................ 13 6.7 Additional Problems .............................................................................................. 13 6.7.1 Current and Current Density........................................................................... 13 6.7.2 Power Loss and Ohm’s Law........................................................................... 13 6.7.3 Resistance of a Cone....................................................................................... 14 6.7.4 Current Density and Drift Speed..................................................................... 14 6.7.5 Current Sheet .................................................................................................. 15 6.7.6 Resistance and Resistivity............................................................................... 15 6.7.7 Power, Current, and Voltage........................................................................... 16 6.7.8 Charge Accumulation at the Interface ............................................................ 16 0Current and Resistance 6.1 Electric Current Electric currents are flows of electric charge. Suppose a collection of charges is moving perpendicular to a surface of area A, as shown in Figure 6.1.1. Figure 6.1.1 Charges moving through a cross section. The electric current is defined to be the rate at which charges flow across any cross-sectional area. If an amount of charge ∆Q passes through a surface in a time interval ∆t, then the average current avgI is given by avgQIt∆=∆ (6.1.1) The SI unit of current is the ampere (A), with 1 A = 1 coulomb/sec. Common currents range from mega-amperes in lightning to nano-amperes in your nerves. In the limit the instantaneous current I may be defined as 0,t∆→ dQIdt= (6.1.2) Since flow has a direction, we have implicitly introduced a convention that the direction of current corresponds to the direction in which positive charges are flowing. The flowing charges inside wires are negatively charged electrons that move in the opposite direction of the current. Electric currents flow in conductors: solids (metals, semiconductors), liquids (electrolytes, ionized) and gases (ionized), but the flow is impeded in non-conductors or insulators. 6.1.1 Current Density To relate current, a macroscopic quantity, to the microscopic motion of the charges, let’s examine a conductor of cross-sectional area A, as shown in Figure 6.1.2. 1Figure 6.1.2 A microscopic picture of current flowing in a conductor. Let the total current through a surface be written as I d= ⋅∫∫JAGG (6.1.3) where is the current density (the SI unit of current density are ). If q is the charge of each carrier, and n is the number of charge carriers per unit volume, the total amount of charge in this section is then JG2A/m(Q )qnA x∆=∆. Suppose that the charge carriers move with a speed; then the displacement in a time interval ∆t will bedvdxvt∆= ∆, which implies avg dQI nqv At∆==∆ (6.1.4) The speed at which the charge carriers are moving is known as the drift speed. Physically, is the average speed of the charge carriers inside a conductor when an external electric field is applied. Actually an electron inside the conductor does not travel in a straight line; instead, its path is rather erratic, as shown in Figure 6.1.3. dvdv Figure 6.1.3 Motion of an electron in a conductor. From the above equations, the current density JG can be written as dnq=JvGG (6.1.5) Thus, we see that JGand point in the same direction for positive charge carriers, in opposite directions for negative charge carriers. dvG 2To find the drift velocity of the electrons, we first note that an electron in the conductor experiences an electric force which gives an acceleration ee=−FGGE eeeemm==−FEaGGG (6.1.6) Let the velocity of a given electron immediate after a collision be ivG. The velocity of the electron immediately before the next collision is then given by fi ieetm=+=−Evvav tGGG GG (6.1.7) where t is the time traveled. The average of fvG over all time intervals is fieetm=−EvvGG G (6.1.8) which is equal to the drift velocity dvG. Since in the absence of electric field, the velocity of the electron is completely random, it follows that 0i=vG. If tτ= is the average characteristic time between successive collisions (the mean free time), we have dfeemτ==−EvvGG G (6.1.9) The current density in Eq. (6.1.5) becomes 2deeenene nemmττ⎛⎞=−
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