# Current And Resistance Current Density and Drift Velocity Current And Resistance Current Density and Drift Velocity Perfect conductors carry charge instantaneously from here to there Perfect insulators carry no charge from here to there, ever Real substances always have some density n of charges q that can move, however slowly Usually electrons When you turn on an electric field, the charges start to move with average velocity vd Why did I draw Called the drift velocity J to the right? J There is a current density J associated with this motion of charges Current density represents a flow of charge J nqv d Note: J tends to be in the direction of E, even when vd isnt Ohms Law: Microscopic Version In general, the stronger the electric field, the faster the charge carriers drift The relationship is often proportional E J Ohms Law says that it is proportional Ohms Law doesnt always apply The proportionality constant, denoted , is called the resistivity It has nothing to do with charge density, even though it has the same symbol It depends (strongly) on the substance used and (weakly) on the temperature Resistivities vary over many orders of magnitude Silver: = 1.5910-8 m, a nearly perfect conductor Ignore units

Fused Quartz: = 7.51017 m, a nearly perfect for now insulator Silicon: = 640 m, a semi-conductor The Drude Model Why do we (often) have a simple relationship between electric field and current density? In the absence of electric fields, electrons are moving randomly at high speeds Electrons collide with impurities/imperfections/vibrating atoms and change their direction randomly When they collide, their velocity changes to a random velocity vi v v i Between collisions, the velocity is constant On average, the velocity at any given time is zero v d v v i 0 a F m eE m Now turn on an electric field The electron still scatters in a random direction at each collision v v i at But between collisions it accelerates Let be the average time since the last collision 2 J nq v ne a J ne E m

v d v v i a t 0 a d Current It is rare we are interested in the microscopic current density We want to know about the total flow of charge through some object n J I n JdA I JA The total amount of charge flowing out of an object is called the current 2 C m/s m What are the units of I? I JA qnvd A m3 The ampere or amp (A) is 1 C/s Current represents a change in charge I C A s Almost always, this charge is being replaced somehow, so there is no dQ I accumulation of charge anywhere dt Ohms Law for Resistors

Suppose we have a cylinder of material with conducting end caps Length L, cross-sectional L area A The material will be assumed to I JA follow Ohms Microscopic Law L JL V EL I Apply a voltage V across it A L J E E V L R A Define the resistance as V IR Then we have Ohms Law for devices Just like microscopic Ohms Law, doesnt always work Resistance depends on composition, temperature and geometry We can control it by manufacture Circuit diagram Resistance has units of Volts/Amps for resistor V Also called an Ohm () R A

An Ohm isnt much resistance Ohms Law and Temperature Resistivity depends on composition and temperature If you look up the resistivity for a substance, it would have to give it at some reference temperature T0 0 T0 E J Normally 20C For temperatures not too far from 20 C, we can hope that resistivity will be approximately linear in temperature T 0 1 T T0 Look up 0 and in tables For devices, it follows there will also be temperature dependence The constants and T0 will be the same for the device L 0 L R 1 T T0 A A R R0 1 T T0 Non-Ohmic Devices Some of the most interesting devices do not follow Ohms Law Diodes are devices that let current through one way much more easily than the other way Superconductors are cold materials that have no resistance at all They can carry current forever with no electric

field E J 0 Power and Resistors The charges flowing through a resistor are having their potential energy changed Q Where is the energy going? U QV The charge carriers are bumping against atoms They heat the resistor up U Q V t t V dU dQ V P I V dt dt dQ I dt dU P dt V IR V

2 P I R R 2 Uses for Resistors You can make heating devices using resistors Toasters, incandescent light bulbs, fuses You can measure temperature by measuring changes in resistance Resistance-temperature devices Resistors are used whenever you want a linear relationship between potential and current They are cheap They are useful They appear in virtually every electronic circuit V2 12V +V V1 -1m/1mV 1kHz R1 15k C1 R2 0.06uF2.3Meg R3 300k Q1

2N3904 R6 80 C2 30uF R4 C4 R11 25k0.06uF2.3Meg Q2 2N3904 R5 1k R10 300k Q4 2N3904 R9 25k Q3 2N3904 R8 1k Q5 2N3904 Q6 2N3904 Q7 C3 2N39041mF

R7 25 RL 50k Equations for Test 1 Electric Fields: ke q1q2 F2 2 r r ke q E 2 r r Gausss Law: E AE n qin E 0 Potential: Capacitance: Q C V C C1 C2 V E ds U qV ke q V r 1 1 1

C C1 C2 F Eq Units: N V C m C A s C F V V A End of material for Test 1 V Ex x V E y y V Ez z