I 1 Electric Currents and Simple Circuits Electrons can flow along inside a metal wire if there is an E field present to push them along r r F q E The flow of electrons in a wire is similar to the flow of water in a pipe Definition electric current I DQ rate of flow of charge Dt units I coulomb second 1 C 1 s 1 ampere A amp It s not the voltage that kills you it the amps About 0 05 A is enough to kill you If current I 1 A in a wire then 1 coulomb of charge flows past any point every second I r In electrostatic problems E 0 inside a metal but if I 0 then the situation is not static the E field is not zero Electrons flow in metals not protons so charges are moving when there is a current The r r electron feel a force F e E and goes E upstream against the E field The flow of charge in one direction is electrically equivalent to the flow of charge in the opposite direction neutral plates Last update 1 13 2019 or either way get Dubson Phys2020 Notes University of Colorado I 2 By convention we define current I as the flow of imaginary charges when it is really charges flowing the other way I Some texts refer to I as the conventional current to distinguish it from the electron current Example How many electrons flow past per second when the current is 1 A I DQ DN e Dt Dt DN I 1A 1C s 5 6 1018 s 1 19 19 Dt e 1 6 10 C 1 6 10 C About 0 01 A 10 mA flowing though your heart is lethal yet I could grab a wire carrying 1000A and be safe Why Because my body has a much higher electrical resistance than the metal The electrons prefer to flow through the metal wire For most materials the current I is proportional to the voltage difference between the ends r r I E since F q E and D V E so I D V lo V hi V E I V From now on we usually follow the bad convention and write V when we really mean V I V really I D V V constant I Definition of resistance R of a piece of wire or other material R V I The experimental fact that for most materials the ratio R V I is a constant independent of V or I is called Ohm s Law R V constant usually written I Units R volt ampere ohm Last update 1 13 2019 V IR R constant is Greek letter omega Dubson Phys2020 Notes University of Colorado I 3 Ohm s Law should be written V I R but the bad convention is to write V I R Ohm s Law is not really a law because it is not always true For many materials Ohm s Law is approximately true the resistance R is approximately constant independent of V or I Materials that obey Ohm s Law are called ohmic materials But some materials are non ohmic they do not obey Ohm s Law The average speed of electrons in a current carrying wire results from a competition between two r r r effects 1 the E field which causes an acceleration according to F q E m a making the electrons go faster and faster and 2 the scattering of electrons due to impurities and thermal vibrations which act like friction making the electrons slow down For typical currents in real wires the average electron speed often called the drift velocity is actually quite slow typically less than 0 1 mm s Incidentally the term drift velocity is incorrect it should be called the drift speed A material with lots of electron scattering has a high resistance Rwire 1 Rhuman 105 V 10 V 5 10 4 A harmless R 10 W 10 V safe V 100 V 3 I 5 10 A painful R 10 W I 100 V dangerous The resistance R of a piece of material depends on its shape and composition R big Shape long and skinny R small short and fat just like the flow of water through a pipe Long skinny water pipes resist flow of water Turns out that R L A area A L so big L means big R big A means small R Last update 1 13 2019 Dubson Phys2020 Notes University of Colorado I 4 R rL A Greek letter rho resistivity We show were this comes in the next section below Resistivity depends on the material composition not on the shape is a measure of the scattering of electrons in that material like viscosity of fluid in a pipe Big means lots of scattering friction big resistance to flow Units r A L R length length length R length m material Cu Al W tungsten 1 7 10 8 m 2 7 10 8 m 5 3 10 8 m cool use house wiring power lines light bulb filaments Fe glass 60 10 8 m hot 9 8 10 8 m 10 10 m not used in wiring electrical insulator Microscopic view of Ohm s Law Definition current density J I A current per area in a conductor We also define current density vector J where direction of J is the direction of the current J is related to the average speed vdrift of the charge carriers usually electrons by J n q vdrift n is the number of carrier per volume q is the charge per carrier usually q e Proof Consider a wire with carrier density n volume cross sectional area A and drift speed vdrift In a time t all the charges move an average distance x vdrift t area A volume A x J x vdrift t Last update 1 13 2019 Dubson Phys2020 Notes University of Colorado I 5 The charge Q in the length of wire x is DQ number of carriers charge volume n q A D x volume carrier So the current density is J I DQ 1 n q A Dx n q v drift A Dt A A Dt Done In ohmic materials the current density J is proportional to the electric field E v v J s E where the proportionality constant is called the conductivity The resistivity is defined as r 1 and so E J s J s E or E r J where 1 constant is the microscopic version of Ohm s Law We now show that this is equivalent to V I R Consider a cylindrical section of conductor length L cross sectional area A with current density J and E field E Start with E r J now multiply both sides by L and write J A J E as J I A E L r L L we have D V I defined the resistance R as R Ohm s Law V I R where R Last update 1 13 2019 I Notice that V E L So A r …
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