Electric Current and Resistance


 Damian Payne
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1 Electric Current and Resistance Electric Current Electric current is the rate f flw f charge thrugh sme regin f space The SI unit f current is the ampere (A) 1 A = 1 C / s The symbl fr electric current is I Average Electric Current Assume charges are mving perpendicular t a surface f area A If Q is the amunt f charge that passes thrugh A in time t, then the average current is I avg Q t Instantaneus Electric Current If the rate at which the charge flws varies with time, the instantaneus current, I, can be fund I dq dt I Q t Culmb Secnd Directin f Current The charges passing thrugh the area culd be psitive r negative r bth It is cnventinal t assign t the current the same directin as the flw f psitive charges The directin f current flw is ppsite the directin f the flw f electrns It is cmmn t refer t any mving charge as a charge carrier 1
2 Current and Drift Speed Charged particles mve thrugh a cnductr f crsssectinal area A n is the number f charge carriers per unit vlume na x is the ttal number f charge carriers Current and Drift Speed The ttal charge is the number f carriers times the charge per carrier, q Q =(na x)q The drift speed, v d, is the speed at which the carriers mve v d = x / t and x = v d t Current and Drift Speed Rewritten: Q = (nav d t)q Finally, current, I avg Q nav d q t Charge Carrier Mtin in a Cnductr The zigzag black lines represent the mtin f a charge carrier in a cnductr The net drift speed is small The sharp changes in directin are due t cllisins The net mtin f electrns is ppsite the directin f the electric field Use the active figure t change the field and bserve the effect PLAY ACTIVE FIGURE Mtin f Charge Carriers, cnt. Mtin f Charge Carriers, final In the presence f an electric field, in spite f all the cllisins, the charge carriers slwly mve alng the cnductr with a drift velcity, v d The electric field exerts frces n the cnductin electrns in the wire These frces cause the electrns t mve in the wire and create a current The electrns are already in the wire They respnd t the electric field set up by the battery The battery des nt supply the electrns, it nly establishes the electric field 2
3 Drift Velcity, Example Assume a cpper wire, with ne free electrn per atm cntributed t the current The drift velcity fr a 12gauge cpper wire carrying a current f 10.0 A is 2.23 x 104 m/s 0.2 mm/s This is a typical rder f magnitude fr drift velcities Rank the current frm lwest t highest c,a,d,b Current Density J is the current density f a cnductr It is defined as the current per unit area J I / A = nqv d This expressin is valid nly if the current density is unifrm and A is perpendicular t the directin f the current J has SI units f A/m 2 The current density is in the directin f the psitive charge carriers I J da s Cnductivity A current density and an electric field are established in a cnductr whenever a ptential difference is maintained acrss the cnductr Fr sme materials, the current density is directly prprtinal t the field The cnstant f prprtinality, σ, is called the cnductivity f the cnductr Ohm s Law Ohm s law states that fr many materials, the rati f the current density t the electric field is a cnstant σ that is independent f the electric field prducing the current Mst metals bey Ohm s law Materials that bey Ohm s law are said t be hmic Ohm s Law, cnt. Nt all materials fllw Ohm s law Materials that d nt bey Ohm s law are said t be nnhmic Ohm s law is nt a fundamental law f nature Ohm s law is an empirical relatinship valid nly fr certain materials J nv q σe d 3
4 Resistance J E I V A l V l I A V R I A Resistance In a cnductr, the vltage applied acrss the ends f the cnductr is prprtinal t the current thrugh the cnductr The cnstant f prprtinality is called the resistance f the cnductr R V I Resistance, cnt. SI units f resistance are hms (Ω) 1 Ω = 1 V / A Resistance in a circuit arises due t cllisins between the electrns carrying the current with the fixed atms inside the cnductr Resistivity The inverse f the cnductivity is the resistivity: σ 1 ρ Resistivity has SI units f hmmeters (Ω. m) Resistance is als related t resistivity: R ρ A Resistr Values Resistr Values Values f resistrs are cmmnly marked by clred bands 4
5 Resistivity Values Resistance and Resistivity, Summary Every hmic material has a characteristic resistivity that depends n the prperties f the material and n temperature Resistivity is a prperty f substances The resistance f a material depends n its gemetry and its resistivity Resistance is a prperty f an bject An ideal cnductr wuld have zer resistivity An ideal insulatr wuld have infinite resistivity Ohmic Material, Graph In an hmic device the resistance is cnstant ver a wide range f vltages The relatinship between current and vltage is linear The slpe is related t the resistance Nnhmic Material, Graph Nnhmic materials are thse whse resistance changes with vltage r current The currentvltage relatinship is nnlinear A junctin dide is a cmmn example f a nnhmic device Resistance f a Cable, Example Assume the silicn between the cnductrs t be cncentric elements f thickness dr The resistance f the hllw cylinder f silicn is ρ dr dr 2πrL Resistance f a Cable, Example, cnt. The ttal resistance acrss the entire thickness is b ρ b R dr ln a 2πL a This is the radial resistance f the cable This is fairly high, which is desirable since yu want the current t flw alng the cable and nt radially ut f it 5
6 Electrical Cnductin A Mdel Electrical Cnductin A Mdel Treat a cnductr as a regular array f atms plus a cllectin f free electrns The free electrns are ften called cnductin electrns These electrns becme free when the atms are bund in the slid In the absence f an electric field, the mtin f the cnductin electrns is randm. Their speed is n the rder f 10 6 m/s Cnductin Mdel, 2 When an electric field is applied, the cnductin electrns are given a drift velcity Assumptins The electrn s mtin after a cllisin is independent f its mtin befre the cllisin The excess energy acquired by the electrns in the electric field is lst t the atms f the cnductr when the electrns and atms cllide This causes the temperature f the cnductr t increase Cnductin Mdel, 3 The frce experienced by an electrn is F qe Frm Newtn s Secnd Law, the acceleratin is F qe a m m Applying a mtin equatin qe vf= vi + at r vf = vi + t m Since the initial velcities are randm, their average value is zer e e Cnductin Mdel, 4 Let be the average time interval between successive cllisins The average value f the final velcity is the drift velcity q E vfavg, vd m e Cnductin Mdel, 5 v favg, is als related t the current density: nq 2 E J nqvd m n is the number f charge carriers per unit vlume e 6
7 Cnductin Mdel, final Using Ohm s Law, expressins fr the cnductivity and resistivity f a cnductr can be fund: 2 nq 1 me 2 m nq e Nte, the cnductivity and the resistivity d nt depend n the strength f the field This feature is characteristic f a cnductr beying Ohm s Law Resistance and Temperature Over a limited temperature range, the resistivity f a cnductr varies apprximately linearly with the temperature ρ ρ [1 α ( T T )] ρ is the resistivity at sme reference temperature T T is usually taken t be 20 C α is the temperature cefficient f resistivity SI units f α are C 1 Temperature Variatin f Resistance Since the resistance f a cnductr with unifrm crss sectinal area is prprtinal t the resistivity, yu can find the effect f temperature n resistance R R [1 α( T T )] Use f this prperty enables precise temperature measurements thrugh careful mnitring f the resistance f a prbe made frm a particular material Resistivity and Temperature, Graphical View Fr sme metals, the resistivity is nearly prprtinal t the temperature A nnlinear regin always exists at very lw temperatures The resistivity usually reaches sme finite value as the temperature appraches abslute zer Residual Resistivity The residual resistivity near abslute zer is caused primarily by the cllisins f electrns with impurities and imperfectins in the metal High temperature resistivity is predminantly characterized by cllisins between the electrns and the metal atms This is the linear range n the graph Resistivity Values R R [1 α( T T )] 7
8 Semicnductrs Semicnductrs are materials that exhibit a decrease in resistivity with an increase in temperature α is negative There is an increase in the density f charge carriers at higher temperatures Supercnductrs A class f materials and cmpunds whse resistances fall t virtually zer belw a certain temperature, T C T C is called the critical temperature The graph is the same as a nrmal metal abve T C, but suddenly drps t zer at T C Supercnductrs, cnt The value f T C is sensitive t: chemical cmpsitin pressure mlecular structure Once a current is set up in a supercnductr, it persists withut any applied vltage Since R = 0 Nbel prize Supercnductr Applicatin An imprtant applicatin f supercnductrs is a supercnducting magnet The magnitude f the magnetic field is abut 10 times greater than a nrmal electrmagnet Supercnductr Applicatin Used in MRI units 8
9 Electrical Pwer Assume a circuit as shwn As a charge mves frm a t b, the electric ptential energy f the system increases by QV The chemical energy in the battery must decrease by this same amunt PLAY ACTIVE FIGURE Electrical Pwer, 2 As the charge mves thrugh the resistr (c t d), the system lses this electric ptential energy during cllisins f the electrns with the atms f the resistr This energy is transfrmed int internal energy in the resistr Crrespnds t increased vibratinal mtin f the atms in the resistr Electric Pwer, 3 Electric Pwer, 4 The resistr is nrmally in cntact with the air, s its increased temperature will result in a transfer f energy by heat int the air The resistr als emits thermal radiatin After sme time interval, the resistr reaches a cnstant temperature The input f energy frm the battery is balanced by the utput f energy by heat and radiatin The rate at which the system lses ptential energy as the charge passes thrugh the resistr equals the rate at which the system gains internal energy in the resistr U Q V I V t t The pwer is the rate at which the energy is delivered t the resistr Electric Pwer, final The pwer is given by the equatin: IV Applying Ohm s Law, alternative expressins can be fund: IV I R 2 2 V Units: I is in A, R is in Ω, V is in V, and is in W R Electric Pwer Transmissin Real pwer lines have resistance Pwer cmpanies transmit electricity at high vltages and lw currents t minimize pwer lsses 9
10 Sme Final Ntes Abut Current A single electrn is mving at the drift velcity in the circuit It may take hurs fr an electrn t mve cmpletely arund a circuit The current is the same everywhere in the circuit Current is nt used up anywhere in the circuit The charges flw in the same rtatinal sense at all pints in the circuit Gerg Simn Ohm German physicist Frmulated idea f resistance Discvered the prprtinalities nw knwn as frms f Ohm s Law 10
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