of conduction electrons

Transcription

1 Dr. Fritz Wilhel, Physics 3 E:\Excel files\3 lecture\ch7 current.ocx Last save: /3/8 :53:; Last printe:/3/8 :53: of 9 Hoework: See website. Table of Contents: Ch. 7 Electric Current an esistance, 7. Electric Current I an current ensity j, 7.a eview of incopressible liqui flow as exaple for flux of an electric fiel, 7.b Electric current an electric current ensity, 3 7.c Density q of conuction electrons 7. Electric esistance in Conucting Material, conuctivity, resistivity; 4 7.a Calculating for varying cross-sections, Siple classical oel for electrical conuction, esistance an Teperature, teperature coefficient α; 6 7.4a Gauge easures for electric wire, Electrical Power, 8

2 Dr. Fritz Wilhel, Physics 3 E:\Excel files\3 lecture\ch7 current.ocx Last save: /3/8 :53:; Last printe:/3/8 :53: Ch. 7 Electric Current an esistance. 7. Electric Current I an current ensity j, of 9 +qv E V E L Δx We consier a conucting wire with length L an cross-section in a constant electric fiel E, irecte parallel to the cyliner s axis. Free positive charges flow to the right. Q I flow rate of charge =current (7.) Coulob I pere; tie We assign the positive irection of current to the irection of flow of positive charges, i.e. the actual irection of flow (rift velocity v ) of electrons on a wire is opposite to the conventional irection of current. The electric fiel is irecte fro + positive polarity to negative polarity. (Pitfalls: istinguish carefully between v for velocity, V for volue an ΔV for potential ifference; also there are two ensities: the conventional ass ensity ρ or ρ ; charge ensity ρ q ; an we use another sybol ρ later, the resistivity ρ Ω.) 7.a eview of incopressible liqui flow as exaple for flux of an electric fiel: We iscusse this previously at the introuction of the concept of flux through a surface which le us to the Gaussian law for electrostatics: ive (chapter 4, chapter ). Let us review again what we learne about the rate of water flow out of a cylinrical pipe: M The aount of liqui ass flowing out of the pipe per secon is given by v If we ha a nuber N of ientical particles, each with ass, flowing through the pipe, N we coul write: nv, with nv (nuber-ensity) an efine the V ass current ensity j=nv v. The ass current woul then be equal to the scalar prouct M of current ensity an cross-sectional surface: j M (7.) Current : I For a steay state flow of an incopressible liqui, i.e. with constant ass ensity, we foun that the aount of liqui coing in is equal to the aount of liqui coing out. This was referre to as the continuity equation. We can see that the continuity equation is a special case of the law that says that the ivergence of the current ensity is equal to, which eans that the total flux of the current ensity through a close surface is. This result presues that there are no liqui sources or sinks for liqui insie of the volue.

3 Dr. Fritz Wilhel, Physics 3 E:\Excel files\3 lecture\ch7 current.ocx Last save: /3/8 :53:; Last printe:/3/8 :53: iv j j j j Gaussian surface (7.3) 7.b Electric current an electric current ensity: In coplete analogy with this we get for the electric current I of positive electric charges q through a conucting wire: (7.4) Q I x v n qv q q V j Coulob C [ I] apere s s [ j] 3 of 9 (7.5) x nuber of charges I j n eleentary charge cross-section= v Vq q volue q V I n qv where v is the rift velocity of the charges ue to the electric fiel E. V q is the value of the eleentary charge =.6 9 Coulob Conversely, we can efine the agnitue of the current ensity j as the current ivie by the cross-section : I current ensity=j= or in vector for (7.6) I= j j =flux of the charge-ensity vector through the surface (7.7) j nvqv qv 7.c Density q of conuction electrons: The ensity of conucting electrons in copper wire is obtaine by assuing that there is one conucting electron per copper ato. The olar ass of copper is 63.5 g/ol. The ass ensity of copper is 8.95 g/c 3. g (7.8) c ol atos.4.4 N 3 n 3 V g 63.5 c ol 8 3 nuber ensity of conucting electrons in Cu: nv 8.5 electrons per (7.9) 8 9 C conucting charge ensity in copper q nq V Note: Get clear on the various istinctions between current I, an current ensity j, as well as charge ensity, ass ensity, ass current, an ass current ensity.

4 Dr. Fritz Wilhel, Physics 3 E:\Excel files\3 lecture\ch7 current.ocx Last save: /3/8 :53:; Last printe:/3/8 :53: 4 of 9 7. Electric esistance in Conucting Material, conuctivity, resistivity: The Geran physicist Georg Sion Oh ( ) establishe that for conventional etals there is a irect proportionality between the current ensity an the applie electric fiel (Note that this is not always the case). Materials in which the current ensity is irectly proportional to the applie electric fiel are calle Ohic: (7.) j E conuctivity E; Oh's law E j j; with resistivity E V V Oh eter j Pay attention to whether the Greek sybol (siga) refers to the conuctivity or to Q the surface ensity of charges. esistivity is the inverse of conuctivity, thus Oh s law can also be written as: (7.) E j Fro the concept of conuctivity we get to the concept of electric resistance in a wire by applying Oh s law above to a segent of wire with length L to which a voltage ΔV is applie. cross the length of a wire with length L we have V E L (ΔV epens on the initial an final point only, not on the path or shape of the wire) We write for the agnitues: j E (7.) Solving this for ΔV (7.3) j I L L L V E L L I I L L V I I I resistance current This is the ost coon for of Oh s law, which states that the voltage ifference easure across a wire is equal to the electrical resistance of the wire ties the current. The unit of resistance is calle Oh for which we use the Greek capital letter for

5 Dr. Fritz Wilhel, Physics 3 E:\Excel files\3 lecture\ch7 current.ocx Last save: /3/8 :53:; Last printe:/3/8 :53: (7.4) V I resistancecurrent Volts V aperes 5 of 9 E j; j nqv; I j ; V I (7.5) resistivity in Oh-eters L = in Ohs or The resistance of a wire increases with the length an ecreases with the cross-section. The resistivity of copper is.7e-8 Ω. Both, resistivity an conuctivity are aterial constants, which can be looke up in tables. They are usually given at a teperature of C. They both vary with teperature: resistivity an resistance increase with teperature in ost etals. 7.a Calculating for varying cross-sections: Soeties we have to consier a flow of current through a non-constant cross-section, like, for exaple in a cyliner or sphere. I such cases we nee to consier the current flow through infinitesial sections an integrate: Exaple: Fin the raial resistance between the walls of a coaxial wire of length with inner raius an outer raius 4. The walls consist of plastic aterial with resistivity.e3 Ω. (7.6) b r r b ln ln. rl L r L a L a Fin the raial resistance between the interior an exterior walls of a spherical shell: b r r 4 (7.7) 8. r r r a b a 7.3 Siple classical oel for electrical conuction: Conucting electrons rift through a etal ue to the electric force qe. This eans they are being accelerate by the electric fiel. They o not ove in a straight line but bounce off of positively charge areas. Due to this effect electrons reach a final velocity (like terinal velocity), or average rift velocity v which we can estiate in the following way: Between two scattering points the electrons are being accelerate ue to the applie exterior qe electric fiel. F qe x a; a

6 Dr. Fritz Wilhel, Physics 3 E:\Excel files\3 lecture\ch7 current.ocx Last save: /3/8 :53:; Last printe:/3/8 :53: v v a; where is the average tie interval between iniviual scatterings, the corresponing length woul be the ean free path length for conucting electrons. l v average v Note: Do not ix this free path length up with the free path length of rano theroynaic otion. They are siilar but not the sae. v qe at We insert this rift velocity into our forula for the current ensity j; (7.7): nv q E (7.8) j E nv qv 6 of 9 (7.9) nq V an nv q ccoring to this classical oel the conuctivity an resistivity o not epen on the current, (or voltage or electric fiel.) We can calculate the tie between collisions for copper which has a resistivity of.7e-8 Ω an n=8.49e8 electrons/ 3 (See (7.9)). (7.) nvq nvq It turns out to be about.5e-4 s. The rift velocity, on the other han oes epen on the current. I (7.) I j nv qv v n q This eans that the rift velocity v an the istance between scatterings increases with increasing currents, whereas the tie between scatterings reains the sae. For a typical current in a gauge copper wire the cross sectional area is See table (7.7). (Note that the saller the gauge nuber, the larger the iaeter; househol wire is typically an 4 gauge; the iaeter for gauge wire is.53 at egree C, 4 gauge is.68, gauge is.588 ). This results in a rift velocity of. /s. 7.4 esistance an Teperature. Over a liite teperature range the relative change of resistivity is proportional to teperature: T T ; an T are usually taken at Celsius (7.) TT T; ; e Celsius T V

7 Dr. Fritz Wilhel, Physics 3 E:\Excel files\3 lecture\ch7 current.ocx Last save: /3/8 :53:; Last printe:/3/8 :53: α is calle the teperature coefficient of resistivity. (For Pt: α=3.9-3 /Cº) s the resistance in a circuit is proportional to the resistivity, we get the sae forula for. TT (7.3) T; e T Evaluating this between an leas to the expression for the teperature epenence of resistance in a coon etal. T T ln T T (7.4) T T e ; e x ( T T ) TT x 7 of 9 (7.5) T T; Celsius 7.4a Gauge easures for electric wire, The ost accurate theroeters are base on the changing resistivity an resistance of a particular kin of wire (often platinu). We use such a theroeter in the echanical equivalent of heat experient, where we eterine the relationship between an T to be of a logarithic nature, in its best approxiation. Here are soe constants: 8 3 Cu :.7 ; 3.9 / C (7.6) 8 3 Pt : ; 3.9 / C 6 3 Nichroe :.5 ;.4 / C Note that Nichroe wire has about ties the resistivity of copper but only / of the teperature coefficient. This leas to the coon use of Nichroe wire in heating eleents like toasters, irons, an electric heaters. The thickness of electric wire is rate by gauge: the higher the gauge nuber, the saller the iaeter of the wire. #.9".5886 (7.7) #.88".53 #6.58".93 #.3".88 #.54".6456 Fin the resistance per eter of #6 Nichroe wire: 6.5 (7.8).5 L /4 If a voltage ifference of Volt is aintaine across of such wire, fin the current in the wire: V V (7.9) I The resistance of a siilar copper wire woul be about ties saller. potential ifference of only about V woul result in the sae current in the wire. s we will see in the next section,

8 Dr. Fritz Wilhel, Physics 3 E:\Excel files\3 lecture\ch7 current.ocx Last save: /3/8 :53:; Last printe:/3/8 :53: power elivere to an electric circuit is equal to P=I, an in a wire this power is turne into heat, which, in turn will increase the resistance. 8 of 9 Soe aterials, calle superconuctors, lose suenly all resistance at a low critical teperature T c. The phenoenon was first iscovere in ercury, which has a critical teperature of 4.5ºK. Soe of the highest critical teperatures can be foun in ceraics with T c up to 34ºK. 7.6 Electrical Power. When current travels through a resistor, heat is being generate an electric potential energy lost, (Oh loss.) The power supply provies the sae potential ifference across the bounaries of a resisting aterial; ΔV is therefore constant. U Q QV V IV I I I (7.3) V U V s I= this is also equal to Power Power elivere to a resistive wire or appliance: (7.3) Power V I Exaple: househol appliance is rate accoring to the axiu power it consues. s appliances typically run at Volts, we can calculate the axiu current rawn by the appliance. Watt light bulb, for exaple, raws a current of Watts/Volts =.9 ps. Every house is subivie into areas of an 5 ps, each protecte by a circuit breaker. Whenever too any appliances are running at the sae tie an the total current rawn excees or 5 ps, the circuit breaker gets activate an stops the flow of electricity. If an unprotecte circuit gets overloae, the wiring is in anger of overheating an eventually causing a fire. (Even though househols are using ac current, the sae concepts of current an power apply.) Proble: coil of Nichroe wire is 5. long. The wire has a iaeter of.4 at ºC. If it carries a current of.5, calculate the electric fiel in the wire an the power elivere to it. If the teperature is increase to 34ºC an the voltage across the wire reains constant, what is the power elivere at the higher teperature / C;.5? (7.3) l V El I; ; 7 V I ; 99 ; E 5.97 V / 4 l l 5 (7.33) P I IV 74.6W with V I 5V If the teperature is increase the resistance increases. To aintain the sae voltage rop, the current ust ecrease.

9 Dr. Fritz Wilhel, Physics 3 E:\Excel files\3 lecture\ch7 current.ocx Last save: /3/8 :53:; Last printe:/3/8 :53: e e ( TT ).4 3 ' (not 337 which you get with +x) (7.34) (7.35) The power elivere is therefore: 5V I'.444 ' P' ' I ' 66.W (7.36) 9 of 9