Current, Resistance and

Transcription

1 Cuent, Resistance and Electomotive Foce Chapte 25 Octobe 2, 2012 Octobe 2, 2012 Physics 208 1

2 Leaning Goals The meaning of electic cuent, and how chages move in a conducto. What is meant by esistivity and conductivity of a substance. How to calculate esistance of a conducto fom its dimensions i and its esistivity. it How an electomotive foce (EMF) makes it possible fo cuent to flow in a cicuit. How to do calculations involving enegy and powe in cicuits. Octobe 2, 2012 Physics 208 2

3 Switch fom static to moving chages Up until now we have been dealing with chages in equilibium (static). Now we will investigate the popeties of systems whee electic chages ae in motion. This will begin ou discussion of DC cicuits.. Octobe 2, 2012 Physics 208 3

4 The dift velocity of chage in cuent flow Let s look in a bit moe detail at the chages in motion inside a cuent. We can intoduce a concept called the dift velocity that descibes the aveage velocity of the chages that ae moving along in this cuent. V dift = Δx/Δt Octobe 2, 2012 Physics 208 4

5 Moe on dift velocity.. Let s wok out the elationship between the dift velocity and the cuent being caied in the wie. ΔQ I = whee ΔQ = Δt so ΔQ = ( n we will chose a Δ Q = ( na ( v I = ( na( v chage pe unit volum e dift dift Δt o we can wite volume Δt )( e )) Δt)( e)) ( numbe of chages ) X ( chage pe paticle ) of V V )( e) = A( v = ( neav I v dift = = nea dift ne j dift ) Δt) whee so is the cuent density. Octobe 2, 2012 Physics j

6 Moe on cuent density. The cuent density is elated to the cuent flowing in the cicuit in the following way. The cuent is the flux of the cuent density though the suface of inteest. J da = I though the suface Octobe 2, 2012 Physics 208 6

7 Typical dift velocities fo coppe the numbe of chage caies pe unit volume is N N n = = = 8.4X10 V m then ρ density 28 chages If we have a wie1.0mm in diamete caying a cuent of 10A v dift nea X10 π( (0.5 X10 ) )( 1.6X10 ) m 3 I 10 = = = 9.47X10 4 m/s Octobe 2, 2012 Physics 208 7

8 Back to Electic Cuent.. We defined the concept of electic cuent as the amount of chage pe unit time that passes a point in a cicuit. I aveage = ΔQ/Δt I instantaneous = dq/dt Cuent has units of Coulombs/second 1 Ampee = 1 Coulomb/second Octobe 2, 2012 Physics 208 8

9 Moe comments on cuent. Cuent is not a vecto. The convention fo cuent diection is that it is in the diection that positive chages would move in the cicuit. Cuent can be poduced by moving positive chages, moving negative chages, o both simultaneously. Octobe 2, 2012 Physics 208 9

10 Example.. How much chage passes by a point in a cicuit in 60 sec when caying a cuent of 1.0 Ampee?? ΔQ = I Δt = 1.0 A X 60 sec = 60 C o 60 C/ (1.6 X C/electon) = 3.75 X electons!! Octobe 2, 2012 Physics

11 Resistivity of Ohmic mateials An altenate and equivalent fom of Ohm s Law elates the electic field applied to a piece of ohmic mateial to the cuent density (cuent/aea) flowing though the mateial. This elationship gives. E = ρ J o J = σe with σ = 1/ρ Whee ρ is the esistivity it and σ is the conductivity of the mateial. Both of these quantities ae constants fo ohmic mateials Octobe 2, 2012 Physics

12 Ohm s Law and esistivity ΔV Octobe 2, 2012 Physics

13 Ohm s Law The linea elationship between the voltage acoss a piece of ohmic mateial and the cuent passing though it is called Ohm s Law and defines the tem esistance. ΔV = I R o R = ΔV/I Octobe 2, 2012 Physics

14 Units fo esistance Resistance will have units of Volt/Amp which we define as an OHM and is epesented by the symbol Ω. The esistance of a piece of mateial with a voltage of 10 V acoss it and a cuent of.5 A though it is R = 10V/.5A = 20 Ω Octobe 2, 2012 Physics

15 Typical esistivities of mateials at oom tempeatue Octobe 2, 2012 Physics

16 Ohmic vs Non-Ohmic Not all mateials obey Ohm s Law. Those that do ae called ohmic and those that do not ae called non-ohmic. Examples of non-ohmic mateials ae semi-conductos that ae used to make diodes and tansistos. Octobe 2, 2012 Physics

17 Ohmic vs Non-Ohmic Octobe 2, 2012 Physics

18 Tempeatue Dependence of Resistivity Note, negative!! Octobe 2, 2012 Physics

19 Calculating esistance Fo any of these configuations we will use the definition of esistance R = ΔV/I and then find ΔV intemsofiand solve. Anothe impotant ingedient is that cuent is conseved as it flows fom one electode to anothe in the esisto. Octobe 2, 2012 Physics

20 Calculating the esistance of a piece of mateial We must use the definition of esistance to calculate the esistance of a cylindical piece of ohmic mateial. ΔV R = with ΔV = E dl and I = J da I in a unifom cylinde the E = ρj and unifom so we can wite... ρj dl ρj dl ρl R = = = J da J da A Octobe 2, 2012 Physics

21 Coaxial cylindes outwad 2 ) ( = = I aea I J l π ] [ 2 2 ) ( so 1 2 = = = n n I d I I d J R l l l l l π ρ π ρ ρ inne adius oute adius and note, = = = n R l l π ρ Cuent flows fom inne cylinde to oute cylinde Octobe 2, 2012 Physics

22 Concentic sphees (anothe example). = = 2 outwad 4 ) ( d I I aea I J ρ π = = = ) ( so I d I d J R π ρ π ρ ρ l = R π ρ Cuent flows fom the inne sphee to the oute sphee adially outwad. Octobe 2, 2012 Physics

23 The battey was essential!!! Until 1800 the only way of poducing chage was though electo-static means..like flying kites into stoms! With the invention of the battey by Volta, scientists we able to do expeiments equiing a constant steam of electic chage. Octobe 2, 2012 Physics

24 Volta s oignal battey design Octobe 2, 2012 Physics

25 Moden batteies Octobe 2, 2012 Physics

26 How do batteies wok??? The combination of dissimila metals in a solution with ions fee to move aound povides a device that is diven by a chemical eaction. Octobe 2, 2012 Physics

27 Real Batteies (souces of EMF) Intenal esistance adds additional esistance to the cicuit. Fo good batteies this is usually small, <.1Ω. Teminal Voltage. Since the battey has an intenal esistance, the voltage at the teminals will be diffeent than the value of the voltage witten on the side of the battey. Octobe 2, 2012 Physics

28 Real Batteies Octobe 2, 2012 Physics

29 Real Batteies Octobe 2, 2012 Physics

30 Real batteies continued The figue at the ight shows a schematic vesion of a eal battey. We show the intenal esistance as being lumped and in seies with the battey s EMF, but in eality the intenal esistance is distibuted thoughout the volume of the battey. Octobe 2, 2012 Physics

31 A simple example. Conside a simple seies cicuit with a battey and a esisto. The battey has a voltage ating of 12.0 V and an intenal esistance of 2.0 Ω. The seies esisto has a value of 4.0 Ω. Find the cuent flowing in the cicuit and the teminal voltage of the battey unde these conditions. Octobe 2, 2012 Physics

32 The solution I = and ε 12.0 = = 2.0 Amps R Vteminal = ε I = 12.0 (2.0)(2.0) = 8.0 Volts Octobe 2, 2012 Physics

33 Potential aound the cicuit Octobe 2, 2012 Physics

34 Cicuits with esistos and capacitos fo this week in lab We will deal with a simple seies RC cicuit and look at how the cuents and voltages in this cicuit change with time. Octobe 4, 2012 Physics

35 Let s conside chaging g the capacito fist Using the definition of capacitance, whee C = Q/ΔV. We can wite down an expession fo the voltage dops aound this simple RC cicuit in the following fom. V battey I R Q/C = 0 whee I = dq/dt Octobe 4, 2012 Physics

36 The final fom of this loop equation. The final fom of this loop equation is as follows: V battey (dq/dt) R Q/C = 0 This is a linea, fist ode, diffeential equation!! But don t despai just yet, we don t need Math 308 to solve this!! Octobe 4, 2012 Physics

37 Solving fo Q(t).. stating We can one last fom... dq ( t) Q( t) V battey R = 0 dt C do some algeba to ewite this as... dq ( t) V = dt bit of algeba Q( t) dq ( t ) V battey C Q( t) RC and then we can integate. battey dt = C RC... Octobe 4, 2012 Physics

38 dq( t) dt = Q( t) VbatteyC RC Integating both sides we get... t ln(q(t) VbatteyC) = RC exponentiating both sides this becomes... o... t Q( t) V C Ae RC battey = Q( t) = Ae t RC Q(t) = V batteyc( 1- e + VbatteyC whee A is a constant to be detemined by the conditions of the poblem.. in this case if we ae chaging the capacito fom zeo chage then Q(t) = 0 so.. t RC ) Octobe 4, 2012 Physics

39 RC time constant Q(t) = V C( 1- e battey t RC ) Octobe 4, 2012 Physics

40 Cuent flow in the system.. dq dt = V battey C(e t RC ) Octobe 4, 2012 Physics