DC Circuits. Crossing the emf in this direction +ΔV

Transcription

1 DC Crcuts Delverng a steady flow of electrc charge to a crcut requres an emf devce such as a battery, solar cell or electrc generator for example. mf stands for electromotve force, but an emf devce transforms one type of energy nto electrc potental energy thereby establshng a + charge separaton and provdng steady currents for the crcut. Crossng the emf n ths drecton +ΔV The battery was nvented by Alessandro Volta n 8. The fundamental component s an electrc cell consstng of two metallc electrodes submersed n a dlute acd soluton, convertng chemcal energy nto electrc energy: Dlute sulfurc acd removes Znc ons from the Zn strp leavng the anode negatvely charged. lectrons are drawn away from the Carbon strp leavng a postvely charged C cathode. The process contnues untl the Zn strp s consumed, the connecton s broken or the acd s neutralzed. Wthout a connecton between the two electrodes, the cell reacton contnues untl equlbrum s reached at the cell potental.

2 mf s the work done per unt charge that the emf devce does n movng charge from the lower potental termnal to ts hgher potental termnal. mf dw dq n volts Ideally, a source of emf has zero nternal resstance 'r', but n realty, an nternal resstance wll always be present. The "emf" s the termnal potental dfference when no current flows n the devce. esstors n Crcuts As charge carrers n a crcut encounter a 'devce', moton through the devce s mpeded by collsons wth atoms of the partcular devce materal. lectrc potental energy s converted nto thermal energy and emf energy s transferred to the resstor. Prevously we have seen that resstors are crcut elements obeyng Ohm's law.

3 The lnear relatonshp V I means charge carrer potental drops as a crcut Δ V I resstor s crossed n the drecton of the conventonal current s Conversely, f we evaluate the change n potental across a crcut resstor when lookng n a drecton opposte to the conventonal current, we have Δ V + I Krchhoff's ules: Current Junctons & Voltage Loops Krchhoff's st ule for crcuts s a statement of the conservaton of electrc charge at a convergence pont or juncton wthn a crcut. At any juncton pont, the sum of all currents nto that juncton equals the sum of all currents leavng the juncton. I + I I I I I

4 Krchhoff's nd ule for crcuts s a statement about the change n a charge's electrc potental as t moves n a closed loop path such as a crcut. f ΔV dr where as charge s dsplaced wthn ecall the path ntegral an electrc feld from ntal to fnal poston, the work done per unt charge s Δ V V f V ΔV Snce the electrostatc force s conservatve, s ndependent of the path taken from start to fnsh and only depends on the electrc potental at the end ponts. ΔV In partcular for the closed path, dr For crcuts, when movng through a closed loop n any crcut the dscrete sum of potental varatons as each crcut element n the loop s crossed wll be zero. Ths s Krchhoff's nd ule. ΔV

5 .g., MF wth nternal resstance: In the crcut shown, fnd the potental dfference between the termnals of battery #. 4.4V.V r.3ω r.8ω 5. 5Ω

6 Wrte down Krchhoff's Voltage ule startng and fnshng at pont 'a' ΔV r r + Solvng for the current then 4.4V.V.4A 4mA + r + r 5.5Ω +.3Ω +.8Ω The Potental dfference across battery s ΔV r + 4mA(.3Ω) + 4.4V 3. 8V Notce ths s not the MF of Battery# whch s 4.4V. Parallel and Seres esstor Networks As was done wth capactors, t may be possble to reduce resstor networks nto an equvalent resstance for the crcut by approprately addng resstor combnatons. In terms of resstvty, resstance s L ρ A Here L s the length of the resstve materal such that f we put two of these back-toback,.e., a seres combnaton, then L doubles and the resstance doubles.

7 Formally, we can also fnd ths by consderng the followng crcut: ΔV ( + + 3) quvalent Or, for resstors n seres, quvalent For resstors n parallel, we use Krchhoff's st ule and fnd an equvalent resstance as:

8 We know from Ohm's Law that: 3 3 In addton, we want to wrte somethng lke: quvalent From Krchhoff's st ule at juncton a, quvalent * quvalent In general for resstors n parallel, quvalent

9 .g., esstor Network educton:

10 .g., Current n a esstor Network: Fnd the current n the battery and through the 5Ω resstor for the crcut shown below. V quvalent 4V Ω + 8Ω // 47Ω 4V Ω + + 8Ω 47Ω V quvalent 4V 4Ω.A ma Ths s current through the battery and the equvalent crcut For the current n the 5Ω resstor, note that the voltage at pont A s V A 4 V (Ω) 3V Whch s also the voltage on the parallel connecton of the 8Ω resstor wth the 5Ω, Ω combnaton.

11 The current n the 5Ω, Ω combnaton s therefore 3V 5 Ω _ Ω 7. 7mA 47Ω.g., Mult-loop Network: Fnd the currents usng Krchhoff's ules. 3 + At Juncton A; 3 Clockwse around the 'left' loop; 4V (Ω) (8Ω) + 6V Clockwse around the 'rght' loop; V 3 (5Ω) 6V + (8Ω),, Leavng us three equatons wth three unknowns 3

12 From the frst and second equatons: 4V ( + 3) *(Ω) (8Ω) + 6V From ths equaton and the 'rght' loop equaton: 4V + (8Ω) 4V ( + )*(Ω) (8Ω) + 6V 5Ω esultng n: 3mA mA 78mA C Seres Crcuts The DC chargng / dschargng characterstcs of seres C crcuts may be determned usng Krchhoff's voltage loop rule: In the chargng case, as the swtch s closed, the charge on the capactor contnues to ncrease untl the potental dfference across the capactor s dentcal to the MF. Krchhoff's Voltage ule gves: q C dq dt + q C

13 The homogeneous equaton has dq dt + C q dq q dt C t ln q + C k q( t) ke t C The partcular equaton has q ( t) C as ts soluton so a general soluton s: t t C τ q( t) C( e ) C( e ) τ C Capactve _ Tme _ Const. The current n the crcut may be found by consderng ( t) dq dt e t τ Note the capactor becomes an open as tme goes to. The voltage on the capactor as t charges s q( t) τ V ( t) ( e t ) C

14 Dschargng case: The capactor starts out ntally wth a voltage of V and current s dsspated as heat loss to the resstor as the capactor dscharges. q C

15 dq dt + C q t C q( t) ke q() q k CV q( t) t C CVe CVe t τ ( t) dq dt V e t τ V ( t) q( t) Ve C t τ At t τ, V~.37V At t 5τ, V~.V

16 Ammeters, Voltmeters and Ohmmeters. Makng measurements of current, voltage and resstance wthn a crcut requres dfferent technques dependng on the measured quantty. In the ammeter, s small such that lttle to no potental drop occurs across. In the voltmeter, s large such that lttle to no current s drawn through. In both cases shown, G s a galvanometer wth a col of wre placed nsde a magnetc feld to gauge the current, voltage or resstance through / across a wre or element. A current-carryng col experences a torque n the magnetc feld whch, when calbrated, rotates an analog meter needle to the readout value. Note the hook-up: n-lne for the ammeter or across the element for the voltage and resstance measurements..

17 Power Dstrbuton lectrc power suppled from power plants s generated manly by burnng coal, ol, or gas, nuclear reactons, and hydroelectrc. lectrcty s alternatng current (AC) V and fused wthn the home. Shorts, surges, devce malfuncton or other crcumstances that create excessve current wthn the wrng are protected aganst wth a fuse or crcut breaker desgned to create an open n the crcut when currents are unsafe. Fuse szes are determned by the amount of resstve heatng the wrng can safely handle. Ths n turn depends on wre gauges such that uppng a fuse arbtrarly s not a good dea. -Gauge Copper Wre 8-Gauge Copper Wre 6-Gauge Copper Wre -Gauge Copper Wre.5-mm Dameter-A max-lghtng 3.6-mm Dameter-Applances 4.-mm Dameter-Applances 6.54-mm Dameter-Man Power Lnes AC current / voltage transmssons have the advantage of not beng as costly n terms of dsspatve resstve loss. Snce power s P IV I the average AC power ncludes averagng over a Sne squared functon whch gves a factor of V VSn( ωt) ω Hz Where π 6 I I Sn( ω ) t

18 V and I Peak values of the voltage and current. P I Sn ωt I Measured at the wall wth a DVM are the MS voltage and current: V rms V I rms I P I rms V rms I V I rms I V rms V Further, AC transmsson at hgh voltage (~'s kv) and low currents, further reducng the resstve loss, s possble snce transformng hgh voltages wth step-down transformers for consumer use s easer wth AC.