Kirchhoff's Laws I 2 I 3. junc. loop. loop -IR +IR 2 2 V P I V I R R R R R C C C. eff R R R C C C. eff 3.0

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1 V Kirchhoff's Laws junc j V + +V - + loop V j V P V - + loop eff eff C C C eff C C C eff

2 Charges in motion Potential difference V + E Metal wire cross-section Effective drag force that resists motion charge q flows through in t = q t Pb+SO 4-2 PbSO 4 +2 e - e- H + Pb H PbO 2 + SO 4-2 current = PbO 2 +SO H + +2e - PbSO 4 + H 2 O e coulombs =ampere sec Electric Potential Source Battery dry cell 1.5V Chemical energy + + Hg cell 1.35V (+ current by def.) 3.1 really charged e - moving but define like + electrical energy

3 3-2

4 Example: 10 resistor V= = (10) V=10 theory. units 3-3

5 Amp meter (very little resistance A DC (direct current ) multi-meters measures volts & amps. V Volt meter (very large resistance) (draws ~NO current.) Don t make a mistake on settings! [especially don t try to measure V on amp setting] Many meters also measures ( ) A Electrical Measurements V battery = Measures for known V bat so.. meter V battery = V bat 3-4

6 Power disipated to heat in resistor Units of power P=V C J J A V = = sec C sec =watts 3-5

7 or s s 3-6

8 Circuit Analysis Kirchhoff s Laws Conservation of current charge in 1 in 2 3 out =0 in in The sum of the currents into any junction must equal zero. (charge does not build up). (+=in -=out) You can choose any direction for, just stay with your choice throughout. out in 1 in Equivalently 2 3 out = 3 in in out 3-7

9 Electric potential energy conservation The sum of the potential difference around any closed loop is zero! Equivalently V cdab = V cyxb = V cb Potential difference between 2 points same for all possible paths! 3-8

10 esistors in Series A V B in general 1 2 V = V + V V = 1 2 n same everywhere V= V 1 + V 2 V 1 = 1 V= V 2 = 2 V= [ ] V= eff eff = eff eff = n 3-9

11 esistors in Parallel V 1 2 = in general V = ef f V V= 1 1 = V V 1 = 2 = = = V V V[ 1 1 ] V = V 1 ef f [ 1 1 ] eff [... ] 3-10 eff n 2 eff

12 Example: reff =? [ ]{ } e 1 [.01.03]{ } e e e 1 1 (10).25(10) Example: f 2A=, what is 33 & 100? What is in each leg? 33 V = ( 100)= (33) = = = = (.33) = ( 1.33) = 33 33= = =1.5A = 2A - 1.5A= 0.5A

13 Symmetry approach for = parallel s 3-11a

14 n general current ratio 1 1 small larger power dissipated 2 = = = P P = = atio of current inverse ratio of. P2 2 1 = ( ) P Watts P P = atio of power inverse ratio of. 3-12

15 eff in parallel-general observation =1000Ω =100Ω = + = eff = + eff eff. = =91Ω 11 1 big 2 small eff. A little less than 2 (small). 3-13

16 Capacitors Series Q Q 1 2 V V1 V2 V C 1 C 2 V V 2 1 Q C Q C V Q Q equal charge Q!! V Q ( ) Q C C C 1 2 eff C C C eff 1 2 n general: C C C C eff

17 Capacitors Parallel V C 1 C 2 equal voltages Vs!!! V C eff. = Q tot. = Q 1 +Q 2 V C eff. =VC 1 +VC 2 C eff =C 1 +C 2 Parallel capacitors add in general C eff = C 1 + C 2 + C 3 (opposite from resistors) like just increasing area of C 3-15

18 1.0 f 1.5 f 1.0 f 5.0 f Example: Example: =( + + )( ) Ceff μf =(1+ +.2) =(1+ +.2) 3 μf 3 μf eff 2 1 =( ) μf 1 1 =1.86 C eff =0.54μf C μf 1.5 f C eff =( )(μf)=7.5 μf 5.0 f 3-16

19 Exponential Function f () t t e = time constant -1 f(t=τ) = e = 1/e = 1/2.718 Big! t df 1 =- f dt τ df 1 + f=0 dt τ Δf 1 or =- f Δt τ 3-17 Note: gen. soln. df 1 =- dt f τ t ln(f)=- τ f(t)= f e 0 df 1 =- dt f τ f () t t - τ t e (f 0 = constant)

20 Time varying electrical current t=0 close switch C +Q -Q 0=V c +V t=0 Q=Q 0 Q 0 C Example: Capacitor discharge Q 0 t dq = dt dq dt Q + =0 [C] τ=c=time constant Q=[?] e t - c t=0 Q=Q [?]= Q Q=Q e 0 t - C 0 0 Q Q ecall = = - C C Q = Q e t - C

21 t=0 close switch. V Very similar to before but not identical t=0 Q=0: t= =0 ie. Q o VC Q = VC (1- e - t C ) = - Q C = V e C t V V (1- e - + V C ) t C Capacitor Charging ΔQ Again: -V + + Q = C 0 Δt ΔQ Q ΔQ Q V dq Q V + =V + = + = Δt C Δt C dt C 3-19

22 Appendix : Temperature dependence of resistivity & superconductivity e - bounces through + atomic position (+ ions) vibrate) Scattering of e - creates resistance to forward motion. esistance is energy loss mechanism motion heat vibrating Heavy atoms scatter little e - strongly when atoms are vibrating (at finite T). Atoms vibrate less at low T Less electron scattering Lower resistivity 3--1

23 Super conductivity 0 at critical temp. T c in some materials 1911-Hg T c =4.2K Nb 3 Sn T c =23K 1989 Y 1 Ba 2 Cu 3 O 7 T c =95K T c =121 K highest yet!!!! e - attractive ++ + e - 1) 2 e - attracted to + ion 2) effective attraction between e - 3) e - -e - pairs form 4) pairs don t scatter so no resistance attractive 3--2 An infinite number of mathematicians walk into a bar. The first one tells the bartender he wants a beer. The second one says he wants half a beer. The third one says he wants a fourth of a beer. The bartender puts two beers on the bar and says You guys need to learn your limits.

24 Appendix : Formal Kirchhoff's Laws approach/concepts ules junctions junction sign convention + in :: - out (or opposite your choice but stick to it) O = current conservation!! = no charge build-up esistors: magnitude and sign convention loop

25 3--2 ules Battery sign convention + - -V + +V - Voltage drop around loop = 0 (Energy conservation!!)

26 3--3 eff

27 esistors in parallel

28 3--5

Direct-Current Circuits. Physics 231 Lecture 6-1

Direct-Current Circuits. Physics 231 Lecture 6-1 Direct-Current Circuits Physics 231 Lecture 6-1 esistors in Series and Parallel As with capacitors, resistors are often in series and parallel configurations in circuits Series Parallel The question then