V Kirchhoff's Laws junc j 0 1 2 3 - -V + +V - + loop V j 0 2 2 V P V - + loop eff 1 2 1 1 1 eff 1 2 1 1 1 C C C eff C C C eff 1 2 1 2 3.0
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 -2 4 +4H + +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
3-2
Example: 10 resistor V= = (10) V=10 theory. units 3-3
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
Power disipated to heat in resistor Units of power P=V C J J A V = = sec C sec =watts 3-5
or s s 3-6
Circuit Analysis Kirchhoff s Laws Conservation of current charge in 1 in 2 3 out 1 + 2-3 =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 1 + 2 = 3 in in out 3-7
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
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= 1 + 2 V 2 = 2 V= [ 1 + 2 ] V= eff eff = 1 + 2 eff eff = 1 + 2 +... + n 3-9
esistors in Parallel V 1 2 = 1 + 2 in general V = ef f V V= 1 1 = 2 2 1 2 2 1 1 2 V V 1 = 2 = 2 1 1 2 3 = = V V V[ 1 1 ] V 1 2 1 2 = V 1 ef f [ 1 1 ] eff 1 2 1 1 1 1 1 [... ] 3-10 eff n 2 eff
Example: reff =? 100 33 1 1 1 1 [ ]{ } 100 33 e 1 [.01.03]{ } 1 1.04 e e e 1 1 (10).25(10).04 4 25 3-11 2 2 Example: f 2A=, what is 33 & 100? 100 100 33 What is in each leg? 33 V = ( 100)= (33) = =.33 100 = + 100 33 100 33 33 100 33 = (.33) + 33 33 = ( 1.33) 33 2 2 3 = 33 33= = =1.5A 1.33 4 2 3 = 2A - 1.5A= 0.5A
Symmetry approach for = parallel s 3-11a
n general current ratio 1 1 small larger power dissipated 2 = = = 2 2 1 1 1 1 2 2 2 1 1 2 2 P P = 2 1 1 2 2 2 2 2 = 2 1 1 1 atio of current inverse ratio of. P2 2 1 = ( ) P 1 1 2 2 74.25 Watts P P = 2 1 1 2 atio of power inverse ratio of. 3-12
eff in parallel-general observation =1000Ω =100Ω 1 2 1 1 1 11 = + = 1000 100 1000 eff. 1 2 1 1 1 = + eff. 1 2 1000 eff. = =91Ω 11 1 big 2 small eff. A little less than 2 (small). 3-13
Capacitors Series Q Q 1 2 V V1 V2 V C 1 C 2 V V 2 1 Q C Q C 1 1 2 2 V - - - - -Q ++++ +Q equal charge Q!! 1 1 1 V Q ( ) Q C C C 1 2 eff. 1 1 1 C C C eff 1 2 n general: 1 1 1 1 C C C C eff 1 2 3... 3-14
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
1.0 f 1.5 f 1.0 f 5.0 f Example: Example: 1 1 1 1 1 =( + + )( ) Ceff 1.0 1.5 5.0 μf 1 1 2 1 =(1+ +.2) =(1+ +.2) 3 μf 3 μf eff 2 1 =(1+.66+.2) μf 1 1 =1.86 C eff =0.54μf C μf 1.5 f C eff =(1.0+1.5+5.0)(μf)=7.5 μf 5.0 f 3-16
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)
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 3-18 0 t - C
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
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
Super conductivity 0 at critical temp. T c in some materials 1911-Hg T c =4.2K 1950-1970 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.
Appendix : Formal Kirchhoff's Laws approach/concepts ules junctions junction sign convention + in :: - out (or opposite your choice but stick to it) O = 1 + 2 current conservation!! = no charge build-up esistors: magnitude and sign convention loop 3--1 - +
3--2 ules Battery sign convention + - -V + +V - Voltage drop around loop = 0 (Energy conservation!!)
3--3 eff
esistors in parallel + - + - + - 3--4
3--5