A student n 1120 emaled me to ask how much extra he should expect to pay on hs electrc bll when he strngs up a standard 1-strand box of ccle holday lghts outsde hs house. (total, cumulatve cost)? Try to make a real estmate, don t just guess! Energy n Colorado costs about 10 /kw hr. A: ess than 1 cent B: Between 1 cent and 10 cents C: Between $.10 and $1.00 D: Between $1.00 and $10.00 E: More than $10.00 50-100 Watts? (ke ONE bulb). 12 hrs/day? 30 days? ~100 W * 10 hrs/day * 30 days = 30,000 W*hrs = 30 kw hrs. $3 (for 100 mllon Joules! Energy s cheap ) 1 Welcome back!! CAPA #13 s due Frday New onlne partcpaton survey s up! Pretest tonght (and Tut hw for tomorrow) eadng: catch up f you re behnd! E.g. 35.6 (1st 2 pp) (We ll fnsh up Ch. 33 ths week -all sectons, ncludng 33.6) ast: Transformers and Inducton Today: Inductors n crcuts Next: AC crcuts 2 Inductor = (col of wre) Important fact: Magnetc Flux Φ B s proportonal to the current makng the Φ B All our equatons for B-felds show that B α v v µ ˆ 0 dl! r v v db = 2 " B! dl = µ 0 4" r Bot-Savart Ampere thru 3 1
Flux " B! B! Flux "! B Assumes leavng everythng else the same. If we double the current, we wll double the magnetc flux through any surface. 4 Self-Inductance () of a col of wre " B! "! B Ths equaton defnes self-nductance. Note that snce Φ B α, must be ndependent of the current. has unts [] = [Tesla meter 2 ]/[Amperes] New unt for nductance = [Henry]. 5 An nductor s just a col of wre. The Magnetc Flux created by the col, through the col tself: v v # =! B " da B Ths s qute hard to calculate for a sngle loop. Earler we calculated B at the center, but t vares over the area. 6 2
Inductor 1 conssts of a sngle loop of wre. Inductor 2 s dentcal to 1 except t has two loops on top of each other. How do the self-nductances of the two loops compare? A) 2 = 2 1 B) 2 > 2 1 C) 2 < 2 1 I Col 1 Col 2 I HINT 1: What s the B feld at the center of col 2, B2, compared to the feld n the center of col 1? HINT 2: nductance = Φ(total)/I Answer: 2 > 21, n fact 2 s roughly 4 1! ecall =Φ/I. When N doubles => B doubles. Φ(each loop) doubles (because B s doubled) But Φ(tot) = 2 Φ(each loop), so double*double = 4 tmes! 7 Consder a smpler case of a solenod B v N nsde = µ 0 n = µ 0 " ecall that the B-feld nsde a solenod s unform! N = number of loops = length of solenod * Be careful wth symbol! 8 B v N nsde = µ 0 n = µ 0 " " B = N $ B v # da v = NBA = N(µ 0 n)a Self-Inductance = " B = µ 0 NnA = µ 0 An 2 " ength 9 3
Clcker Queston Two long solenods, each of nductance, are connected together to form a sngle very long solenod of nductance total. What s total? A) 2 B) 4 C) 8 D) none of these/don't know + = total =? Answer: 2. The nductance of a solenod s = µ o An 2, (n = N/ and s the length.) In ths case, we dd not change n, but (length) doubled, so doubles. 10 What does nductance tell us?! B =! = B d B d =! s ndependent of tme. Depends only on geometry of nductor (lke capactance). 11 d! B d = d!" = d " =! ecall Faraday s aw # = " d! Changng the current n an nductor creates an EMF whch opposes the change n the current. Sometmes called back EMF B 12 4
d " =! It s dffcult (requres bg external Voltage) to change quckly the current n an nductor. The current n an nductor cannot change nstantly. If t dd (or tred to), there would be an nfnte back EMF. Ths nfnte back EMF would be fghtng the change! 13 14 What do these nductors do n crcuts? Just recall that the EMF or Voltage across an nductor s: d " =! So, when we add them to crcuts, we can apply the usual Krchhoff s Voltage aw and nclude the nductors. 15 5
Consder a crcut wth a battery, resstor and nductor ( crcut) Battery=V swtch a b Suppose swtch s n poston (a) for a long tme. 16 Consder a crcut wth a battery, resstor and nductor ( crcut) Battery=V swtch a b Suppose swtch s n poston (a) for a long tme. In steady state (after a long tme), the current wll no longer be changng and thus the nductor looks lke a regular wre! 17 Battery=V swtch a b If after a long tme the nductor acts lke a wre: V " V = V! = 0 = 18 6
Battery=V swtch a b At t=0, move the swtch to (b). Normally one mght expect there to mmedately be zero current. However, nductors don t let the current change nstantly. 19 Battery=V " V =!! d d swtch a = 0 b & # = ' $! We need to solve ths dfferental % " equaton for (t). 20 d & # = ' $! % " & # ' $! t % " ( t) = 0 e where ( t) = e & # ' t / $! % " 0 V 0 = Current exponentally decays wth Tme Constant =τ= / (unts of seconds). 21 7
Clcker Queston ank n order, from largest to smallest, the tme constants n the three crcuts. A. τ 1 > τ 2 > τ 3 B. τ 2 > τ 1 > τ 3 C. τ 2 > τ 3 > τ 1 D. τ 3 > τ 1 > τ 2 E. τ 3 > τ 2 > τ 1 ( t) = e & # ' t / $! % " 0 22 Clcker Queston = 20! The swtch n the crcut below s closed at t=0. What s the ntal rate of change of current d/ n the nductor, mmedately after the swtch s Hnts: What s the ntal current through the crcut? Gven that - what s the ntal voltage across the nductor? 23 Clcker Queston Before t=0, V=? =20 Ω V=? The swtch n the crcut below s closed at t=0. What s the ntal rate of change of current d/ n the nductor, mmedately after the swtch s Hnts: What s the ntal current through the crcut? Gven that - what s the ntal voltage across the nductor? 24 8
Clcker Queston The swtch n the crcut below s closed at t=0. =20 Ω Before t=0, V=? What s the ntal rate of change of current d/ n the nductor, mmedately after the swtch s Hnts: What s the ntal current through the crcut? Gven that - what s the ntal voltage across the nductor? 25 Clcker Queston V=10 =20 Ω V=10! The swtch n the crcut below s closed at t=0. What s the ntal rate of change of current d/ n the nductor, mmedately after the swtch s Hnts: What s the ntal current through the crcut? Gven that - what s the ntal voltage across the nductor? 26 Clcker Queston V=10 = 20! At t=0+, V=? The swtch n the crcut below s closed at t=0. What s the ntal rate of change of current d/ n the nductor, mmedately after the swtch s Hnts: What s the ntal current through the crcut? Gven that - what s the ntal voltage across the nductor? 27 9
Clcker Queston V=10 = 20! V=10!! (brefly) The swtch n the crcut below s closed at t=0. What s the ntal rate of change of current d/ n the nductor, mmedately after the swtch s Hnts: What s the ntal current through the crcut? Gven that - what s the ntal voltage across the nductor? 28 Clcker Queston V=10 = 20! V=10!! (brefly) The swtch n the crcut below s closed at t=0. What s the ntal rate of change of current d/ n the nductor, mmedately after the swtch s Hnts: What s the ntal current through the crcut? Gven that - what s the ntal voltage across the nductor? Answer: I(ntal)=0, so t must stll be 0. Thus, ΔV(across )=0 V(across nductor) = V(batt!), but also V(nductor) = d/. So at t=0+, d/ = V(batt)/ = 10V/10H = 1 A/s. 29 Clcker Queston An crcut s shown below. Intally the swtch s open. At tme t=0, the swtch s closed. What s the current thru the nductor mmedately after the swtch s closed (tme = 0+)? 1 = 10! 2 = 10! A) Zero B) 1 A C) 0.5A D) None of these. 30 10
Clcker Queston An crcut s shown below. Intally the swtch s open. At tme t=0, the swtch s closed. 1 = 10! 2 = 10! After a long tme, what s the current from the battery? A) 0A B) 0.5A C) 1.0A D) 2.0A 31 What do you thnk of today s all powerpont format? A) Better than usual blackboard work - let s keep t! B) Some + s, some - s, t s a wash for me. C) I prefer the usual blackboard work (wth powerpont reserved for just clcker questons) Go back! D) No opnon, the professor should decde E) I have somethng dfferent to say about ths 32 AC Crcuts The Voltage n your wall sockets at home s AC. AC stands for Alternatng Current, but would perhaps more approprately be called Alternatng Voltage. Alternatng = Snusodal wth tme 34 11
V ( t) = Vpeak sn(! t) 2! " = 2! f = T ω s the radal frequency (radans / second) f s the frequency n (cycles / second = Hertz) T s the perod n (seconds),.e. tme for one cycle 35 12