Transcript Lect23
RC Circuits 2 I a I a I I R C e RC Ce1 R b b + + C e 2RC RC Ce 1 1 - - 2RC Q q Cee t / RC f( x ) q 0.5 q Ce 1 e 00 0 1 t t / RC f( xq) 0.5 0.0183156 0 2 x 3 4 0 0 1 t 2 x 3 4 4 Quiz Histogram 35 20 Frequency 15 10 5 Bin 80 95 60 65 40 50 20 35 0 80 0 20 Highest 97% 25 5 Mean 63% Frequency 30 100% RC circuits from yesterday • When circuits contain capacitors, currents and voltages vary with time • Kirchoff’s Laws apply, they give differential equations! • The differential equations have exponential solutions – from the form of differential equation • time constant t = RC • series RC charging solution • series RC discharging solution Q Ce 1 e t Q Ce e RC t RC Question 1 • At t=0 the switch is thrown from position b to position a in the circuit shown: The capacitor is initially uncharged. – At time t=t1=t, the charge Q1 on the capacitor is (1-1/e) of its asymptotic charge Qf=Ce. – What is the relation between Q1 and Q2 , the charge on the capacitor at time t=t2=2t ? (a) Q2 < 2Q1 (b) Q2 = 2Q1 a I b I R e C R (c) Q2 > 2Q1 Question 1 62% 35% c b 4% a 1. a 2. b 3. c Question 1 a • At t=0 the switch is thrown from position b to position a in the circuit shown: The capacitor is initially uncharged. I R b e – At time t=t1=t, the charge Q1 on the capacitor is (1-1/e) of its asymptotic charge Qf=Ce. – What is the relation between Q1 and Q2 , the charge on the capacitor at time t=t2=2t ? (a) Q2 < 2Q1 I C R (c) Q2 > 2Q1 (b) Q2 = 2Q1 • Charge increases according to: Q Ce (1 e 2 RC ) 2Q1 Q2 Q1 Q • From the graph the charge increase 1 is not as fast as linear. • The rate of increase is just f( x ) 0.5 proportional to the current (dQ/dt) Q which decreases with time. • Therefore, Q2 < 2Q1. 0 t 0 t1 2t2 3 4 Capacitor time constants •The two circuits shown below contain identical fully charged capacitors at t=0. Circuit 2 has twice as much resistance as circuit 1. •How does the charge on the two capacitors compare a short time after t = 0? •Initially, the charges on the two capacitors are the same. But the two circuits have different time constants •t1 = RC and t2 = 2RC. Since t2 > t1 it takes circuit 2 longer to discharge its capacitor (twice as long). •Therefore, at any given time, the charge on capacitor 2 is bigger than that on capacitor 1. Question 2 a Ce1 0 t01 2 x t/RC C Ce 1 (c) (b) (a) 00 2R e Ce1 3 4 f( x )q 0.5 t 00 f ( x ) 0 .5q Q f( x )q 0.5 b R – At t = t0, the switch is thrown from position a to position b. – Which of the following graphs best represents the time dependence of the charge on C? Q Q • At t=0 the switch is connected to position a in the circuit shown: The capacitor is initially uncharged. 0 t01 2 x t/RC 3 4 t 00 0 t0 1 2 x t/ RC t 3 Question 2 1. a 2. b 3. c 54% b a c 23% 23% Question 2 a Ce1 0 t01 2 x t/RC C Ce 1 (c) (b) (a) 00 2R e Ce1 3 4 f( x )q 0.5 t 00 f ( x ) 0 .5q Q f( x )q 0.5 b R – At t = t0, the switch is thrown from position a to position b. – Which of the following graphs best represents the time dependence of the charge on C? Q Q • At t=0 the switch is connected to position a in the circuit shown: The capacitor is initially uncharged. 0 t01 2 x t/RC 3 4 t 00 0 t0 1 2 x t/ RC • For 0 < t < t0, the capacitor is charging with time constant t = RC • For t > t0, the capacitor is discharging with time constant t = 2RC • (a) has equal charging and discharging time constants • (b) has a larger discharging t than a charging t • (c) has a smaller discharging t than a charging t t 3 A very interesting RC circuit I1 I2 I3 e C R2 R1 •First consider the short and long term behavior of this circuit. •Initially the capacitor acts like an ideal wire. Hence, and Question 3 •After a long time the current I1 is a) I1 = 0 b) I1 = e/R1 c) I1 = e/(R1+ R2) I1 I2 I3 e C R1 R2 Question 3 1. a 2. b 3. c 91% c 2% b a 7% Question 3 •After a long time the current I1 is I1 I3 e a) I1 = 0 b) I1 = e/R1 c) I1 = e/(R1+ R2) I2 C R1 •After a long time the capacitor becomes fully charged, curent I3 becomes zero (the capacitor acts like an open switch •So, I1 = I2, and we have a one-loop circuit with two e resistors in series, I1 I 2 R1 R2 •C is in parallel with R2, therefore the voltage across each is the same Vc I 2 R2 e R2 R1 R2 e R2 Very interesting RC circuit continued Loop 2 • Loop 1: • Loop 2: • Node: Q e I1 R1 0 C e I 2 R2 I1R1 0 I1 e dQ I1 I 3 I 2 I2 dt Loop 1 I2 I3 C R1 • Eliminate I1 in L1 and L2 using Node equation: • Loop 1: Q dQ e R1 I2 0 C dt eliminate I2 from this dQ e I 2 R2 R1 I2 0 dt e dQ Q • Final differential eqn: R1 dt R1 R2 C R1 R2 • Loop 2: R2 Very interesting RC circuit Loop 2 • Final differential eqn: dQ Q e dt R1 R2 R1 C R1 R2 dQ Q e dt RC R1 • Try solution of the form: continued I1 e I2 I3 Loop 1 C R1 Q (t ) A 1 e t t – and plug into differential equation to get parameters A and τ A t e t t t A(1 e CR t ) e R1 t CR C A e CR R1 R1 R2 R1 R2 A Ce R2 R1 R2 R2 Very interesting RC circuit continued Loop 2 • Final differential eqn: dQ Q e dt R1 R2 R1 C R1 R2 I1 e time constant: t parallel combinationof R1 and R2 •Solution t Q (t ) A 1 e t R2 A Ce R R 1 2 Loop 1 I2 I3 C R2 R1 R R 1 2 t R R C 2 1 •C and R2 are in parallel •After a long time the capacitor will be charged to the same potential as the potential drop across R2. I3 will be 0 and the current will just flow in the resistance loop Very interesting RC circuit continued • What about discharging? – Open the switch... e • Loop 1 does not exist any more • I2 is only current • only one loop – start at x marks the spot... •Solution Q (t ) Ae I2 t C R2 R1 Q dQ Q I 2 R2 0 R2 C dt C R2C •Initial condition, Q(0)=A, is the same as the final condition before Different time constant for discharging RC Circuits • That is all about circuits for the time being • After we have done magnetism we will come back to consider circuits with inductance as well as capacitance Magnetism The Magnetic Force F qE qv B B x x x x x x x x x x x x v x x x x x x q F B B v q F v q F=0 Magnetism • Magnetic effects from natural magnets have been known for a long time. Recorded observations from the Greeks more than 2500 years ago. • The word magnetism comes from the Greek word for a certain type of stone (lodestone) containing iron oxide found in Magnesia, a district in northern Greece. • Properties of lodestones: could exert forces on similar stones and could impart this property (magnetize) to a piece of iron it touched. • Small sliver of lodestone suspended with a string will always align itself in a north-south direction—it detects the earth’s magnetic field. Bar Magnet • Bar magnet ... two poles: N and S Like poles repel; Unlike poles attract. • Magnetic Field lines: (defined in same way as electric field lines, direction and density) S N • Looks like dipole in electrostatics? Electric Field Lines of an Electric Dipole Magnetic Field Lines of a bar magnet S N Magnetic Monopoles • Perhaps there exist magnetic charges, just like electric charges. Such an entity would be called a magnetic monopole (having + or - magnetic charge). • How can you isolate this magnetic charge? Try cutting a bar magnet in half: S N S N S N Even an individual electron has a magnetic “dipole”! • Many searches for magnetic monopoles—the existence of which would explain (within framework of QM) the quantization of electric charge (argument of Dirac) • No monopoles have ever been found: B dS 0 Source of Magnetic Fields? • What is the source of magnetic fields, if not magnetic charge? • Answer: electric charge in motion! – e.g., current in wire surrounding cylinder (solenoid) produces very similar field to that of bar magnet. • Therefore, understanding source of field generated by bar magnet lies in understanding currents at atomic level within bulk matter. Orbits of electrons about nuclei Intrinsic “spin” of electrons (more important effect) Summary • Read Fishbane Chapter 28 on Magnetic Fields • Do Fishbane problems Chapter 27, #57, 59, 78 • Have a good Spring Break