Chapter 26 – Direct-Current and Circuits- Resistors in Series and Parallel- Kirchhoff’s Rules- Electric Measuring Instruments- R-C Circuits1. Resistors in Series and ParallelResistors in Series:Vax= I R1 Vxy= I R2Vyb= I R3Vab= Vax+ Vxy+ Vyb= I (R1+ R2+ R3)Vab/I = R1+ R2+ R3= ReqResistors in Parallel:I1= Vab/R1I2= Vab/R2 I3= Vab/R3I = I1+ I2+ I3= Vab(1/R1+ 1/R2+ 1/R3) I/Vab= 1/Req...321+++=RRRReq...1111321+++=RRRReq2. Kirchhoff’s RulesJunction: point where three or more conductors meet (nodes, branch points).Loop: closed conducting path.Kirchhoff’s junction rule: the algebraic sum of the currents into any junction is zero.0=ΣIKirchhoff’s loop rule: the algebraic sum of the potential difference in any loop, Including those associated with emfs and those of resistive elements, must equal zero.0=ΣV-The junction rule is based on conservation of electric charge. No charge can accumulate at a junction total charge entering the junction per unit time = total charge leaving.(electrostatic force is conservative)Sign Conventions for Loop Rule: emf source (ε) positive (travel from – to +)negative (travel from + to -)resistor (IR) negative (travel in same direction as I decreasing V)positive (travel in contrary direction to I increasing V)- “Travel” is the direction that we imagine going around the loop, not necessarily the direction of the current.3. Electrical Measuring Instrumentsd’Arsonval galvanometer (meter):Coil of wire mounted next to a permanent magnet. Attached to the coil is a spring. Any current passing through the coil will induce magnetism in the coil (magnetic field exerts a torque on the coil ~current). When the coil turns, spring makes restoring torque ~ angular displacement ~ current.V = IfsRcIfs= current full scale (coil)Rc= resistance of coilAmmeter: device that measures current, R = 0- It can be adapted to measure currents larger thanits full scale range by connecting Rsh(shunt resistor) in parallel (some I bypasses meter coil).Ia= Ish+ IfsIfs= current through coilIsh= current through RshIa= current measured by ammeterVab= IfsRc= IshRsh= (Ia-Ifs) RshVoltmeter: device that measures voltage, R = ∞- It can be adapted to measure voltages larger than its full scale range by connecting Rsin series with the coil .Vv = Vab= Ifs(Rc+Rs)Ohmeter: device that measures resistance.- The series resistance Rsis adjusted so that when the terminals x-y are short-circuited (R = 0),the meter deflects full scale (zero). When nothing is connected between x-y (open circuit, R = ∞) there is no current (no deflection). For intermediate R values, meter scale is calibrated to read R.Potenciometer: device that measures emf of a source without drawing any current from it.- Rabconnected to terminals of known emf (ε1). A sliding contact (c) is connected through galvano-meter (G) to unknown source (ε2). As contact (c) is moved along Rab, Rcbvaries proportional to wire length (c-b). To find ε2 (c) is moved until G shows no deflection (IG= 0):- G calibrated by replacing ε2by source of known emf.ε = I Rcb4. R-C CircuitsCharging a Capacitor:CqviRvvvbcabbcab==+=ε0=−−CqiRεRCQRf=εRCqRi −=εRCQf=- Capital letters: V, Q, I (constant)- Lowercase letters: v, i, q (varying)RRvIab//0ε==t = 0 q = 0 vbc= 0 t = tf I = 0 vab= 0 vbc= ε = Qf/CAt an intermediate time, t:At t = tf i = 0 Charging a Capacitor:()()RCtfRCteQeCq//11−−−=−=εRCtRCteIeRdtdqi/0/ −−===εRCteCCq/−=−−εεRCtCCq−=−−εεlnRCdtCqdq−=−ε∫∫−=−tqRCdtCqdq00'''ε( )εεCqRCRCqRdtdqi −−=−==1(solve by changing variablex = q’ – Cε)Discharging a Capacitor:Time Constant: relaxation time of the circuit timeafter which the current in the circuit has decreased to 1/e of I0and charge has reached (1-1/e) of Qf= Cε.RC=τRCqdtdqi −==∫∫−=tqQdtRCqdq0'1''0- If RC small circuit charges quickly.- i never becomes exactly 0, and q never becomes exactly Qf(asymptotic behavior).RCtQq−=0lnt = 0 q = Q0, ε = 0 (capacitor discharges through R)0=−−CqiRDischarging a Capacitor:RCteQq/0−=RCtRCteIeRCQdtdqi/0/0−−=−==- During charging: CiqRii +=⋅2εInstantaneous rate at which battery delivers energy to circuiti2R = power dissipated in Ri q/C = power stored in CTotal energy supplied by battery: ε QfTotal energy stored in capacitor:
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