Lesson 31 Study Sheet

Circuits Resistors can be combined in Series or in Parallel. Series Assume that we are adding resistors together in a line. That is, we are just placing one resistor after another, with one wire going into and out of each resistor. This configuration of resistors is called series wiring. R1 R2 R3 The current through the resistors is the same. Resistors can only oppose current flow and cannot store charge. By charge conservation, the current that goes in must come out. So I1 = I2 = I3 = … Then ∆V1 = IR1; ∆V2 = IR2; ∆V3= IR3 , etc. Consider the circuit below. R1 ε R2 Equivalent resistance ε = ∆Veff = ∆V1 + ∆V2 ∆Veff = IReff = IR1 + IR2 = I(R1 + R2) So Reff = R1 + R2 Parallel In another type of wiring, parallel wiring, there is a spot (or junction) where the wire splits into different possible routes for the current to take. By conservation of charge, charge cannot be created or lost. Therefore, when the wire splits, the current splits along with it. The total charge passing through remains the same. That is, the current before the split equals the current after the split. Therefore, I = I1 + I2 + I3 +… R1 R2 R3 The resistors share the same junctions, so the potential differences across all resistors must be the same. Then V1 = V2 = V3 = …….. Consider the circuit below. We use Ohm’s Law, written as I = V/R, to find each current. The currents through the two resistors must add up to equal the total current in the circuit, Ieff. R1 R2 ε The equivalent resistance is found from Ieff = I1 + I2. Ieff = ∆V/Reff ; I1 = ∆V/R1; I2 = ∆V/R2 So ∆Veff/Reff = ∆V/R1 + ∆V/R2 1 1 1 1 = + + + .... Reff R1 R2 R3 Kirchhoff’s Rules 1. Junction rule: The sum of currents entering a junction must equal sum of currents leaving. 2. Loop Rule: The total potential difference of all elements in a loop must be zero. Capacitors in Series and in Parallel Parallel When capacitors are connected in parallel the same potential difference is applied to each. q1 = C1V q2 = C2V q3 = C3V C1 The effective capacitance can be found by using conservation of charge: Total charge, q = Ceff V = q1 + q2 + q3 = C1V + C2V + C3V = (C1 + C2 + C3 )V C2 Thus, Ceff = C1 + C2 + C3 C3 Series For the series circuit, the sum of the voltages across the capacitors equals the voltage supplied by the source. The magnitude of the charge on each plate of each capacitor is the same, because the charge cannot flow across the plates of a capacitor. V1 = q q q V2 = V3 = C2 C3 C1 V1 −q V2 +q −q V V= q q q q 1 1 1 = V1 + V2 + V3 = + + =( + + )q Ceff C1 C2 C3 C1 C2 C3 1 1 1 1 = + + + ... Ceff C1 C2 C3 V3 +q −q +q RC Circuits While an ideal emf source does not have any internal resistance, all real emfs do. This resistance is usually denoted in circuit diagrams by the following notation. ε R In most cases, this internal resistance must be measured via experiment. A power supply provides charge to a capacitor. Even when the circuit is disconnected, the capacitor retains the charge. However, if there is a resistor in the circuit, the resistor can drain charge from the capacitor once the power supply is disconnected. The rate of discharge is related to the amount of capacitance and to the amount of resistance. Obviously if C is large, it will take a long time. If R is large, it will also drain the charge slower. The charge is drained exponentially instead of linearly because ∆V changes with q. The rate of discharge is given by q (t ) = q0 e −t RC , where e is the base of the natural logarithm. RC in the equation is called the time constant of the circuit. An RC circuit charging up behaves similarly: q (t ) = q0 (1 − e −t RC )