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Chapter 25 Electric Circuits Direct Current • When the current in a circuit has a constant direction, the current is called direct current • Most of the circuits analyzed will be assumed to be in steady state, with constant magnitude and direction • Because the potential difference between the terminals of a battery is constant, the battery produces direct current • The battery is known as a source of emf Sources of emf • The source that maintains the current in a closed circuit is called a source of emf • Any devices that increase the potential energy of charges circulating in circuits are sources of emf (e.g. batteries, generators, etc.) • The emf (ε) is the work done per unit charge • Emf of a source is the maximum possible voltage that the source can provide between its terminals • SI units: Volts Electric Circuits • A circuit is a collection of objects usually containing a source of electrical energy connected to elements that convert electrical energy to other forms • A circuit diagram – a simplified representation of an actual circuit – is used to show the path of the real circuit • Circuit symbols are used to represent various elements Resistors in Series • When two or more resistors are connected end-to-end, they are said to be in series • The current is the same in all resistors because any charge that flows through one resistor flows through the other • The sum of the potential differences across the resistors is equal to the total potential difference across the combination I1 I 2 I V IR1 IR 2 I ( R1 R2 ) IReq Resistors in Series V IReq Req R1 R2 • The equivalent resistance has the effect on the circuit as the original combination of resistors (consequence of conservation of energy) • For more resistors in series: Req R1 R 2 R3 ... • The equivalent resistance of a series combination of resistors is greater than any of the individual resistors emf and Internal Resistance • A real battery has some internal resistance r; therefore, the terminal voltage is not equal to the emf • The terminal voltage: ΔV = Vb – Va ΔV = ε – Ir • For the entire circuit (R – load resistance): ε = ΔV + Ir = IR + Ir emf and Internal Resistance ε = ΔV + Ir = IR + Ir • ε is equal to the terminal voltage when the current is zero – open-circuit voltage I = ε / (R + r) • The current depends on both the resistance external to the battery and the internal resistance • When R >> r, r can be ignored • Power relationship: I ε = I2 R + I2 r • When R >> r, most of the power delivered by the battery is transferred to the load resistor Resistors in Parallel I1 R1 I 2 R2 V V V I I1 I 2 R1 R2 • The potential difference across each resistor is the same because each is connected directly across the battery terminals • The current, I, that enters a point must be equal to the total current leaving that point (conservation of charge) • The currents are generally not the same Resistors in Parallel I1 R1 I 2 R2 V V V I I1 I 2 R1 R2 1 1 V V R1 R2 Req 1 1 1 Req R1 R2 Resistors in Parallel • For more resistors in parallel: 1 1 1 1 Req R1 R2 R3 • The inverse of the equivalent resistance of two or more resistors connected in parallel is the algebraic sum of the inverses of the individual resistance • The equivalent is always less than the smallest resistor in the group Problem-Solving Strategy • Combine all resistors in series • They carry the same current • The potential differences across them are not necessarily the same • The resistors add directly to give the equivalent resistance of the combination: Req = R1 + R2 + … Problem-Solving Strategy • Combine all resistors in parallel • The potential differences across them are the same • The currents through them are not necessarily the same • The equivalent resistance of a parallel combination is found through reciprocal addition: 1 1 1 ... Req R1 R2 Problem-Solving Strategy • A complicated circuit consisting of several resistors and batteries can often be reduced to a simple circuit with only one resistor • Replace resistors in series or in parallel with a single resistor • Sketch the new circuit after these changes have been made • Continue to replace any series or parallel combinations • Continue until one equivalent resistance is found Problem-Solving Strategy • If the current in or the potential difference across a resistor in the complicated circuit is to be identified, start with the final circuit and gradually work back through the circuits (use formula ΔV = I R and the procedures describe above) Chapter 25 Problem 20 What resistance should you place in parallel with a 56-kΩ resistor to make an equivalent resistance of 45 kΩ? Kirchhoff’s Rules • There are ways in which resistors can be connected so that the circuits formed cannot be reduced to a single equivalent resistor • Two rules, called Kirchhoff’s Rules (Laws) can be used instead: • 1) Junction (Node) Rule • 2) Loop Rule Gustav Kirchhoff 1824 – 1887 Kirchhoff’s Rules • Junction (Node) Rule (A statement of Conservation of Charge): The sum of the currents entering any junction must equal the sum of the currents leaving that junction • Loop Rule (A statement of Conservation of Energy): The sum of the potential differences across all the elements around any closed circuit loop must be zero Junction Rule I1 = I2 + I3 • Assign symbols and directions to the currents in all branches of the circuit • If a direction is chosen incorrectly, the resulting answer will be negative, but the magnitude will be correct Loop Rule • When applying the loop rule, choose a direction for transversing the loop • Record voltage drops and rises as they occur • If a resistor is transversed in the direction of the current, the potential across the resistor is – IR • If a resistor is transversed in the direction opposite of the current, the potential across the resistor is +IR Loop Rule • If a source of emf is transversed in the direction of the emf (from – to +), the change in the electric potential is +ε • If a source of emf is transversed in the direction opposite of the emf (from + to -), the change in the electric potential is – ε Equations from Kirchhoff’s Rules • Use the junction rule as often as needed, so long as, each time you write an equation, you include in it a current that has not been used in a previous junction rule equation • The number of times the junction rule can be used is one fewer than the number of junction points in the circuit • The loop rule can be used as often as needed so long as a new circuit element (resistor or battery) or a new current appears in each new equation • You need as many independent equations as you have unknowns Equations from Kirchhoff’s Rules Problem-Solving Strategy • Draw the circuit diagram and assign labels and symbols to all known and unknown quantities • Assign directions to the currents • Apply the junction rule to any junction in the circuit • Apply the loop rule to as many loops as are needed to solve for the unknowns • Solve the equations simultaneously for the unknown quantities • Check your answers Chapter 25 Problem 72 The figure shows a portion of a circuit used to model muscle cells and neurons. All resistors have the same value R = 1.5 MΩ, and the emfs are ε1 = 75 mV, ε2 = 45 mV and ε3 = 20 mV. Find the current through ε3, including its direction. RC Circuits • If a direct current circuit contains capacitors and resistors, the current will vary with time • At the instant the switch is closed, the charge on the capacitor is zero • When the circuit is completed, the capacitor starts to charge, and the potential difference across the capacitor increases, until the charge reaches its maximum (Q = Cε) Charging Capacitor in an RC Circuit • Once the capacitor is fully charged (maximum charge is reached), the current in the circuit is zero, and the potential difference across the capacitor matches that supplied by the battery q IR 0 C q I R RC dq q C q dt R RC RC Charging Capacitor in an RC Circuit dq C q dt RC dq dt Q q RC dq dt Q q RC t ln q Q Const ' q Q e RC t q(t ) Q e RC Const 0 RC q(0) Q e Q Const 0 q(t ) Q(1 e Const Const Q t RC ) t RC Const Charging Capacitor in an RC Circuit • The charge on the capacitor varies with time q = Q (1 – e -t/RC ) • The time constant, = RC, represents the time required for the charge to increase from zero to 63.2% of its maximum • In a circuit with a large (small) time constant, the capacitor charges very slowly (quickly) • After t = 10 , the capacitor is over 99.99% charged Discharging Capacitor in an RC Circuit • When a charged capacitor is placed in the circuit, it can be discharged q q dq q IR 0 I C RC dt RC t dq dt RC q ( t ) e Const q RC t q (0) Const Q q(t ) Qe RC d I (t ) Qe dt t RC Q e RC t RC Discharging Capacitor in an RC Circuit • The charge on the capacitor varies with time q = Qe -t/RC • The charge decreases exponentially • At t = = RC, the charge decreases to 0.368 Qmax; i.e., in one time constant, the capacitor loses 63.2% of its initial charge Chapter 25 Problem 34 An uncharged 10-µF capacitor and a 470-kΩ resistor are in series, and 250 V is applied across the combination. How long does it take the capacitor voltage to reach 200 V? Meters in a Circuit – Ammeter, Voltmeter • An ammeter is used to measure current in line with the bulb – all the charge passing through the bulb also must pass through the meter • A voltmeter is used to measure voltage (potential difference) – connects to the two ends of the bulb Electrical Safety • Electric shock can result in fatal burns • Electric shock can cause the muscles of vital organs (such as the heart) to malfunction • The degree of damage depends on – the magnitude of the current – the length of time it acts – the part of the body through which it passes Effects of Various Currents • 5 mA or less – Can cause a sensation of shock – Generally little or no damage • 10 mA – Hand muscles contract – May be unable to let go a of live wire • 100 mA – If passes through the body for just a few seconds, can be fatal Answers to Even Numbered Problems Chapter 25: Problem 18 200 kJ Answers to Even Numbered Problems Chapter 25: Problem 26 2A Answers to Even Numbered Problems Chapter 25: Problem 30 1 kW Answers to Even Numbered Problems Chapter 25: Problem 58 7.5 mJ