Chapter 7 DC Circuits 7.1 Introduction. 7-3 Example 7.1.1:Junctions,branches and loops 7-4 7.2 Electromotive Force..... 7-5 7.3 Electrical Energy and Power........ 7-9 7.4 Resistors in Series and in Parallel. .7-10 7.5 Kirchhoff's Circuit Rules................ .7-12 7.6 Voltage-Current Measurements..... .7-14 7.7 Capacitors in Electric Circuits .7-15 7.7.1 Parallel Connection ..7-16 7.7.2 Series Connection.......... .7-17 Example 7.7.1:Equivalent Capacitance... .7-19 Example 7.7.2:Capacitance with Dielectrics 7-19 7.8 RC Circuit… 7-20 7.8.1 Charging a Capacitor. .7-20 7.8.2 Discharging a Capacitor.7-24 7.9 Summary. 。。 7-26 7.10 Problem-Solving Strategy:Applying Kirchhoffs Rules..................7.-27 7.11 Solved Problems… 7-31 7.11.1 Equivalent Resistance................. .7-31 7.11.2 Variable Resistance................ .7-32 7.11.3 RCCircuit.. .7-33 7.11.4 Parallel vs.Series Connections .7-34 7.11.5 Resistor Network........ 7-36 7.11.6 Equivalent Capacitance 7-37 7.12 Conceptual Questions 7-38 7.13 Additional Problems.... 7-39 7-1
7-1 Chapter 7 DC Circuits 7.1 Introduction.......................................................................................................... 7-3 Example 7.1.1: Junctions, branches and loops ........................................................ 7-4 7.2 Electromotive Force............................................................................................. 7-5 7.3 Electrical Energy and Power................................................................................ 7-9 7.4 Resistors in Series and in Parallel...................................................................... 7-10 7.5 Kirchhoff’s Circuit Rules................................................................................... 7-12 7.6 Voltage-Current Measurements......................................................................... 7-14 7.7 Capacitors in Electric Circuits........................................................................... 7-15 7.7.1 Parallel Connection .................................................................................... 7-16 7.7.2 Series Connection....................................................................................... 7-17 Example 7.7.1: Equivalent Capacitance ................................................................ 7-19 Example 7.7.2: Capacitance with Dielectrics........................................................ 7-19 7.8 RC Circuit.......................................................................................................... 7-20 7.8.1 Charging a Capacitor.................................................................................. 7-20 7.8.2 Discharging a Capacitor............................................................................. 7-24 7.9 Summary............................................................................................................ 7-26 7.10 Problem-Solving Strategy: Applying Kirchhoff’s Rules................................... 7-27 7.11 Solved Problems................................................................................................ 7-31 7.11.1 Equivalent Resistance................................................................................. 7-31 7.11.2 Variable Resistance .................................................................................... 7-32 7.11.3 RC Circuit................................................................................................... 7-33 7.11.4 Parallel vs. Series Connections .................................................................. 7-34 7.11.5 Resistor Network........................................................................................ 7-36 7.11.6 Equivalent Capacitance .............................................................................. 7-37 7.12 Conceptual Questions ........................................................................................ 7-38 7.13 Additional Problems.......................................................................................... 7-39
7.13.1 Resistive Circuits. 7-39 7.13.2 Multi-loop Circuit .7-39 7.13.3 Power Delivered to the Resistors. 7-40 7.13.4 Resistor Network........... 7-40 7.13.5 RC Circuit.............. .7-40 7.13.6 Resistors in Series and Parallel..... 7-41 7.13.7 Capacitors in Series and in Parallel 7-41 7.13.8 Power Loss and Ohm's Law .................. .7-42 7.13.9 Power,Current,and Potential difference .7-42 7-2
7-2 7.13.1 Resistive Circuits........................................................................................ 7-39 7.13.2 Multi-loop Circuit....................................................................................... 7-39 7.13.3 Power Delivered to the Resistors ............................................................... 7-40 7.13.4 Resistor Network........................................................................................ 7-40 7.13.5 RC Circuit................................................................................................... 7-40 7.13.6 Resistors in Series and Parallel .................................................................. 7-41 7.13.7 Capacitors in Series and in Parallel............................................................ 7-41 7.13.8 Power Loss and Ohm’s Law ...................................................................... 7-42 7.13.9 Power, Current, and Potential difference ................................................... 7-42
Chapter 7 Direct-Current Circuits 7.1 Introduction Electrical circuits connect power supplies to loads such as resistors,capacitors,motors, heaters,or lamps.The connection between the supply and the load is made by soldering with wires that are often called leads,or with many kinds of connectors and terminals. Energy is delivered from the source to the user on demand at the flick of a switch Sometimes many circuit elements are connected to the same lead,which is the called a common lead for those elements.Various parts of the circuits are called circuit elements, which can be in series or in parallel,as we have already seen in the case of capacitors. A node is a point in a circuit where three or more elements are soldered together.A branch is a current path between two nodes.Each branch in a circuit can have only one current in it although a branch may have no current.A loop is a closed path that may consist of different branches with different currents in each branch. A direct current (DC)circuit is a circuit is which the current through each branch in the circuit is always in the same direction.When the power supply is steady in time,and then the circuit is a purely resistive network then the current in each branch will be steady,that is the currents will not vary in time.In later chapters,when we introduce inductors into circuits with capacitance,transient power supplies initiate free oscillating currents. Finally when the power supply itself oscillates in time,then an alternating current(AC) is set up in the circuit. bulb 2 R2 junction A bulb 1 Junction B Figure 7.1.1 Elements connected (a)in parallel,and (b)in series. 7-3
7-3 Chapter 7 Direct-Current Circuits 7.1 Introduction Electrical circuits connect power supplies to loads such as resistors, capacitors, motors, heaters, or lamps. The connection between the supply and the load is made by soldering with wires that are often called leads, or with many kinds of connectors and terminals. Energy is delivered from the source to the user on demand at the flick of a switch. Sometimes many circuit elements are connected to the same lead, which is the called a common lead for those elements. Various parts of the circuits are called circuit elements, which can be in series or in parallel, as we have already seen in the case of capacitors. A node is a point in a circuit where three or more elements are soldered together. A branch is a current path between two nodes. Each branch in a circuit can have only one current in it although a branch may have no current. A loop is a closed path that may consist of different branches with different currents in each branch. A direct current (DC) circuit is a circuit is which the current through each branch in the circuit is always in the same direction. When the power supply is steady in time, and then the circuit is a purely resistive network then the current in each branch will be steady, that is the currents will not vary in time. In later chapters, when we introduce inductors into circuits with capacitance, transient power supplies initiate free oscillating currents. Finally when the power supply itself oscillates in time, then an alternating current (AC) is set up in the circuit. Figure 7.1.1 Elements connected (a) in parallel, and (b) in series
Example 7.1.1:Junctions,branches and loops In the circuit shown in Figure 7.1.1(a),there are two junctions,A and B,on either side of light bulb 1.There are three branches:branch 1 goes from A to B through the battery, branch 2 goes from A to B through light bulb 1,and branch 3 goes from A to B through light bulb 2.There are three closed loops.We shall describe the loops by arbitrarily starting at junction A.Loop 1 consists of branches 1 and 2;it starts at junction A,passes through the battery to junction B,and then from junction B back to junction A through light bulb 1.Loop 2 consists of branches 2 and 3;it starts at junction A,passes through light bulb I to junction B,then continues through light bulb 2 back to junction A.Loop 3 consists of branches 1 and 3;it stars at junction A,passes through the battery to junction B,then continues through bulb 2 back to junction A.The circuit shown in Figure 7.1.1(b) has no junctions,one branch and one closed loop. Elements are said to be in parallel when they are connected across the same potential difference.Both light bulbs in Figure 7.1.1(a)are connected across the battery.Generally, loads are connected in parallel across the power supply.On the other hand,when the elements are connected one after another in a branch,the same current passes through each element,and the elements are in series (see Figure 7.1.1b). There are pictorial diagrams that show wires and components roughly as they appear,and schematic diagrams that use conventional symbols,somewhat like road maps.Some frequently used symbols are shown in the Table below. Often there is a switch in series;when the switch is open the load is disconnected;when the switch is closed,the load is connected. Electromotive Source Seat of emf Resistor Switch One can have closed circuits,through which current flows,or open circuits in which there are no currents.Usually by accident,wires may touch,causing a short circuit.Most of the current flows through the short,very little will flow through the load.This may burn out a piece of electrical equipment such as a transformer.To prevent damage,a fuse or circuit breaker is put in series.When there is a short the fuse blows,or the breaker opens. 7-4
7-4 Example 7.1.1: Junctions, branches and loops In the circuit shown in Figure 7.1.1(a), there are two junctions, A and B, on either side of light bulb 1. There are three branches: branch 1 goes from A to B through the battery, branch 2 goes from A to B through light bulb 1, and branch 3 goes from A to B through light bulb 2. There are three closed loops. We shall describe the loops by arbitrarily starting at junction A. Loop 1 consists of branches 1 and 2; it starts at junction A, passes through the battery to junction B, and then from junction B back to junction A through light bulb 1. Loop 2 consists of branches 2 and 3; it starts at junction A, passes through light bulb 1 to junction B, then continues through light bulb 2 back to junction A. Loop 3 consists of branches 1 and 3; it stars at junction A, passes through the battery to junction B, then continues through bulb 2 back to junction A. The circuit shown in Figure 7.1.1(b) has no junctions, one branch and one closed loop. Elements are said to be in parallel when they are connected across the same potential difference. Both light bulbs in Figure 7.1.1(a) are connected across the battery. Generally, loads are connected in parallel across the power supply. On the other hand, when the elements are connected one after another in a branch, the same current passes through each element, and the elements are in series (see Figure 7.1.1b). There are pictorial diagrams that show wires and components roughly as they appear, and schematic diagrams that use conventional symbols, somewhat like road maps. Some frequently used symbols are shown in the Table below. Often there is a switch in series; when the switch is open the load is disconnected; when the switch is closed, the load is connected. Electromotive Source Seat of emf Resistor Switch One can have closed circuits, through which current flows, or open circuits in which there are no currents. Usually by accident, wires may touch, causing a short circuit. Most of the current flows through the short, very little will flow through the load. This may burn out a piece of electrical equipment such as a transformer. To prevent damage, a fuse or circuit breaker is put in series. When there is a short the fuse blows, or the breaker opens
In electrical circuits,a point(or some common lead)is chosen as the ground.This point is assigned an arbitrary voltage,usually zero,and the voltage Iat any point in the circuit is defined as the potential difference between that point and ground. 7.2 Electromotive Force Consider an electric circuit shown in Figure 7.2.1(a).To drive the current around the circuit,the battery undergoes a discharging process that converts chemical energy into electric energy that eventually gets dissipated as heat in the resistor. In the external circuit,the electrostatic field,E,is directed from the positive terminal of the battery to the negative terminal of the battery,exerting a force on the charges in the wire to produce a current from the positive to the negative terminal.The electrostatic field also insures that the current in the wire is uniform.Recall that the electrostatic field is a conservative vector field and so the line integral around the loop in Figure 7.2.1(b)is zero, ∫Es=0 (7.2.1) loop [E.ds=0 Figure 7.2.1(a)A simple circuit consisting of a battery and a resistor Figure 7.2.1(b)Integral of electrostatic field is zero around loop. Inside the battery,in the region close to the positive terminal,the electrostatic field points away from the positive terminal.In the region close to the negative terminal,the electrostatic field points towards the negative terminal.(In the region in between,the electrostatic field may point in either direction depending on the nature of the battery.) The current is directed from the negative to the positive terminals.Near both terminals, the electrostatic field points in the opposite direction of the current. In order to maintain the current,there must that be some force that transports charge carriers in the opposite direction in which the electrostatic field is trying to move them The origin of this source force,F,in batteries is a chemical force.In the regions near the terminal where chemical reactions are taking place,chemical forces move charge carriers in the opposite direction in which the electrostatic field is trying to move them.The work done per unit charge by this source force in moving a charge carrier with charge q from the negative to the positive terminal is given by the expression, 7-5
7-5 In electrical circuits, a point (or some common lead) is chosen as the ground. This point is assigned an arbitrary voltage, usually zero, and the voltage V at any point in the circuit is defined as the potential difference between that point and ground. 7.2 Electromotive Force Consider an electric circuit shown in Figure 7.2.1(a). To drive the current around the circuit, the battery undergoes a discharging process that converts chemical energy into electric energy that eventually gets dissipated as heat in the resistor. In the external circuit, the electrostatic field, E, is directed from the positive terminal of the battery to the negative terminal of the battery, exerting a force on the charges in the wire to produce a current from the positive to the negative terminal. The electrostatic field also insures that the current in the wire is uniform. Recall that the electrostatic field is a conservative vector field and so the line integral around the loop in Figure 7.2.1(b) is zero, E⋅d s = 0 loop ∫ (7.2.1) Figure 7.2.1(a) A simple circuit consisting of a battery and a resistor Figure 7.2.1(b) Integral of electrostatic field is zero around loop. Inside the battery, in the region close to the positive terminal, the electrostatic field points away from the positive terminal. In the region close to the negative terminal, the electrostatic field points towards the negative terminal. (In the region in between, the electrostatic field may point in either direction depending on the nature of the battery.) The current is directed from the negative to the positive terminals. Near both terminals, the electrostatic field points in the opposite direction of the current. In order to maintain the current, there must that be some force that transports charge carriers in the opposite direction in which the electrostatic field is trying to move them. The origin of this source force, Fs , in batteries is a chemical force. In the regions near the terminal where chemical reactions are taking place, chemical forces move charge carriers in the opposite direction in which the electrostatic field is trying to move them. The work done per unit charge by this source force in moving a charge carrier with charge q from the negative to the positive terminal is given by the expression
了s=了is (7.2.2) neg g neg where f is the source force per unit charge.Outside the battery f=0,so we extend the path in Eq.(7.2.2)to the entire loop.In that case,the work done per unit charge by the non-electrostatic force,F,around a closed path is commonly referred to as the electromotive force,or emf (symbol s). i-j语6-js (7.2.3) path This is a poor choice of name because it is not a force but work done per unit charge.The SI unit for emf is the volt (V). Inside our ideal battery without any internal resistance,the sum of the electrostatic force and the source force on the charge is zero, qE+qf=0 (7.2.4) Therefore the electrostatic field is equal in magnitude to the source force per unit charge but opposite in direction, E=-f. (7.2.5) The electric potential difference between the terminals is defined in terms of the electrostatic field r(+)-(-)=-∫Es=∫is=e (7.2.6) The potential difference Al between the positive and the negative terminals of the battery is called the terminal voltage,and in this case is equal to the emf. Electromotive force is not restricted to chemical forces.In Figure 7.2.2,the inner working of a Van de Graaff generator are displayed.An electric motor drives a non- conducting belt that transports charge carriers in a direction opposite the electric field. The positive charge carriers are moved from lower to higher potential,and negative charge carriers are moved from higher to lower potential.Strong local fields at the brushes of the terminals both add and remove charge carriers from the belt.An electric motor provides the energy to move the belt and hence is the source of the electromotive force. 7-6
7-6 Fs q ⋅ d s neg pos ∫ = fs ⋅ d s neg pos ∫ , (7.2.2) where fs is the source force per unit charge. Outside the battery fs = 0 , so we extend the path in Eq. (7.2.2) to the entire loop. In that case, the work done per unit charge by the non-electrostatic force, Fs , around a closed path is commonly referred to as the electromotive force, or emf (symbol ε ). ε ≡ fs ⋅ d s closed path ∫ = Fs q ⋅ d s − + ∫ = fs ⋅ d s − + ∫ . (7.2.3) This is a poor choice of name because it is not a force but work done per unit charge. The SI unit for emf is the volt (V). Inside our ideal battery without any internal resistance, the sum of the electrostatic force and the source force on the charge is zero, q E + q fs = 0 . (7.2.4) Therefore the electrostatic field is equal in magnitude to the source force per unit charge but opposite in direction, E = − fs . (7.2.5) The electric potential difference between the terminals is defined in terms of the electrostatic field V (+) −V (−) = − E⋅ d s − + ∫ = fs ⋅ d s − + ∫ = ε . (7.2.6) The potential difference ΔV between the positive and the negative terminals of the battery is called the terminal voltage, and in this case is equal to the emf . Electromotive force is not restricted to chemical forces. In Figure 7.2.2, the inner working of a Van de Graaff generator are displayed. An electric motor drives a nonconducting belt that transports charge carriers in a direction opposite the electric field. The positive charge carriers are moved from lower to higher potential, and negative charge carriers are moved from higher to lower potential. Strong local fields at the brushes of the terminals both add and remove charge carriers from the belt. An electric motor provides the energy to move the belt and hence is the source of the electromotive force
V=V V=0 Figure 7.2.2 Van de Graaff generator Solar cells and thermocouples are also examples of emf source.They can also be thought of as a"charge pump"that moves charges from lower potential to higher potential. Consider a simple circuit consisting of a battery as the emf source and a resistor of resistance R,as shown in Figure 7.2.3 D R 0=-∫Es=0=-jEs-jEs leg 2 leg 1 Figure 7.2.3(a)Electric potential Figure 7.2.3(b)Circuit diagram. difference for leg 1 and leg 2 sum to zero. The circuit diagram in Figure 7.2.3(b)corresponds to the circuit in Figure 7.2.3(a).The electric potential difference around the loop is zero because the electrostatic field is conservative,Eq.(7.2.1).We can divide the loop into two legs;leg 1 goes from the positive terminal to the negative terminal through the external circuit,and leg 2 goes from the negative terminal to the positive terminal through the battery, 0=-∫Es=0=-jEs-jEs (7.2.7) oop The integral via leg 1 in the external circuit is just the potential difference across the resistor,which is given by Ohm's Law,where we have assumed that the wires have negligible resistance, 7-7
7-7 Figure 7.2.2 Van de Graaff generator Solar cells and thermocouples are also examples of emf source. They can also be thought of as a “charge pump” that moves charges from lower potential to higher potential. Consider a simple circuit consisting of a battery as the emf source and a resistor of resistance R, as shown in Figure 7.2.3. Figure 7.2.3(a) Electric potential difference for leg 1 and leg 2 sum to zero. Figure 7.2.3(b) Circuit diagram. The circuit diagram in Figure 7.2.3(b) corresponds to the circuit in Figure 7.2.3(a). The electric potential difference around the loop is zero because the electrostatic field is conservative, Eq. (7.2.1). We can divide the loop into two legs; leg 1 goes from the positive terminal to the negative terminal through the external circuit, and leg 2 goes from the negative terminal to the positive terminal through the battery, 0 = − E⋅ d s = 0 loop ∫ = − E⋅ d s 1 + − ∫ − E⋅ d s2 − + ∫ , (7.2.7) The integral via leg 1 in the external circuit is just the potential difference across the resistor, which is given by Ohm’s Law, where we have assumed that the wires have negligible resistance
△r=-∫E.d=-lIR (7.2.8) The integral via leg 2 through the battery is the emf (Eq.(7.2.6), An--jE.di=e. (7.2.9) Eq.(7.2.7)becomes 0=△V+△V,=-IR+e. (7.2.10) Therefore the current in the loop is given by I=e (7.2.11) R However,a real battery always carries an internal resistance r(Figure 7.2.4a),and the potential difference across the battery terminals becomes △V=e-Ir. (7.2.12) d Ir IR b Figure 7.2.4 (a)Circuit with an emf source having an internal resistance r and a resistor of resistance R.(b)Change in electric potential around the circuit. Because there is no net change in potential difference around a closed loop,we have 8-Ir-IR=0. (7.2.13) Therefore the current through the circuit is 7-8
7-8 ΔV1 = − E⋅ d s 1 + − ∫ = −IR . (7.2.8) The integral via leg 2 through the battery is the emf (Eq. (7.2.6), ΔV2 = − E⋅ d s − + ∫ = ε . (7.2.9) Eq. (7.2.7) becomes 0 = ΔV1 + ΔV2 = −IR + ε . (7.2.10) Therefore the current in the loop is given by I R ε = . (7.2.11) However, a real battery always carries an internal resistance r (Figure 7.2.4a), and the potential difference across the battery terminals becomes ΔV = ε − Ir . (7.2.12) Figure 7.2.4 (a) Circuit with an emf source having an internal resistance r and a resistor of resistance R. (b) Change in electric potential around the circuit. Because there is no net change in potential difference around a closed loop, we have ε − Ir − IR = 0 . (7.2.13) Therefore the current through the circuit is
I= (7.2.14) R+r Figure 7.2.4(b)depicts the change in electric potential as we traverse the circuit clockwise.From the figure,we see that the highest potential is immediately after the battery.The potential drops as each resistor is crossed.Note that the potential is essentially constant along the wires.This is because the wires have a negligibly small resistance compared to the resistors. 7.3 Electrical Energy and Power Consider a circuit consisting of an ideal battery (zero internal resistance)and a resistor with resistance R(Figure 7.3.1).The potential difference between two points a and b be g=V-V>0.If a charge Ag is moved through the battery,its electric potential energy is increased by AU=Age.On the other hand,as the charge moves across the resistor, the potential energy is decreased due to collisions with atoms in the resistor.If we neglect the internal resistance of the battery and the connecting wires,upon returning to a,the change in potential energy of Ag is zero. R Figure 7.3.1 A circuit consisting of an ideal battery with emf g and a resistor of resistance R. The rate of energy loss through the resistor is given by P= △U △ (7.3.1) △1△1 This is equal to the power supplied by the battery.Using &=IR in Eq.(7.3.1),one may rewrite the rate of energy loss through the resistor as P=IR (7.3.2) Using I=8/R in Eq.(7.3.1),the power delivered by the battery is P=82/R (7.3.3) 7-9
7-9 I R r ε = + . (7.2.14) Figure 7.2.4(b) depicts the change in electric potential as we traverse the circuit clockwise. From the figure, we see that the highest potential is immediately after the battery. The potential drops as each resistor is crossed. Note that the potential is essentially constant along the wires. This is because the wires have a negligibly small resistance compared to the resistors. 7.3 Electrical Energy and Power Consider a circuit consisting of an ideal battery (zero internal resistance) and a resistor with resistance R (Figure 7.3.1). The potential difference between two points a and b be ε =Vb −Va > 0 . If a charge Δq is moved through the battery, its electric potential energy is increased by ΔU = Δqε . On the other hand, as the charge moves across the resistor, the potential energy is decreased due to collisions with atoms in the resistor. If we neglect the internal resistance of the battery and the connecting wires, upon returning to a, the change in potential energy of Δq is zero. Figure 7.3.1 A circuit consisting of an ideal battery with emf ε and a resistor of resistance R. The rate of energy loss through the resistor is given by P = ΔU Δt = Δq Δt ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ε = Iε . (7.3.1) This is equal to the power supplied by the battery. Using ε = IR in Eq. (7.3.1), one may rewrite the rate of energy loss through the resistor as P = I 2 R . (7.3.2) Using I = ε / R in Eq. (7.3.1), the power delivered by the battery is P = ε2 / R. (7.3.3)
a Figure 7.3.2 A circuit consisting of a battery with emf g and a resistor of resistance R. For a battery with emf g and internal resistance r(Figure 7.3.2),the power or the rate at which chemical energy is delivered to the circuit is P=I8=1(IR+Ir)=IR+Ir. (7.3.4) The power of the source emf is equal to the sum of the power dissipated in both the internal and load resistance as required by energy conservation. 7.4 Resistors in Series and in Parallel The two resistors with resistance R and R,in Figure 7.4.1 are connected in series to a source of emf g.By current conservation,the same current,I,is in each resistor. R W Rea Figure 7.4.1 (a)Resistors in series.(b)Equivalent circuit. The total voltage drop from a to c across both elements is the sum of the voltage drops across the individual resistors: e=△V=IR+IR=I(R+R) (7.4.1) 7-10
7-10 Figure 7.3.2 A circuit consisting of a battery with emf ε and a resistor of resistance R. For a battery with emf ε and internal resistance r (Figure 7.3.2), the power or the rate at which chemical energy is delivered to the circuit is 2 2 P = Iε = I(IR + Ir) = I R + I r . (7.3.4) The power of the source emf is equal to the sum of the power dissipated in both the internal and load resistance as required by energy conservation. 7.4 Resistors in Series and in Parallel The two resistors with resistance R1 and R2 in Figure 7.4.1 are connected in series to a source of emf ε . By current conservation, the same current, I , is in each resistor. Figure 7.4.1 (a) Resistors in series. (b) Equivalent circuit. The total voltage drop from a to c across both elements is the sum of the voltage drops across the individual resistors: ε = ΔV = I R1 + I R2 = I(R1 + R2 ) . (7.4.1)