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Chapter 6 Review

Key Terms

ammeter
instrument that measures current



electromotive force (emf)
energy produced per unit charge, drawn from a source that produces an electrical current



equivalent resistance
resistance of a combination of resistors; it can be thought of as the resistance of a single resistor that can replace a combination of resistors in a series and/or parallel circuit



internal resistance
amount of resistance to the flow of current within the voltage source



junction rule
sum of all currents entering a junction must equal the sum of all currents leaving the junction



Kirchhoff’s rules
set of two rules governing current and changes in potential in an electric circuit



loop rule
algebraic sum of changes in potential around any closed circuit path (loop) must be zero



potential difference
difference in electric potential between two points in an electric circuit, measured in volts



potential drop
loss of electric potential energy as a current travels across a resistor, wire, or other component



RC circuit
circuit that contains both a resistor and a capacitor



shock hazard
hazard in which an electric current passes through a person



terminal voltage
potential difference measured across the terminals of a source when there is no load attached



thermal hazard
hazard in which an excessive electric current causes undesired thermal effects



three-wire system
wiring system used at present for safety reasons, with live, neutral, and ground wires



voltmeter
instrument that measures voltage

Key Equations

Terminal voltage of a single voltage source
Equivalent resistance of a series circuit
Equivalent resistance of a parallel circuit
Junction rule
Loop rule
Terminal voltage of N voltage sources in series
Terminal voltage of N voltage sources in parallel
Charge on a charging capacitor
Time constant
Current during charging of a capacitor
Charge on a discharging capacitor
Current during discharging of a capacitor

Summary

6.1 Electromotive Force

  • All voltage sources have two fundamental parts: a source of electrical energy that has a characteristic electromotive force (emf), and an internal resistance . The emf is the work done per charge to keep the potential difference of a source constant. The emf is equal to the potential difference across the terminals when no current is flowing. The internal resistance of a voltage source affects the output voltage when a current flows.
  • The voltage output of a device is called its terminal voltage and is given by where is the electric current and is positive when flowing away from the positive terminal of the voltage source and is the internal resistance.

6.2 Resistors in Series and Parallel

  • The equivalent resistance of an electrical circuit with resistors wired in a series is the sum of the individual resistances: .
  • Each resistor in a series circuit has the same amount of current flowing through it.
  • The potential drop, or power dissipation, across each individual resistor in a series is different, and their combined total is the power source input.
  • The equivalent resistance of an electrical circuit with resistors wired in parallel is less than the lowest resistance of any of the components and can be determined using the formula .
  • Each resistor in a parallel circuit has the same full voltage of the source applied to it.
  • The current flowing through each resistor in a parallel circuit is different, depending on the resistance.
  • If a more complex connection of resistors is a combination of series and parallel, it can be reduced to a single equivalent resistance by identifying its various parts as series or parallel, reducing each to its equivalent, and continuing until a single resistance is eventually reached.

6.3 Kirchhoff’s Rules

  • Kirchhoff’s rules can be used to analyze any circuit, simple or complex. The simpler series and parallel connection rules are special cases of Kirchhoff’s rules.
  • Kirchhoff’s first rule, also known as the junction rule, applies to the charge to a junction. Current is the flow of charge; thus, whatever charge flows into the junction must flow out.
  • Kirchhoff’s second rule, also known as the loop rule, states that the voltage drop around a loop is zero.
  • When calculating potential and current using Kirchhoff’s rules, a set of conventions must be followed for determining the correct signs of various terms.
  • When multiple voltage sources are in series, their internal resistances add together and their emfs add together to get the total values.
  • When multiple voltage sources are in parallel, their internal resistances combine to an equivalent resistance that is less than the individual resistance and provides a higher current than a single cell.
  • Solar cells can be wired in series or parallel to provide increased voltage or current, respectively.

6.4 Electrical Measuring Instruments

  • Voltmeters measure voltage, and ammeters measure current. Analog meters are based on the combination of a resistor and a galvanometer, a device that gives an analog reading of current or voltage. Digital meters are based on analog-to-digital converters and provide a discrete or digital measurement of the current or voltage.
  • A voltmeter is placed in parallel with the voltage source to receive full voltage and must have a large resistance to limit its effect on the circuit.
  • An ammeter is placed in series to get the full current flowing through a branch and must have a small resistance to limit its effect on the circuit.
  • Standard voltmeters and ammeters alter the circuit they are connected to and are thus limited in accuracy.
  • Ohmmeters are used to measure resistance. The component in which the resistance is to be measured should be isolated (removed) from the circuit.

6.5 RC Circuits

  • An circuit is one that has both a resistor and a capacitor.
  • The time constant for an circuit is
  • When an initially uncharged ( at ) capacitor in series with a resistor is charged by a dc voltage source, the capacitor asymptotically approaches the maximum charge.
  • As the charge on the capacitor increases, the current exponentially decreases from the initial current:
  • If a capacitor with an initial charge is discharged through a resistor starting at then its charge decreases exponentially. The current flows in the opposite direction, compared to when it charges, and the magnitude of the charge decreases with time.

6.6 Household Wiring and Electrical Safety

  • The two types of electric hazards are thermal (excessive power) and shock (current through a person). Electrical safety systems and devices are employed to prevent thermal and shock hazards.
  • Shock severity is determined by current, path, duration, and ac frequency.
  • Circuit breakers and fuses interrupt excessive currents to prevent thermal hazards.
  • The three-wire system guards against thermal and shock hazards, utilizing live/hot, neutral, and ground wires, and grounding the neutral wire and case of the appliance.
  • A ground fault circuit interrupter (GFCI) prevents shock by detecting the loss of current to unintentional paths.

Answers to Check Your Understanding

6.1 If a wire is connected across the terminals, the load resistance is close to zero, or at least considerably less than the internal resistance of the battery. Since the internal resistance is small, the current through the circuit will be large,

The large current causes a high power to be dissipated by the internal resistance (

). The power is dissipated as heat.



6.2 The equivalent resistance of nine bulbs connected in series is

The current is

If one bulb burns out, the equivalent resistance is

and the voltage does not change, but the current increases (

). As more bulbs burn out, the current becomes even higher. Eventually, the current becomes too high, burning out the shunt.



6.3 The equivalent of the series circuit would be

which is higher than the equivalent resistance of the parallel circuit

The equivalent resistor of any number of resistors is always higher than the equivalent resistance of the same resistors connected in parallel. The current through for the series circuit would be

which is lower than the sum of the currents through each resistor in the parallel circuit,

This is not surprising since the equivalent resistance of the series circuit is higher. The current through a series connection of any number of resistors will always be lower than the current into a parallel connection of the same resistors, since the equivalent resistance of the series circuit will be higher than the parallel circuit. The power dissipated by the resistors in series would be

which is lower than the power dissipated in the parallel circuit



6.4 A river, flowing horizontally at a constant rate, splits in two and flows over two waterfalls. The water molecules are analogous to the electrons in the parallel circuits. The number of water molecules that flow in the river and falls must be equal to the number of molecules that flow over each waterfall, just like sum of the current through each resistor must be equal to the current flowing into the parallel circuit. The water molecules in the river have energy due to their motion and height. The potential energy of the water molecules in the river is constant due to their equal heights. This is analogous to the constant change in voltage across a parallel circuit. Voltage is the potential energy across each resistor.



The analogy quickly breaks down when considering the energy. In the waterfall, the potential energy is converted into kinetic energy of the water molecules. In the case of electrons flowing through a resistor, the potential drop is converted into heat and light, not into the kinetic energy of the electrons.



6.5 1. All the overhead lighting circuits are in parallel and connected to the main supply line, so when one bulb burns out, all the overhead lighting does not go dark. Each overhead light will have at least one switch in series with the light, so you can turn it on and off. 2. A refrigerator has a compressor and a light that goes on when the door opens. There is usually only one cord for the refrigerator to plug into the wall. The circuit containing the compressor and the circuit containing the lighting circuit are in parallel, but there is a switch in series with the light. A thermostat controls a switch that is in series with the compressor to control the temperature of the refrigerator.



6.6 The circuit can be analyzed using Kirchhoff’s loop rule. The first voltage source supplies power:

The second voltage source consumes power:



6.7 The current calculated would be equal to

instead of

The sum of the power dissipated and the power consumed would still equal the power supplied.



6.8 Since digital meters require less current than analog meters, they alter the circuit less than analog meters. Their resistance as a voltmeter can be far greater than an analog meter, and their resistance as an ammeter can be far less than an analog meter. Consult Figure 6.4.3 and Figure 6.4.2 and their discussion in the text.

Conceptual Questions

6.1 Electromotive Force

1. What effect will the internal resistance of a rechargeable battery have on the energy being used to recharge the battery?



2. A battery with an internal resistance of

and an emf of

is connected to a load resistor

As the battery ages, the internal resistance triples. How much is the current through the load resistor reduced?



3. Show that the power dissipated by the load resistor is maximum when the resistance of the load resistor is equal to the internal resistance of the battery.



6.2 Resistors in Series and Parallel

4. A voltage occurs across an open switch. What is the power dissipated by the open switch?



5. The severity of a shock depends on the magnitude of the current through your body. Would you prefer to be in series or in parallel with a resistance, such as the heating element of a toaster, if you were shocked by it? Explain.



6. Suppose you are doing a physics lab that asks you to put a resistor into a circuit, but all the resistors supplied have a larger resistance than the requested value. How would you connect the available resistances to attempt to get the smaller value asked for?



7. Some light bulbs have three power settings (not including zero), obtained from multiple filaments that are individually switched and wired in parallel. What is the minimum number of filaments needed for three power settings?



6.3 Kirchhoff’s Rules

8. Can all of the currents going into the junction shown below be positive? Explain.

9. Consider the circuit shown below. Does the analysis of the circuit require Kirchhoff’s method, or can it be redrawn to simplify the circuit? If it is a circuit of series and parallel connections, what is the equivalent resistance?

10. Do batteries in a circuit always supply power to a circuit, or can they absorb power in a circuit? Give an example.



11. What are the advantages and disadvantages of connecting batteries in series? In parallel?



12. Semi-tractor trucks use four large

batteries. The starter system requires

while normal operation of the truck’s other electrical components utilizes

How could the four batteries be connected to produce

? To produce

? Why is

better than

for starting the truck’s engine (a very heavy load)?



6.4 Electrical Measuring Instruments

13. What would happen if you placed a voltmeter in series with a component to be tested?



14. What is the basic operation of an ohmmeter as it measures a resistor?



15. Why should you not connect an ammeter directly across a voltage source as shown below?

6.5 RC Circuits

16. A battery, switch, capacitor, and lamp are connected in series. Describe what happens to the lamp when the switch is closed.



17. When making an ECG measurement, it is important to measure voltage variations over small time intervals. The time is limited by the

constant of the circuit—it is not possible to measure time variations shorter than

How would you manipulate

and

in the circuit to allow the necessary measurements?



6.6 Household Wiring and Electrical Safety

18. Why isn’t a short circuit necessarily a shock hazard?



19. We are often advised to not flick electric switches with wet hands, dry your hand first. We are also advised to never throw water on an electric fire. Why?

Problems

6.1 Electromotive Force

20. A car battery with a

emf and an internal resistance of

is being charged with a current of

Note that in this process, the battery is being charged. (a) What is the potential difference across its terminals? (b) At what rate is thermal energy being dissipated in the battery? (c) At what rate is electric energy being converted into chemical energy?



21. The label on a battery-powered radio recommends the use of rechargeable nickel-cadmium cells (nicads), although they have a

emf, whereas alkaline cells have a

emf. The radio has a

resistance. (a) Draw a circuit diagram of the radio and its batteries. Now, calculate the power delivered to the radio (b) when using nicad cells, each having an internal resistance of

and (c) when using alkaline cells, each having an internal resistance of

(d) Does this difference seem significant, considering that the radio’s effective resistance is lowered when its volume is turned up?



22. An automobile starter motor has an equivalent resistance of

and is supplied by a

battery with a

internal resistance. (a) What is the current to the motor? (b) What voltage is applied to it? (c) What power is supplied to the motor? (d) Repeat these calculations for when the battery connections are corroded and add

to the circuit. (Significant problems are caused by even small amounts of unwanted resistance in low-voltage, high-current applications.)



23. (a) What is the internal resistance of a voltage source if its terminal potential drops by

when the current supplied increases by

? (b) Can the emf of the voltage source be found with the information supplied?



24. A person with body resistance between his hands of

accidentally grasps the terminals of a

power supply. (Do NOT do this!) (a) Draw a circuit diagram to represent the situation. (b) If the internal resistance of the power supply is

what is the current through his body? (c) What is the power dissipated in his body? (d) If the power supply is to be made safe by increasing its internal resistance, what should the internal resistance be for the maximum current in this situation to be

or less? (e) Will this modification compromise the effectiveness of the power supply for driving low-resistance devices? Explain your reasoning.



25. A

emf automobile battery has a terminal voltage of

when being charged by a current of

(a) What is the battery’s internal resistance? (b) What power is dissipated inside the battery? (c) At what rate (in

) will its temperature increase if its mass is

and it has a specific heat of

assuming no heat escapes?



6.2 Resistors in Series and Parallel

26. (a) What is the resistance of a

a

and a

resistor connected in series? (b) In parallel?



27. What are the largest and smallest resistances you can obtain by connecting a

a

and a

resistor together?



28. An

toaster, a

speaker, and a

lamp are plugged into the same outlet in a

fuse and

circuit. (The three devices are in parallel when plugged into the same socket.) (a) What current is drawn by each device? (b) Will this combination blow the

fuse?



29. Your car’s

headlight and

starter are ordinarily connected in parallel in a

system. What power would one headlight and the starter consume if connected in series to a

battery? (Neglect any other resistance in the circuit and any change in resistance in the two devices.)



30. (a) Given a

battery and

and

resistors, find the current and power for each when connected in series. (b) Repeat when the resistances are in parallel.



31. Referring to the example combining series and parallel circuits and Figure 6.2.6, calculate

in the following two different ways: (a) from the known values of

and

; (b) using Ohm’s law for

In both parts, explicitly show how you follow the steps in the Figure 6.2.7.



32. Referring to Figure 6.2.6, (a) Calculate

and note how it compares with

found in the first two example problems in this module. (b) Find the total power supplied by the source and compare it with the sum of the powers dissipated by the resistors.



33. Refer to Figure 6.2.7 and the discussion of lights dimming when a heavy appliance comes on. (a) Given the voltage source is

the wire resistance is