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	$V_{\mathrm{terminal}}=\mathcal{E}-Ir_{\mathrm{eq}}$
Equivalent resistance of a series circuit	$R_{\mathrm{eq}}=R_1+R_2+R_3+\ldots+R_{N-1}+R_N=\sum_{i=1}^NR_i$
Equivalent resistance of a parallel circuit	$R_{\mathrm{eq}}=\left(\frac{1}{R_1}+\frac{1}{R_2}+\frac{1}{R_3}+\ldots+\frac{1}{R_{N-1}}+\frac{1}{R_N}\right)^{-1}=\left(\sum_{i=1}^NR_i\right)^{-1}$
Junction rule	$\sum I_\mathrm{in}=\sum I_{\mathrm{out}}$
Loop rule	$\sum V=0$
Terminal voltage of N voltage sources in series	$V_{\mathrm{terminal}}=\sum_{i=1}^N\mathcal{E}_i-I\sum_{i=1}^NR_i=\sum_{i=1}^N\mathcal{E}_i-IR_{\mathrm{eq}}$
Terminal voltage of N voltage sources in parallel	$V_{\mathrm{terminal}}=\mathcal{E}-I\sum_{i=1}^N\left(\frac{1}{R_i}\right)^{-1}=\mathcal{E}-IR_{\mathrm{eq}}$
Charge on a charging capacitor	$q(t)=C\mathcal{E}\left(1-e^{-\frac{t}{RC}}\right)=Q\left(1-e^{-\frac{t}{\tau}}\right)$
Time constant	$\tau=RC$
Current during charging of a capacitor	$I=\frac{\mathcal{E}}{R}e^{-\frac{t}{RC}}=I_0e^{-\frac{t}{\tau}}$
Charge on a discharging capacitor	$q(t)=Qe^{-\frac{t}{\tau}}$
Current during discharging of a capacitor	$I(t)=-\frac{Q}{RC}e^{-\frac{t}{\tau}}$

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 $r$ . 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 $r$ of a voltage source affects the output voltage when a current flows.
The voltage output of a device is called its terminal voltage $V_{\mathrm{terminal}}$ and is given by $V_{\mathrm{terminal}}=\mathcal{E}-Ir,$ where $I$ is the electric current and is positive when flowing away from the positive terminal of the voltage source and $r$ 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: $R_{\mathrm{eq}}=R_1+R_2+R_3+\ldots+R_{N-1}+R_N=\sum_{i=1}^NR_i$ .
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 $R_{\mathrm{eq}}=\left(\frac{1}{R_1}+\frac{1}{R_2}+\frac{1}{R_3}+\ldots+\frac{1}{R_{N-1}}+\frac{1}{R_N}\right)^{-1}=\left(\sum_{i=1}^NR_i\right)^{-1}$ .
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 $RC$ circuit is one that has both a resistor and a capacitor.
The time constant $\tau$ for an $RC$ circuit is $\tau=RC.$
When an initially uncharged ( $q=0$ at $t=0$ ) 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: $I_0=\mathcal{E}/R.$
If a capacitor with an initial charge $Q$ is discharged through a resistor starting at $t=0,$ 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.