Ch 20 Electric Circuits

Chapter 20

Electric Circuits

Learning ObjectivesElectric circuits Current, resistance, power

Students should understand the definition of electric current, so they can relate the magnitude and direction of the current to the rate of flow of positive and negative charge.

Students should understand conductivity, resistivity, and resistance, so they can: Relate current and voltage for a resistor. Describe how the resistance of a resistor depends upon

its length and cross-sectional area, and apply this result in comparing current flow in resistors of different material or different geometry.

Apply the relationships for the rate of heat production in a resistor.

Learning Objectives Steady-state direct current circuits with batteries and

resistors only Students should understand the behavior of series and

parallel combinations of resistors, so they can: Identify on a circuit diagram whether resistors are in

series or in parallel. Determine the ratio of the voltages across resistors

connected in series or the ratio of the currents through resistors connected in parallel.

Calculate the equivalent resistance of a network of resistors that can be broken down into series and parallel combinations.

Calculate the voltage, current, and power dissipation for any resistor in such a network of resistors connected to a single power supply.

Design a simple series-parallel circuit that produces a given current through and potential difference across one specified component, and draw a diagram for the circuit using conventional symbols.

Learning Objectives Steady-state direct current circuits with batteries and

resistors only Students should understand the properties of ideal and real

batteries, so they can: Calculate the terminal voltage of a battery of specified

emf and internal resistance from which a known current is flowing.

Students should be able to apply Ohm’s law and Kirchhoff’s rules to direct-current circuits, in order to: determine a single unknown current, voltage, or resistance.

Students should understand the properties of voltmeters and ammeters, so they can: State whether the resistance of each is high or low. Identify or show correct methods of connecting meters

into circuits in order to measure voltage or current.

Learning Objectives Capacitors in circuits

Students should understand the t = 0 and steady-state behavior of capacitors connected in series or in parallel, so they can: Calculate the equivalent capacitance of a series or

parallel combination. Describe how stored charge is divided between

capacitors connected in parallel. Determine the ratio of voltages for capacitors connected

in series. Calculate the voltage or stored charge, under steady-

state conditions, for a capacitor connected to a circuit consisting of a battery and resistors.

Table of Contents1. Electromotive Force and Current2. Ohm’s Law3. Resistance & Resistivity4. Electric Power5. Alternating Current (Not AP)6. Series Wiring7. Parallel Wiring8. Series and Parallel Wiring9. Internal Resistance10. Kirchoff’s Rules11. The Measurement of Current and Voltage12. Capacitors in Series and Parallel Circuits13. RC Circuits (Not AP)14. Safety and the Physiological Effects of Current (Not AP)

Chapter 20:Electric Circuits

Section 1:

Electromotive Force & Current

Electric Circuits

In an electric circuit, an energy source and an energy consuming device are connected by conducting wires through which electric charges move.

Electromotive Force Within a battery, a chemical reaction occurs that transfers

electrons from one terminal to another terminal. The maximum potential difference across the terminals is

called the electromotive force (emf).

The electric current is the amount of charge per unit time that passes through a surface that is perpendicular to the motion of the charges.

t

QIavg

One coulomb per second equals one ampere (A).

Electric Current

Types of Current

If the charges move around the circuit in the same direction at

all times, the current is said to be direct current (dc).

If the charges move first one way and then the opposite way,

the current is said to be alternating current (ac).

Example 1 A Pocket Calculator

The current in a 3.0 V battery of a pocket calculator is 0.17 mA. In one hourof operation, (a) how much charge flows in the circuit and (b) how much energydoes the battery deliver to the calculator circuit?

(a)

(b)

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V 0.3C 61.0

t

QIavg

s 3600A1017.0 3 C 61.0

QVU J 8.1

Direction of Current Conventional current is the hypothetical flow of positive

charges that would have the same effect in the circuit as the movement of negative charges that actually does occur.

20.1.1. In which one of the following situations does a conventional electric current flow due north?

a) Protons in a beam are moving due south.

b) A water molecule is moving due north.

c) Electrons in a beam are moving due south.

d) Electrons in a wire connected to a battery are moving from south to north.

20.1.2. The battery capacity of a lithium ion battery in a digital music player is 750 mA-h. The manufacturer claims that the player can operate for eight hours if the battery is initially fully charged. Given this information, determine the number of electrons that flow through the player as you listen to your favorite songs for three hours.

a) 6.2 1018 electrons

b) 1.0 103 electrons

c) 2.4 109 electrons

d) 6.3 1021 electrons

e) 8.1 1028 electrons

Chapter 20:Electric Circuits

Section 2:

Ohm’s Law

Resistance

The resistance (R) is defined as

the ratio of the voltage V applied

across a piece of material to the

current I through the material.

To the extent that a wire or an

electrical device offers resistance

to electrical flow, it is called a

resistor.

SI Unit of Resistance: volt/ampere (V/A) = ohm (Ω)

IV

Ohm’s Law The V is proportional to I,

where V is the voltage applied across a piece of material and I is the current through the material:

IRV

Example 2 A Flashlight

The filament in a light bulb is a resistor in the formof a thin piece of wire. The wire becomes hot enoughto emit light because of the current in it. The flashlightuses two 1.5-V batteries to provide a current of0.40 A in the filament. Determine the resistance ofthe glowing filament.

I

VR

IRV

A 0.40

V 0.3 5.7

20.2.1. In a certain circuit containing a battery and a resistor, Ohm’s law is obeyed.An instrument to measure the current in the circuit, an ammeter, is connected in between one of the terminals of the battery and one end of the resistor. The ammeter indicates that the current in the circuit is I. The battery is then removed and replaced with another battery. This time, the ammeter indicates the current is 2I. Which one of the following statements concerning the resistor is true?

a) When the second battery was placed in the circuit, the resistance increased to twice its initial value.

b) When the second battery was placed in the circuit, the resistance decreased to one half its initial value.

c) When the second battery was placed in the circuit, the resistance increased to four times its initial value.

d) When the second battery was placed in the circuit, the resistance increased to one fourth its initial value.

e) When the second battery was placed in the circuit, the resistance did not change.

20.2.2. Consider the circuit containing a battery and a resistor shown. For which one of the following combinations of current and voltage does R have the smallest value?

a) V = 9 V and I = 0.002 A

b) V = 12 V and I = 0.5 A

c) V = 1.5 V and I = 0.075 A

d) V = 6 V and I = 0.1 A

e) V = 4.5 V and I = 0.009 A

20.2.3. A certain circuit contains a battery and a resistor. An instrument to measure the current in the circuit, an ammeter, is connected in between one of the terminals of the battery and one end of the resistor. The graph shows the current in the circuit as the voltage is increased. Which one of the following statements best describes the resistor in this circuit?

a) The resistor does not obey Ohm’s law.

b) The resistor obeys Ohm’s law for voltages between zero and twenty-five volts.

c) The resistor obeys Ohm’s law for voltages between zero and thirty-five volts.

d) The resistor obeys Ohm’s law for voltages between zero and forty volts.

e) The resistor obeys Ohm’s law for voltages between thirty and forty volts.

Chapter 20:Electric Circuits

Section 3:

Resistance and Resitivity

A

LR

resistivity in units of ohm·meter

Resistance in Materials

For a wide range of materials, the resistance of a piece of material of length L and cross-sectional area A is:

Example 3 Longer Extension Cords

The instructions for an electric lawn mower suggest that a 20-gauge extension cord can be used for distances up to 35 m, but a thicker 16-gauge cord should be used for longer distances. The cross sectional area of a 20-gauge wire is 5.2x10-7Ω·m, while that of a 16-gauge wire is 13x10-7 Ω·m. Determine the resistance of (a) 35 m of 20-gauge copper wire and (b) 75 m of 16-gauge copper wire.

A

LR

(a)

(b) 27-

8

m1013

m 75m1072.1

27-

8

m105.2

m 35m1072.1

2.1

A

LR

99.0

oo TT 1

temperature coefficient of resistivity

oo TTRR 1

Temperature Effects

20.3.1. For which combination for the length L and radius R of a wire will the resistance have the smallest value?

a) L = 0.50 m and R = 0.03 m

b) L = 0.25 m and R = 0.08 m

c) L = 0.40 m and R = 0.2 m

d) L = 0.80 m and R = 0.1 m

e) L = 0.10 m and R = 0.05 m

20.3.2. The ends of a wire are connected to the terminals of a battery. For which of the following changes will the resulting current in the circuit have the largest value?

a) Replace the wire with one that has a larger resistivity.

b) Replace the wire with one that has a larger radius.

c) Replace the wire with one that has a longer length.

Chapter 20:Electric Circuits

Section 4:

Electric Power

t

UP

energy

power

time

Electric Power Suppose some charge emerges from a battery and the

potential difference between the battery terminals is V.

t

Vq

Vt

q

IV

IVP

SI Unit of Power: watt (W)

RIIRIP 2

R

VV

R

VP

2

Electric Power When there is current in a circuit as a result of a voltage, the

electric power delivered to the circuit is:

Example 5 The Power and Energy Used in aFlashlight

In the flashlight, the current is 0.40A and the voltageis 3.0 V. Find (a) the power delivered to the bulb and(b) the energy dissipated in the bulb in 5.5 minutesof operation.

(a)

(b)

IVP

PtE

V 0.3A 40.0 W2.1

s 330 W2.1 J100.4 2

20.4.1. An automatic coffee maker uses a resistive heating element to boil the 2.4 kg of water that was poured into it at 21 C. The current delivered to the coffee pot is 8.5 A when it is plugged into a 120 V electrical outlet. If the specific heat capacity of water is 4186 J/kgC, approximately how long does it take to boil all of the water?

a) 5 minutes

b) 8 minutes

c) 10 minutes

d) 13 minutes

e) 15 minutes

20.4.2. The insulated wiring in a house can safely carry a maximum current of 18 A. The electrical outlets in the house provide an alternating voltage of 120 V. A space heater when plugged into the outlet operates at an average power of 1500 W. How many space heaters can safely be plugged into a single electrical outlet and turned on for an extended period of time?

a) zero

b) one

c) two

d) three

e) four

20.4.3. A portable CD player was recently introduced that has a “special power saving technology.” The manufacturer claims that with only two standard AA batteries (together: 3.0 V, 20 kJ energy storage) that the player can be played for about 25 hours. What is the approximate resistance in the CD player’s electrical circuitry?

a) 41

b) 0.010

c) 300

d) 1.5

e) 15

20.4.4. A wire is used as a heating element that has a resistance that is fairly independent of its temperature within its operating range. When a current I is applied to the wire, the energy delivered by the heater each minute is E. For what amount of current will the energy delivered by the heater each minute be 4E?

a) 2I

b) 4I

c) 0.5I

d) 0.25I

e) 8I

Chapter 20:Electric Circuits

Section 5:

Alternating Current

In an AC circuit, the charge flow reverses direction periodically.

ftVV o 2sin

In circuits that contain only resistance, the current reverses direction each time the polarity of the generator reverses.

ftIftR

V

R

VI o

o 2sin2sin

peak current

ftVIIVP oo 2sin 2

ftII o 2sin ftVV o 2sin

rmsrms222

VIVIVI

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RIV rmsrms

rmsrmsIVP

RIP 2rms

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2rms

Example 6 Electrical Power Sent to a Loudspeaker

A stereo receiver applies a peak voltage of34 V to a speaker. The speaker behavesapproximately as if it had a resistance of 8.0 Ω.Determine (a) the rms voltage, (b) the rmscurrent, and (c) the average power for this circuit.

(a)

(b)

(c)

V 242

V 34

2rms oV

V

A 3.0 0.8

V 24rmsrms

R

VI

W72V 24A 3.0rmsrms VIP

Conceptual Example 7 Extension Cords and a Potential Fire Hazard

During the winter, many people use portable electric space heaters to keepwarm. Sometimes, however, the heater must be located far from a 120-V wall receptacle, so an extension cord must be used. However, manufacturers oftenwarn against using an extension cord. If one must be used, they recommenda certain wire gauge, or smaller. Why the warning, and why are smaller-gauge wires better then larger-gauge wires?

20.5.1. The graph shows the current as a function of time for an electrical device plugged into a outlet with an rms voltage of 120 V. What is the resistance of the device?

a) 24

b) 21

c) 17

d) 14

e) 12

20.5.2. Consider the circuits shown in parts A and B in the picture. In part A, a light bulb is plugged into a wall outlet that has an rms voltage of 120 volts. A current I passes through the circuit and the bulb turns on. In part B, a second, identical light bulb is connected in series in the circuit. How does the current in circuit B compare with that in circuit A?

a) The current is the same, I, as in part A.

b) The current is twice as much, 2I, as in part A.

c) The current in part B is zero amperes.

d) The current is one fourth as much, 0.25I, as in part A.

e) The current is one half as much, 0.5I, as in part A.

Chapter 20:Electric Circuits

Section 6:

Series Wiring

Series Wiring There are many circuits in which more than one device is

connected to a voltage source. Series wiring means that the devices are connected in

such a way that there is the same electric current through each device. (One Path)

21 VVV

321 RRRRSSeries resistors

Resistance in a series Circuit As we will discuss later, the sum of all voltage in a circuit must

equal zero. Voltage supplied by battery is lost by resistors

21 IRIR 21 RRI SIR

i

iS RR

Example 8 Resistors in a Series Circuit

A 6.00 Ω resistor and a 3.00 Ω resistor are connected in series with a 12.0 V battery. Assuming the battery contributes no resistance to the circuit, find (a) the current, (b) the power dissipated in each resistor, and (c) the total power delivered to the resistors by the battery.

(a)

(b)

(c)

00.9 00.3 00.6SRSR

VI

RIP 2

RIP 2

W31.5 W6.10 P

00.9

V 0.12A 33.1

00.6A 33.1 2 W6.10

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W9.15

20.6.1. Consider the circuit shown in the drawing. Two identical light bulbs, labeled A and B, are connected in series with a battery and are illuminated equally. There is a switch in the circuit that is initially open. Which one of the following statements concerning the two bulbs is true after the switch is closed?

a) Bulbs A and B will be off.

b) Bulbs A and B will be equally illuminated.

c) Bulb A will be brighter and bulb B will be off.

d) Bulb A will be off and bulb B will be brighter.

e) Both bulbs will be dimmer than before the switch was closed.

Chapter 20:Electric Circuits

Section 7:

Parallel Wiring

Parallel Wiring Parallel wiring means that the

devices are connected in such a way that the same voltage is applied across each device.

Multiple paths are present. When two resistors are connected in

parallel, each receives current from the battery as if the other was not present.

Therefore the two resistors connected in parallel draw more current than does either resistor alone.

Wiring in your home

parallel resistors…

321

1111

RRRRP

Parallel Wiring As we will discuss later, the total

current flowing into any point must equal the total current flowing out.

21 III 21 R

V

R

V

21

11

RRV

PRV

1

i iP RR

11

Simplifying Circuits

i iP RR

11

R1 = 5 ΩR2 = 3 Ω

21

11

RR

21

21

RR

RRR

21

121

RR

RR

R

35

35

89.1R0

5

RTotal

OR

Example 10 Main and Remote Stereo Speakers

Most receivers allow the user to connect to “remote” speakers in additionto the main speakers. At the instant represented in the picture, the voltageacross the speakers is 6.00 V. Determine (a) the equivalent resistanceof the two speakers, (b) the total current supplied by the receiver, (c) thecurrent in each speaker, and (d) the power dissipated in each speaker.

(a)

00.4

1

00.8

11

PR 67.2

(b)

PR

VI rms

rms

R

VI rms

rms (c)

R

VI rms

rms

(d)rmsrmsVIP

rmsrmsVIP

i iP RR

11

00.8

3

67.2

V 00.6A 25.2

00.8

V 00.6A 750.0 A

00.4

V 00.6

A 50.1

V 00.6A 750.0 W50.4

V 00.6A 50.1 W00.9

Conceptual Example 11 A Three-Way Light Bulband Parallel Wiring

Within the bulb there are two separate filaments.When one burns out, the bulb can produce onlyone level of illumination, but not the highest.

Are the filaments connected in series orparallel?

How can two filaments be used to produce threedifferent illumination levels?

20.7.1. Consider the three resistors and the battery in the circuit shown. Which resistors, if any, are connected in parallel?

a) R1 and R2

b) R1 and R3

c) R2 and R3

d) R1 and R2 and R3

e) No resistors are connected in parallel.

20.7.2. Consider the circuits shown in parts A and B in the picture. In part A, a light bulb is plugged into a wall outlet that has an rms voltage of 120 volts. A current I passes through the circuit and the bulb turns on. In part B, a second, identical light bulb is connected in parallel in the circuit. How does the total current in circuit B compare with that in circuit A?

a) The current is the same, I, as in part A.

b) The current is twice as much, 2I, as in part A.

c) The current in part B is zero amperes.

d) The current is one fourth as much, 0.25I, as in part A.

e) The current is one half as much, 0.5I, as in part A.

20.7.3. Two light bulbs, one “50 W” bulb and one “100 W” bulb, are connected in parallel with a standard 120 volt ac electrical outlet. The brightness of a light bulb is directly related to the power it dissipates. Therefore, the 100 W bulb appears brighter. How does the brightness of the two bulbs compare when these same bulbs are connected in series with the same outlet?

a) Both bulbs will be equally bright.

b) The “100 W” bulb will be brighter.

c) The “50 W” bulb will be brighter.

Chapter 20:Electric Circuits

Section 8:

Series and Parallel Wiring

Compound Circuits

20.8.1. Consider the three identical light bulbs shown in the circuit. Bulbs B and C are wired in series with each other and are wired in parallel with bulb A. When the bulbs are connected to the battery as shown, how does the brightness of each bulb compare to the others?

a) Bulbs B and C are equally bright, but bulb A is less bright.

b) Bulbs B and C are equally bright, but less bright than bulb A.

c) All three bulbs are equally bright.

d) Bulbs A and B are equally bright, but bulb C is less bright.

e) Only bulb A is illuminated.

20.8.2. A circuit is formed using a battery, three identical resistors, and connecting wires as shown. How does the current passing through R3 compare with that passing through R1?

a) I3 < I1

b) I3 = I1

c) I3 > I1

d) This cannot be determined without knowing the amount of current passing through R2.

20.8.3. What is the approximate equivalent resistance of the five resistors shown in the circuit?

a) 21

b) 7

c) 11

d) 14

e) 19

R RR

Resistance = 3 R

What is the resistance between x and y?

x y

R RR

Resistance = R

What is the resistance between x and y?

x y

a

R RR

Resistance = R

What is the resistance between x and y?

x yb

R RR

What is the resistance between x and y?

x y

a

b

Rx

R

Ry

ab

R RR

x y

a

b

Redraw the circuit

R

R

R

What is the resistance between x and y?

x

y

ab

3R

Resistance

Chapter 20:Electric Circuits

Section 9:

Internal Resistance

Internal Resistance Batteries and generators add some resistance to a circuit.

This resistance is called internal resistance. The actual voltage between the terminals of a battery is

known as the terminal voltage.

Example 12 The Terminal Voltage of a Battery

The car battery has an emf of 12.0 V and an internal resistance of 0.0100 Ω. What is the terminal voltage when the current drawn from the battery is (a) 10.0 A and (b) 100.0 A?

(a) IrV

V 10.0V 0.12

(b) IrV

V 0.1V 0.12

010.0A 0.10 V 10.0

11.9V

010.0A 0.100 V 0.1

11.0V

20.9.1. In physics lab, two students measured the potential difference between the terminals of a battery and the current in a circuit connected to the battery. The students then made a graph of the two parameters as shown. They then drew a best fit line through the data. From their results, determine the approximate internal resistance of the battery.

a) 0.002

b) 0.08

c) 0.1

d) 0.3

e) 0.6

Chapter 20:Electric Circuits

Section 10:

Kirchoff’s Rules

Loop Rule The loop rule expresses conservation of energy in terms of

the electric potential. States that for a closed circuit loop, the total of all potential

rises is the same as the total of all potential drops.

Junction Rule

Conservation of mass Electrons entering must equal the

electrons leaving The junction rule states that the total

current directed into a junction must equal the total current directed out of the junction.

Example 14 Using Kirchhoff’s Loop Rule

Determine the current in the circuit.

drops potentialrises potential

0.8V 0.6 12V 24 II

A 90.0I

i

iV 0

321 VVVVbattery

Reasoning StrategyApplying Kirchhoff’s Rules

1. Draw the current in each branch of the circuit. Choose any direction. If your choice is incorrect, the value obtained for the current will turn out to be a negative number.

2. Mark each resistor with a + at one end and a – at the other end in a way that is consistent with your choice for current direction in step 1. Outside a battery, conventional current is always directed from a higher potential (the end marked +) to a lower potential (the end marked -).

3. Apply the junction rule and the loop rule to the circuit, obtaining in the process as many independent equations as there are unknown variables.

4. Solve these equations simultaneously for the unknown variables.

20.10.1. What is the current through the 4- resistor in this circuit?

a) 0.67 A

b) 0.75 A

c) 1.0 A

d) 1.3 A

e) 1.5 A

20.10.2. What is the current through the 1- resistor in this circuit?

a) 2.8 A

b) 3.0 A

c) 3.4 A

d) 4.3 A

e) 4.8 A

20.10.3. Which one of the following equations is not correct relative to the other four equations determined by applying Kirchoff’s Rules to the circuit shown?

a) I2 = I1 + I4

b) I2 = I3 + I5

c) 6 V (8 ) I1 (5 ) I2 (4 ) I3 = 0

d) 6 V (6 ) I4 (5 ) I2 (2 ) I5 = 0

e) 6 V (8 ) I1 (6 ) I4 6 V (2 ) I5 (4 ) I3 = 0

Chapter 20:Electric Circuits

Section 11:

The Measurement of

Current and Voltage

A dc galvanometer. The coil ofwire and pointer rotate when thereis a current in the wire.

An ammeter must be inserted intoa circuit so that the current passesdirectly through it.

If a galvanometer with a full-scalelimit of 0.100 mA is to be used to measure the current of 60.0 mA, a shunt resistance must be used so thatthe excess current of 59.9 mA can detour around the galvanometer coil.

To measure the voltage between two pointsin a circuit, a voltmeter is connected betweenthe points.

Chapter 20:Electric Circuits

Section 12:

Capacitors in series and in parallel

Parallel capacitors i

iP CC

21 qqq

Capacitors in Parallel Voltage is the same on each side of the circuit Charges on each capacitor directly add

VCVC 21 VCC 21

21 VVV

Series capacitors i iS CC

11

Capacitors in Series Since there is only one path, charge is the same in all

capacitors regardless of capacitance Voltage drop of each capacitor directly add

21 C

q

C

q

21

11

CCq

20.12.1. A parallel plate capacitor is connected to a battery and becomes fully charged. A voltmeter is used to measure the potential difference across the plates of the capacitor. Then, an uncharged thin metal plate is inserted into the gap between the parallel plates without touching either plate. What affect, if any, does the insertion of the plate have on the potential difference across the plates?

a) The potential difference will not change.

b) The potential difference will increase to twice its initial value.

c) The potential difference will decrease to one half its initial value.

d) The potential difference will increase to a value that cannot be determined without having more information.

e) The potential difference will decrease to a value that cannot be determined without having more information.

20.12.2. Three parallel plate capacitors, each having a capacitance of 1.0 µF are connected in series. The potential difference across the combination is 100 V. What is the charge on any one of the capacitors?

a) 30 C

b) 300 C

c) 3000 C

d) 100 C

e) 1000 C

Chapter 20:Electric Circuits

Section 13:

RC Circuits

Capacitor charging

RCto eqq 1

RCtime constant

RC Circuits

Capacitor discharging

RCtoeqq

RCtime constant

RC Circuits

20.13.1. In physics lab, Rebecca measured the voltage across an unknown capacitor in an RC circuit, every ten seconds after a switch in the circuit that allows the capacitor to discharge is closed. The capacitor was initially fully charged. Using the graph, estimate the time constant.

a) 7.5 s

b) 15 s

c) 30 s

d) 45 s

e) 60 s

20.13.2. An RC circuit contains a battery, a switch, a resistor, and a capacitor – all connected in series. Initially, the switch is open and the capacitor is uncharged. Which one of the following statements correctly describes the current in the circuit during the time the capacitor is charging?

a) The current is increasing with increasing time.

b) The current is constant with increasing time.

c) The current is decreasing with increasing time.

d) The current increases for the first half of the time until the capacitor is fully discharged, and then decreases during the second half of the time.

e) The current can either increase or decrease with increasing time depending on the value of the time constant.

Chapter 20:Electric Circuits

Section 14:

Safety and the Physiological

Effects of Current

20.14 Safety and the Physiological Effects of Current

To reduce the danger inherent in using circuits, proper electrical grounding is necessary.