4: Forces between parallel wires

4



Electrodynamics:

moving charges

and magnetic fields



Qualitative analysis



Try This!

the motor effect

Take a piece of insulated wire

about 5–10 metres long.

Stretch it out between two

retort stands so that there are

two pieces of wire running

parallel within a few

centimetres of each other.

Connect the ends to a 12 V

battery and insert a tapping

key switch at one end of the

circuit. When connected

briefly, the currents will run

antiparallel to each other.

Caution: Connect these wires

for a very short time only, as

they carry a large current.

Predict what will happen when

you press the switch. Now

observe! How did you go?



Let us first consider the situation in which we have two parallel current-carrying

wires with currents that are travelling in the same direction.

In Figure 4.4.1a we use the right-hand grip rule to determine the direction of

the magnetic fields around the wires. Now, to understand what is happening to

each wire, we will consider what is happening to one wire at a time.

In Figure 4.4.1b we are looking at what is happening to wire 2. The magnetic

field generated by the current in wire 1 travels into the page around wire 2.

Using the right-hand palm rule, we can see that wire 2 experiences a force

towards wire 1.

In Figure 4.4.1c we now see what is happening to wire 1. The magnetic field

of wire 2 comes out of the page around wire 1. Therefore the right-hand palm

rule shows that wire 1 experiences a force towards wire 2.

The conclusion we can come to is that when two parallel currentcarrying conductors have currents travelling in the same direction, the two

conductors are forced towards each other.



a



b



II

F

=k 12

l

d



94



I2



F1



wire 1



wire 2



magnetic fields

around parallel wires



Describe qualitatively and

quantitatively the force

between long parallel currentcarrying conductors:



I1



wire 1



wire 2



magnetic field

due to I1



c



I1



I2



F2



wire 1



wire 2



magnetic field

due to I2



Figure 4.4.1 Determining the forces on two parallel wires

with currents flowing in the same direction



Now let’s consider the two parallel current-carrying wires with currents that

are travelling in opposite directions.

In Figure 4.4.2a the right-hand grip rule shows us the direction of the

magnetic fields around the wires. To understand what is happening to each wire,

we will again consider each wire in turn.

Let’s look at what is happening to wire 2 first (Figure 4.4.2b). The right-hand

grip rule shows the magnetic field of wire 1. This field travels into the page

around wire 2. The right-hand palm rule shows that wire 2 experiences a force

away from wire 1.

Figure 4.4.2c shows what is happening to wire 1. The magnetic field of wire 2

goes into the page around wire 1. The right-hand palm rule then shows that

wire 1 experiences a force away from wire 2.

The conclusion we can now come to is that when two parallel currentcarrying conductors have currents travelling in the opposite direction, the two

conductors are forced away from each other.

It may be easy for you to remember the two conclusions above about the

direction of forces on parallel wires, although remembering the result is generally

less important than knowing how you got there. If you forget the conclusions



motors and

generators

a



b



wire 1



I1



wire 2



wire 1



magnetic fields

around antiparallel wires



I2



wire 2



magnetic field

due to I1



c



I1



I2



wire 1



wire 2



magnetic field

due to I2



above and you know how to work them out yourself you can never get them

wrong. You will apply similar methods in other problems later in this module.

So if you are comfortable with these methods now, it will be easier later. If at any

time you have trouble using your right-hand rules, come back to the relevant part

of this chapter and revise. You will meet the skills you have used here several more

times yet and each occasion is a chance to test your knowledge.



Figure 4.4.2 Determining the forces on two

current-carrying wires with

currents in opposite directions



Quantifying the relationship

Using our right-hand rules, we have determined that parallel wires exert forces on

each other. To quantify these forces, let’s start with an equation we have seen

already. Recall the equation for the force on a current-carrying conductor:

F = BIl sin θ

For parallel wires, each wire is at right angles to the magnetic field of the other

wire. The sin θ term in the above equation is therefore equal to 1 (see Figure 4.3.4)

so the equation becomes:

F = BIl

Inspecting this formula we can see that the current I and length l can be

measured relatively easily. To work out F, the size of the force, we now need to

calculate B, the strength of the magnetic field around the wire.

The strength of the magnetic field around a current-carrying conductor

can be determined using the equation:

I

B=k

d

where the proportionality constant k is 2 × 10–7 N A–2, I is the current in amps,

and d is the distance away from the wire in metres.

Let’s consider the situation in Figure 4.4.1b. The magnetic field is being

produced by wire 1, so the current I1 will be used in calculating the magnetic

field strength. The force we are calculating is acting on wire 2, so the current we

should use in this part of the calculation is I2. Combining the two previous

equations and inserting the correct currents in each we get:

F =k



Rearranging this gives:



I1

I l

d 2



II

F

=k 1 2

l

d



where F is the force on each wire in newtons and l is the length the wires are

parallel in metres, so F/l is the force on each metre of wire. k is the proportionality

constant 2 × 10–7 N A–2, I1 and I2 are the currents in the two wires in amps, and

d is the distance between the two wires in metres.

95



4



Electrodynamics:

moving charges

and magnetic fields

1.5 A



1.0 A



Worked example

QUESTION

For the situation shown in Figure 4.4.3 calculate the magnitude of the force acting on

each wire.



0.5 m



SOLUTION

2 cm



Figure 4.4.3 Two parallel current-carrying

wires



Feeling

the pinch



T



he piece of copper

pipe shown in

Figure 4.4.4 was

crushed by lightning.

Just like two parallel

wires carrying currents

in the same direction,

the sides of this pipe

were pulled together

when a current of more

than 100 000 amps

was present.



Figure 4.4.4

Dramatic evidence of

parallel conductors

experiencing a force



From Figure 4.4.3, l = 0.5 m, l1 = 1.5 A, l2 = 1.0 A and d = 2/100 = 0.02 m.

II

F

Use:

=k 1 2

l

d

I1 I 2

l

Rearrange to make F  the subject: F = k

d

Substitute:









F=



2 × 10−7 × 1.5 × 1.0

× 0.5

0.02



= 7.5 × 10–6 N



Note that as this question asked for only the magnitude of the force, you do not have to

include a description of the direction. If asked, you should add that the force on each wire is

directed towards the other wire.



More qualitative analysis

Our last stop in our look at parallel wires is to analyse the relationships expressed

in the equation:

II

F

=k 1 2

l

d

If we follow the process we used in section 4.3, we can see the following

relationships:

• F is directly proportional to the length l. To see this easily we rearrange the

formula to make F the subject. As l increases F increases.

• F is also proportional to both I1 and I2.

• F is inversely proportional to the distance d between the two wires. This

means that as the distance increases F decreases, or as the distance decreases

F increases.



Checkpoint 4.4

1 Identify the two key facts that explain the interactions of two parallel current-carrying conductors.

2 Describe the interactions of two parallel current-carrying conductors.



96



PRACTICAL EXPERIENCES



motors and

generators



CHAPTER 4

This is a starting point to get you thinking about the mandatory practical

experiences outlined in the syllabus. For detailed instructions and advice, use

in2 Physics @ HSC Activity Manual.



Activity 4.1: The motor effect

Observe the effect of a current-carrying wire that is placed in an external magnetic

field and relate it to the mathematical formula:



F = BIl sin θ

Equipment: 2 strong horseshoe magnets or ceramic magnets on an iron yoke,

long wire, power supply, retort stand, clamp.



hanging from

a retort stand



Perform a first-hand

investigation to demonstrate

the motor effect.

Solve problems and analyse

information about the force on

current-carrying conductors in

magnetic fields using:

F = BIl sin θ



flexible wire



ceramic magnets

on an iron yoke



low voltage DC

power supply



Figure 4.5.1 Experiment set-up



Discussion questions

1 Describe the motor effect.

2 Discuss what happened when the current direction was changed.



97



4

•



•



•

•

•



•



•



Chapter summary



Electrodynamics:

moving charges

and magnetic fields



Current is the rate of flow of charge through a region,

and in circuits the direction of a current is that of a

positive charge, called conventional current.

A current in which the charges only flow in one

direction is called direct current (DC). A current in

which the charges move back and forth is an alternating

current (AC).

Resistance is a measure of how easily a current flows and

it is defined as the ratio of voltage over current.

Electrical power is the rate at which energy is transferred

within a circuit component.

Electric currents produce a magnetic field around a

wire. The direction of this field can be determined by

the right-hand grip rule.

A magnetic field is depicted by lines with an arrow

indicating the direction in which the north pole of a

magnet points within the field.

The symbols × and • are used to show the direction of

a magnetic field into and out of the page respectively.



•

•



•



•

•



•





The symbols  and  are used to show the direction

of a current into and out of the page respectively.

Charged particles moving in a magnetic field experience

a force. When these charges are moving in a wire, the

wire experiences a force called the motor effect.

The right-hand palm rule relates the perpendicular

directions of force, magnetic field and motion in the

motor effect.

A loudspeaker is a great example of an application of the

motor effect.

The motor effect is quantified by the equation

F = BIl sin θ where the force is proportional to the

magnetic field strength (B), current (I), length of the

wire within the magnetic field (l ) and the angle between

the wire and the magnetic field (θ).

Parallel current-carrying wires experience the motor

effect due to each other’s magnetic fields and this

II

F

phenomenon is quantified by the equation = k 1 2 .

l

d



Review questions

Physically speaking

1



Across

4 Equation to determine the force between

two current-carrying wires (7, 3)



2

4



3

5



7 emf stands for this (13, 5)

9 Application of the motor effect that

converts electrical energy to sound

10 Quantity related to the energy given

to electrons in a circuit



6

8



7



9



11 Force experienced by current-carrying

wire in an external magnetic field (5, 6)



Down

1 Electrons moving in one direction (6, 7)

2 Unit of power



10



11



98



3 The branch of electricity that deals with

moving charges and magnetic fields

5 Rate of use of electrical energy

6 Opposition to the flow of electrons

8 Device that converts electrical energy

to kinetic energy using the motor effect



motors and

generators



Reviewing

1

2



Describe the difference between DC and AC.



3



Explain why a bird can sit on an electrical power cable

and not get electrocuted.



4



Describe what is meant by metals having a ‘sea of

electrons’.



5



a Recall the factors that affect the resistance of

a wire.

b State how they affect it.



6



Given that the definition of power is P = W /t, show

that the equation for electrical power is P = VI.



7



Describe what happens to a charged particle in

a magnetic field.



8



Compare the paths of two charged particles entering

magnetic fields. The first is in a constant magnetic

field and the second is in a magnetic field that is

gradually increased.



16 Determine the direction the particle will move when it

enters the magnetic field that is shown.

a



Outline the journey of an electron through the circuit

in Figure 4.1.1, noting the energy transformation.



9 State the motor effect in words.

10 Sketch a graph to show how changing the angle

of the wire in a magnetic field changes the force

experienced.



11 Explain how current can create sound in

a loudspeaker.



12 Draw a labelled energy transformation diagram

of a loudspeaker.



+

b



+



17 Determine the force on a current-carrying wire

(I = 2 A) of length 0.5 m that is placed in a magnetic

field of 3 T.



18 A physics student did an experiment to measure

the force on a wire placed in an external magnetic

field. The field is altered and the results are

recorded below.



Force (N)



Magnetic field (T)



0.05

0.08

0.11

0.17



0.1

0.2

0.3

0.4



a Graph the results.

b Determine the value of the gradient of the graph

and what the gradient represents.

c Given that the length of the wire is l = 0.2 m and

the current is I = 2 A, comment on the accuracy

of the student’s results.



solving Problems

13 Calculate the resistance in a circuit that has a battery

supplying 10 V and current flow of 2.3 A.



14 Determine what happens to current in a circuit when

the thickness of the wire is doubled and the voltage is

increased to four times the original.



15 Using the right-hand palm rule, determine the

direction of the unknown quantity B, I or F.

a



iew



Q uesti o



n



s



Re



c



v



b



99



5

electromagnetic induction,

Faraday’s law, magnetic flux,

magnetic field strength B, magnetic flux

density, emf, perpendicular area, Lenz’s

law, law of conservation of energy, eddy

currents, induction cooktops, resistive

heating, eddy current braking



Induction:

the influence

of changing

magnetism

Electromagnetic induction

The discovery of electromagnetic induction was a giant step on the path

to modern technology. Our understanding of this phenomenon required

a great deal of new physics and involved the work of many individuals.

One man, Michael Faraday, led the way with his experimental genius

and intuitive diagrammatic reasoning.

Faraday lacked the mathematical skills to numerically describe his

discoveries, but James Clerk Maxwell took Faraday’s understanding and

eventually quantified all electromagnetism. Faraday’s law and Lenz’s

law provide us with the tools to explain and predict

eddy currents and to understand their applications.

Later they will help us in our quest to uncover the

secrets of motors, generators and transformers.



5.1 Michael Faraday discovers

electromagnetic induction



Outline Michael Faraday’s

discovery of the generation

of an electric current by a

moving magnet.



100



Michael Faraday (1791–1867) was born into a working-class family in London

in 1791. He received little formal schooling and started work at the age of 12.

At 14 he became a bookbinder’s apprentice and set about educating himself with

the books he was able to access. Over this time he developed a keen interest in

science and began attending scientific lectures. In 1813 he became a research

assistant for the prominent scientist Sir Humphry Davy (1788–1829). In the

following years Faraday became renowned as one of the greatest experimental

scientists.

One of his numerous experimental discoveries was the phenomenon of

Electromagnetic induction is the

electromagnetic induction in 1831.

generation of an electric current by a changing magnetic field. Faraday showed

that when he moved a magnet near a wire coil, a current flowed within the coil.

He first moved a magnet into one end of a wire coil (Figure 5.1.1a). As he did

this he measured a current in the coil on a galvanometer (a type of ammeter).



motors and

generators

He noticed that this current was only induced while the magnet was moving.

He moved the magnet out of the coil, and this time he measured a current in

Following many

the opposite direction within the coil (Figure 5.1.1b).

other experiments, he put forward his general principle of electromagnetic

induction that a changing magnetic field can cause a current to be generated in

a wire. This change can be caused by either the relative motion of the field and

the coil or by a change in the strength of the magnetic field.

In light of Faraday’s conclusion, let’s offer a simple way to understand his

observations. In Figure 5.1.1 we can consider the change in the magnetic field

through the coil to be represented by the number of magnetic field lines passing

within the coil. In Figure 5.1.1a, notice that the magnetic field around the bar

magnet is not uniform. You see that as you get closer to the poles of the magnet

the magnetic field lines get closer together, indicating that the field is stronger.

Therefore, as the magnet gets closer to the coil more field lines pass within the

coil, indicating that the magnetic field within the coil gets stronger. So, as the

magnet gets closer to the coil the strength of the field within the coil is changing

and this induces the current. This gives us a general explanation for

electromagnetic induction. Now let’s take things a bit further.



Faraday’s law: explaining electromagnetic induction

Electromagnetic induction can be summarised by Faraday’s law.

The induced emf in a coil is proportional to the product of the number

of turns and the rate at which the magnetic field changes within the turns.

Faraday’s law is quantified by an equation, and it is very useful to analyse the

equation to understand the relationships involved. This equation is:

ε = n(∆ΦB/∆t)

Let us spend a little time now looking at what each of the variables in this

equation means and then we can understand the relationships.



Magnetic flux ΦB

Magnetic flux is a measure of the ‘amount’ of magnetic field passing

through a given area. There are two variables that determine the value of

magnetic flux: the strength of the magnetic field B and the area the field is

passing through A. Magnetic flux ΦB is measured in weber (Wb) and can be

expressed by the equation:

ΦB = BA⊥

Magnetic field strength B, a quantity we are already familiar with,

is also called magnetic flux density. This quantity is measured in tesla (T), or

equivalently in webers per square metre (Wb m–2). This is a measure of the field

strength per square metre.

A ⊥ is an area that is perpendicular to the magnetic field lines. If a magnetic

field passes through a circular wire loop of area A (Figure 5.1.2a) and the loop is

at an angle to the field (see Figure 5.1.2b), then the field passes through an

‘effective’ area A⊥ that is perpendicular to the field and is smaller than area A (see

Figure 5.1.2c). Note that flux could also be calculated using the perpendicular

component of the magnetic field strength, B⊥, and the total area A.



a

N



A

b

N



A



Figure 5.1.1 The changing magnetic field of

a moving magnet can induce a

current in a coil of wire.



Our sun’s

magnetic

influence



L



arge outbursts from the Sun cause

changes in the strength of the

magnetic field at the Earth’s surface.

This changing magnetic field induces

currents in long metal pipelines and the

wires of power grids, especially at high

latitudes. Currents of 1000 amps have

been measured in pipelines in Alaska,

causing accelerated corrosion. Large

currents in power grids have overloaded

circuits and left millions of people in

the dark for hours.



PRACTICAL

EXPERIENCES

Activity 5.1



Activity Manual, Page

33



Perform an investigation to model

the generation of an electric

current by moving a magnet in

a coil or a coil near a magnet.

Define magnetic field strength B

as magnetic flux density.

Describe the concept of magnetic

flux in terms of magnetic flux

density and surface area.



101