2: Forces on charged particles in magnetic fields

4



Electrodynamics:

moving charges

and magnetic fields



Try this!

Bending beams of particles

Ask your teacher if they can show you a beam of electrons

in a Crookes magnetic deflection tube (see Figure 4.2.2).

With teacher supervision, bring the north pole of a magnet

close to the front of the tube and observe the effect on the

beam of electrons. By the right-hand palm rule, negative

particles experience a force out of the back of your hand.

Try to predict the direction in which electrons will be

deflected when you place the south pole of the magnet

close to the front of the tube. Now try it and then explain

it to a friend. Do they agree with your explanation?



Figure 4.2.2 A Crookes magnetic deflection tube

produces a beam of electrons.



Checkpoint 4.2

1 Explain why a stationary charged particle experiences a force when you move a magnet past it.

2 Identify the direction in which the proton in Figure 4.2.1a would be moving for it to experience a force into the page.



4.3 The motor effect



Identify that the motor effect

is due to the force acting on

a current-carrying conductor

in a magnetic field.



PRACTICAL

EXPERIENCES

Activity 4.1



Activity Manual, Page

27



90



In the previous section we saw that charges moving in a magnetic field

If a current-carrying conductor is placed in an external

experience a force.

magnetic field, the wire also experiences a force and this is called the motor

effect. This effect occurs because the charges within the wire are travelling

through the magnetic field and experience a force, just as they would if they were

Remember, we are considering

free charged particles (see Figure 4.2.1a).

current to be a flow of positive particles. It is actually the negative electrons

that experience the force from the magnetic field. It works just fine to use

conventional current and consider the force is acting on positive particles to keep

things simple.

We can now use the right-hand palm rule we saw in section 4.2 to work out

the direction of the force on a current-carrying wire in an external magnetic field

To find the direction of the force that acts on a current(see Figure 4.3.1).

carrying wire that is perpendicular to an external magnetic field:

1 Place your open right hand with the fingers pointing in the direction of the

magnetic field (north to south).

2 Place your thumb at right angles to your fingers and in the direction in which

the conventional current is flowing (from the positive to negative terminals

in the circuit).

3 The force experienced by the wire will be directed out of your palm at right

angles to your hand.



motors and

generators

When we allow current to flow through a wire

within the magnetic field in Figure 4.3.1, we see

that the wire moves out of the page, at right angles

to both the magnetic field and the direction of the

conventional current. Each positive particle of the

conventional current within the wire in Figure 4.3.1

experiences a force due to its motion within the

external magnetic field. Since these positive particles

are within the wire, the force acts on the wire.



I

S



+



N



direction in which the I

current is flowing



F



B

direction of the

magnetic field

F



Figure 4.3.1 The right-hand palm



PHYSICS FEATURE



Identify data sources, gather and process

information to qualitatively describe the

application of the motor effect in:

– the galvanometer

– the loudspeaker.



Loudspeakers



A



n excellent example of an application of the motor

effect is a loudspeaker. This device is a key part

of telephones, televisions and any other appliance in

which an electrical signal needs to be converted into

sound for us to hear. Figure 4.3.2a shows the most

fundamental parts of a typical loudspeaker labelled

A to D. Figure 4.3.2b illustrates its operation via the

motor effect.

A loudspeaker contains a current-carrying coil (C),

which is commonly called the voice coil. This coil is

wound around a hollow cardboard tube (B) and the

tube is fixed to the cone of the speaker (A). The voice

coil is suspended inside a cylindrical permanent

magnet (D) that provides a uniform magnetic field at

right angles to the coil. An alternating current is passed

through the voice coil, causing the cone to rise and fall

due to the motor effect. Each time the speaker cone

pushes outwards on the air, it creates a wave of

pressure that travels away from the speaker. These

waves of air pressure are sound waves. By varying the

frequency of the alternating current in the voice coil,

the frequency (or pitch) of the sound can be varied.

This means that speakers can generate a variety of

sounds and reproduce sounds recorded elsewhere.

At the moment in time shown in Figure 4.3.2b,

the current is travelling out of the page on the left

side of the voice coil. Using the right-hand palm rule



Figure 4.3.2 A loudspeaker converts electrical energy into sound.

(a) A cut-away section showing the parts of the

loudspeaker and (b) a simplified cross section showing

the direction of the magnetic field



direction of the force

on the positive particle



rule for a current



you can see that a force will be exerted upwards on

the voice coil on this side. This force is exerted in the

same direction around the circumference of the voice

coil and causes the speaker cone to move upwards.

So we can see that now we know about the motor

effect and the right-hand palm rule we can explain

how some everyday things work.

a



A



B

C



D



b



direction of force

on the voice coil



N



S



A cardboard cone



N



B cardboard tube fixed

to cardboard cone and

wrapped in voice coil

C wire coil (called the

voice coil)

D permanent cylindrical

magnet



91



4



Electrodynamics:

moving charges

and magnetic fields



Quantifying the motor effect

If we place a current-carrying wire within a magnetic field (see Figure 4.3.3) the

force on the conductor is given by:

Solve problems and analyse

information about the force on

current-carrying conductors in

magnetic fields using:

F = BIl sin θ



F = BIl sin θ

where F is magnitude of the force on the wire in newtons (N), l represents the

length of the wire inside the magnetic field in metres (m), I is the size of the

current in amps (A), B is the strength of the magnetic field in tesla (T) and θ is

the acute angle between the magnetic field and the wire.

Note that you can rearrange this equation and make any of the variables the

subject to find their values.



Worked example

Question

If the wire in Figure 4.3.3 has a length of 5 cm within a 0.2 T magnetic field, the wire is at

30º to the field and it contains a current of 0.5 A, what is the force exerted on the wire?



I

B

l



S



a



SOLUTION



N



First we convert the length of the wire into the appropriate units: 5 cm = 5/100 = 0.05 m



θ

sin θ =



Figure 4.3.3



a

l



a = l sin θ



Then substitute:



F = BIl sin θ







= 0.2 × 0.5 × 0.05 × sin 30







= 2.5 × 10–3 N out of the page



Qualitative analysis of factors affecting the motor effect

Discuss the effect on the

magnitude of the force on a

current-carrying conductor of

variations in:

• the strength of the magnetic

field in which it is located

• the magnitude of the current

in the conductor

• the length of the conductor

in the external magnetic field

• the angle between the

direction of the external

magnetic field and the

direction of the length of

the conductor.



92



If you have an equation that describes a relationship, the easiest way to see

how the variables affect the subject of the equation is to place some numbers in

the equation and see what happens.

Remembering that the motor effect is the force F that a current-carrying

conductor experiences in a magnetic field, let’s look at how the other variables in

the equation affect the magnitude of F. As you read through this section keep

referring back to the equation and check that you come to the same conclusions.



Magnetic field strength, B

If the value of B was very small, then the right side of the equation would be

multiplied by a very small number. Conversely, if the value of B was very large,

the right side of the equation would be multiplied by a very large number. Since

the magnitude of the force F is equal to the right-hand side of the equation,

F is clearly directly related to B.

To take our analysis one step further, consider multiplying the value of B by 2.

Looking at the equation, we see that if we do this, the effect on the value of F is

We can now say the force F is directly

the same as multiplying F by 2.

proportional to magnetic field strength B; that is, as B increases by some factor

(say 2 times) F also increases by that same factor.

Having understood the analysis above, it should be easy now to see the

following relationships.



motors and

generators

Current, I

As for magnetic field strength, by inspecting the formula we can see

that force F will be directly proportional to current I.

Length, l

Similarly, F is directly proportional to the length l of the currentcarrying conductor within the magnetic field. Be particularly careful to

remember that l is the total length of the wire within the magnetic field.

It is noteworthy that l is regarded as a vector, but current I is not.

Angle, θ

When the wire is parallel to the magnetic field, the angle θ is zero. Inspecting

Figure 4.3.4 you can see that if θ is zero degrees then sin θ is also zero. When

you substitute zero for sin θ in the motor effect equation, you see that the

This shows us the interesting situation that the force

force must be zero.

on a current-carrying conductor in a magnetic field is zero when the

conductor is parallel to the magnetic field lines.

When the current-carrying conductor is perpendicular to the magnetic

field, θ is 90°. From Figure 4.3.4 you can see that sin 90 is 1. Since 1 is the

maximum value for sin θ, when we substitute 1 into the equation the force F

will be the maximum value it can be for each set of the other variables.

This shows that the force F is a maximum value when a current-carrying

conductor is perpendicular to the magnetic field B.

Inspecting Figure 4.3.4 you can see that as θ increases from 0° to 90° the

value of sin θ increases towards a value of 1. The rate of this increase is not

So we can only say that

constant (i.e. the graph is not a straight line).

force F depends on θ, as it is not directly proportional to θ.



Nanotube

Loudspeakers:

No Magnets



A



group of Chinese researchers has

developed a loudspeaker that

consists only of a thin film of carbon

nanotubes driven by an AC input

signal. The sound generation is

attributed to a thermoacoustic effect.

Changes in the current flowing

through the film are reflected in the

film’s temperature. Those temperature

changes excite pressure waves in the

surrounding air and these are sound

waves. The film is flexible and can be

stretched and still operate unimpeded.

Perhaps loudspeakers won’t have

magnets in the near future!



sin θ

1



90



180



270



360 θ



–1



Checkpoint 4.3

1 Describe the relative directions of the force, the current and the

magnetic field when a current-carrying wire experiences the

maximum possible force due to the motor effect.

2 Compare the relationships of B, I, l and θ to F in the equation

F = BIl sin θ.

3 Explain the motor effect.



Figure 4.3.4 Graph of θ

versus sin θ



4.4 Forces between parallel wires

In many applications of electric circuits there are wires bundled tightly together

and running parallel to each other. If we want to explore the interaction of

these wires, we need to bring together two of the facts we have learned so far.

The first is that current-carrying conductors produce a magnetic field.

The second is that a current-carrying conductor experiences a force when inside

a magnetic field. We will also need to apply two of the right-hand rules we have

learned to determine the direction of the magnetic fields and the forces.

93



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