Objective Theory - Yonsei Universityphylab.yonsei.ac.kr/exp_ref/204_MagneticForce_ENG.pdf ·...

General Physics Lab (International Campus) Department of PHYSICS YONSEI University

Lab Manual

Magnetic Force Ver.20171027

Lab Office (Int’l Campus)

Room 301, Building 301 (Libertas Hall B), Yonsei University 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA (☏ +82 32 749 3430) / 11

[International Campus]

Magnetic Force

Investigate the effects of current, length of wire and magnetic field strength on a magnetic force.

1. Current, Drift Velocity, and Current Density

A current is any motion of charge from one region to another.

Electrons in motion involve currents in conducting materials

such as copper wire.

In electrostatic situations, the electric field is zero every-

where within the conductor, and there is no current. However,

this does not mean that all charges within the conductor are

at rest. In an ordinary metal, some of electrons are free to

move, as shown in Fig.1. These free electrons move random-

ly in all direction, so there is no net flow of charge in any di-

rection and hence no current.

If an electric field 𝑬𝑬��⃗ is established inside a conductor, a free

electron is then subjected to a force 𝑭𝑭��⃗ = 𝑞𝑞𝑬𝑬��⃗ and accelerated

in the direction of 𝑭𝑭��⃗ . But an electron moving in a conductor

undergoes frequent collision with the massive stationary ions

of the material, so the electron’s direction of motion under-

goes a random change. The net effect of 𝑬𝑬��⃗ is that in addition

to the random motion of the electron, there is also a very slow

net motion of drift of moving electrons as a group in the direc-

tion of 𝑭𝑭��⃗ . This motion is described in terms of the drift veloc-ity 𝒗𝒗��⃗ d of the electrons. As a result, there is a net current in

the conductor. While the random motion of the electrons has

a very fast average speed of about 106 m/s, the drift speed is

very slow, often on the order of 10−4 m/s.

Fig 1 If there is no electric field inside a conductor, an electron

moves randomly from 𝑃𝑃1 to 𝑃𝑃2 in 𝛥𝛥𝛥𝛥. If 𝑬𝑬��⃗ is present, 𝑭𝑭��⃗ = 𝑞𝑞𝑬𝑬��⃗ imposes a small drift that takes the electron to 𝑃𝑃2′, a distance 𝑣𝑣d𝛥𝛥𝛥𝛥 from 𝑃𝑃2 in the direction of the force.

Objective

Theory

----------------------------- Reference --------------------------

Young & Freedman, University Physics (14th ed.), Pearson, 2016

25.1 Current (p.841-844)

27.2 Magnetic Field (p.907-911)

27.6 Magnetic Force on a Current-Carrying Conductor (p.920-923)

-----------------------------------------------------------------------------


Lab Manual




We define the conventional current 𝐼𝐼 to be in the direction

in which there is a flow of positive charge, even if the actual

current is due to electrons. Fig. 2 shows a segment of a con-

ductor in which a current is flowing. We define the current 𝐼𝐼

through the cross-sectional area 𝐴𝐴 to be the net charge 𝑑𝑑𝑑𝑑

flowing through the area per unit time 𝑑𝑑𝛥𝛥.

𝐼𝐼 =𝑑𝑑𝑑𝑑𝑑𝑑𝛥𝛥 (1)

The SI unit of current is the ampere. (1 A = 1 C s⁄ )

We can express current in terms of the drift velocity of the

moving charges. In Fig. 2, an electric field 𝑬𝑬��⃗ is directed to

right in a conductor with cross-sectional area 𝐴𝐴. Assume that

the free charges in the conductor are positive; then the drift

velocity is in the same direction as the field. Suppose there

are 𝑛𝑛 moving charged particles per unit volume and all the

particles move with the same drift velocity with magnitude 𝑣𝑣d.

We call 𝑛𝑛 the concentration of particles; its SI unit is m−3.

In a time interval 𝑑𝑑𝛥𝛥, each particle moves a distance 𝑣𝑣d𝑑𝑑𝛥𝛥.

The particles in the shaded cylinder with length 𝑣𝑣d𝑑𝑑𝛥𝛥 flow out

of the right end of the cylinder during 𝑑𝑑𝛥𝛥. The volume of the

cylinder is 𝐴𝐴𝑣𝑣d𝑑𝑑𝛥𝛥, and the number of particles within it is

𝑛𝑛𝐴𝐴𝑣𝑣d𝑑𝑑𝛥𝛥.

Fig 2 The current 𝐼𝐼 is the time rate of charge transfer through

the cross-sectional area 𝐴𝐴. The random component of each moving charged particle’s motion averages to zero, and the current is in the same direction as 𝑬𝑬��⃗ whether the moving charges are positive (as shown here) or negative.

If each particle has a charge 𝑞𝑞, the charge 𝑑𝑑𝑑𝑑 that flows

out of the end of the cylinder during time 𝑑𝑑𝛥𝛥 is

𝑑𝑑𝑑𝑑 = 𝑞𝑞(𝑛𝑛𝐴𝐴𝑣𝑣d𝑑𝑑𝛥𝛥) = 𝑛𝑛𝑞𝑞𝑣𝑣d𝐴𝐴𝑑𝑑𝛥𝛥 (2)

and the current is

𝐼𝐼 =𝑑𝑑𝑑𝑑𝑑𝑑𝛥𝛥 = 𝑛𝑛𝑞𝑞𝑣𝑣d𝐴𝐴 (3)

The current per unit cross-sectional area is called the cur-

rent density 𝐽𝐽:

𝐽𝐽 =𝐼𝐼𝐴𝐴 = 𝑛𝑛𝑞𝑞𝑣𝑣d (A m2⁄ ) (4)

2. Magnetic Force on a Current-Carrying Conductor

The magnetic force 𝑭𝑭��⃗ on a charge 𝑞𝑞 moving with velocity

𝒗𝒗��⃗ in a magnetic field 𝑩𝑩��⃗ is given by

𝑭𝑭��⃗ = 𝑞𝑞𝒗𝒗��⃗ × 𝑩𝑩��⃗ (5)

Fig 3 The magnetic force 𝑭𝑭��⃗ acting on a positive charge 𝑞𝑞

moving with velocity 𝒗𝒗��⃗ is perpendicular to both 𝒗𝒗��⃗ and the magnetic field 𝑩𝑩��⃗ .


Lab Manual




There is a magnetic force on a current-carrying conductor in

a magnetic field because charges in the conductor are in

motion. We can compute the force on it with equation (5).

Figure 4 shows a straight segment of a conducting wire, with

length 𝑙𝑙 and cross-sectional area 𝐴𝐴. The wire is in a uniform

magnetic field 𝑩𝑩��⃗ , perpendicular to the plane of the diagram

and directed into the plane. Let’s assume that the moving

charges are positive.

The drift velocity 𝒗𝒗��⃗ d is upward, perpendicular to 𝑩𝑩��⃗ . The

magnitude of the average force on each charge is then

𝐹𝐹 = 𝑞𝑞𝑣𝑣d𝐵𝐵 from equation (5).

We can derive an expression for the total force on all the

moving charges in a length 𝑙𝑙 of conductor with cross-

sectional area 𝐴𝐴 using the same language we used in Eqs.

(3) and (4). The number of charges per unit volume is 𝑛𝑛; a

segment of conductor with length 𝑙𝑙 has volume 𝐴𝐴𝑙𝑙 and con-

tains a number of charges equal to 𝑛𝑛𝐴𝐴𝑙𝑙. The total force 𝑭𝑭��⃗ on

all the moving charges in this segment has magnitude

𝐹𝐹 = (𝑛𝑛𝐴𝐴𝑙𝑙)(𝑞𝑞𝑣𝑣d𝐵𝐵) = (𝑛𝑛𝑞𝑞𝑣𝑣d𝐴𝐴)(𝑙𝑙𝐵𝐵) (6)

From Eq. (4) the current density is 𝐽𝐽 = 𝑛𝑛𝑞𝑞𝑣𝑣d. The product

𝐽𝐽𝐴𝐴 is the total current 𝐼𝐼, so we can rewrite Eq. (6) as

𝐹𝐹 = 𝐼𝐼𝑙𝑙𝐵𝐵 (7)

Fig 4 Forces on a moving positive charge

in a current-carrying conductor.

If the 𝑩𝑩��⃗ is not perpendicular to the wire but makes and an-

gle 𝜙𝜙 with it, only the component of 𝑩𝑩��⃗ perpendicular to the

wire exerts a force; this component is 𝐵𝐵⊥ = 𝐵𝐵 sin𝜙𝜙 . The

magnetic force on the wire segment is then

𝐹𝐹 = 𝐼𝐼𝑙𝑙𝐵𝐵⊥ = 𝐼𝐼𝑙𝑙𝐵𝐵 sin𝜙𝜙 (8)

The force is always perpendicular to both the conductor and

the field, with the direction determined by the right-hand rule.

Hence this force can be expressed as a vector product. We

represent the segment of wire a vector 𝒍𝒍 along the wire in

the direction of the current; then the force 𝑭𝑭��⃗ on this segment

is

𝑭𝑭��⃗ = 𝐼𝐼𝒍𝒍 × 𝑩𝑩��⃗ (9)

Figure 5 and 6 illustrate the direction of 𝑩𝑩��⃗ , 𝒍𝒍, and 𝑭𝑭��⃗ for

several cases.

Fig 5 A straight wire segment of length 𝒍𝒍 carries a current 𝐼𝐼

in the direction of 𝒍𝒍. The magnetic force on this segment is perpendicular to both 𝒍𝒍 and the magnetic field 𝑩𝑩��⃗

Fig 6 Magnetic field 𝑩𝑩��⃗ , length 𝒍𝒍, and force 𝑭𝑭��⃗ vectors

for a straight wire carrying a current 𝐼𝐼.


Lab Manual




1. List

Items Qty. Description

Conductor Boards set

1 set Includes 6 PC boards with different length of straight conductive foil.

Rotatable Conductor Assembly with Goniometer

1 Rotates the direction of conductive wires.

Conductor Holder

1 Holds the Conductor Boards or the Rotatable Conduc-tor Assembly.

Magnet Assemblies

1 set Produces a magnetic field. Magnet assembly #1: for Conductor Boards Magnet assembly #2: for Rotatable Conductor Ass.

Power Supply (Power Cable included)

1 Produces regulated DC power up to 10 A in a voltage range 0 to 30 V.

Electronic Balance (DC Adapter included)

1 Measures mass of an object with a precision to 0.01 g. in a range of 0~300g.

Ruler

1 Measures length.

Patch Cords (with banana plugs)

2 Carry electric current.

A-shaped Base Support Rod (300mm)

1 1

Provides stable support for experiment set-ups.

Equipment


Lab Manual




Prior to the experiment, measure the length of horizontal

conductive foil of the Conductor Boards.

Single-sided 𝑙𝑙

Double-sided 𝑙𝑙 = 𝑙𝑙1 + 𝑙𝑙2

(front) (back)

Make sure the north poles of the magnets of the Magnet

Assembly #1 are all on the same side. You can disassemble

the Magnet Assembly if required.

Experiment 1. Force vs. Current

(1) Set up your equipment.

① Mount the Conductor Holder on an A-shaped Base.

② Select any Conductor Board, and plug it into the end of

the Holder, with the conductive foil extending down.

③ Place the Magnet Assembly on the balance. Position the

A-shaped Base so the horizontal position of the conductive

foil passes through the pole region of the magnets. The con-

ductive foil shouldn’t touch the magnets.

Procedure


Lab Manual




(2) Turn on the power supply.

Before turning on the power supply, rotate the voltage and

current adjustment knobs fully counterclockwise for no output

settings.

Turn on the power supply, and then rotate the voltage ad-

justment knob fully clockwise. Do not increase the current yet.

You can now adjust the current 𝐼𝐼 through the conductor us-

ing the current adjustment knob.

(3) Zero the electronic balance.

Place the Magnet Assembly on the balance. With no current

flowing, zero the balance by pressing [영점] or [용기] button.

Caution

Do not touch the Conductor Board or the metal arms of

the Conductor Holder while current is flowing through

them.

Note

If you cannot obtain the desired output current:

① Check the connections. Make sure the power supply

is properly connected to the conductor.

② Check if the CV lamp is on. It indicates that the DC

output is in constant voltage mode, i.e. the voltage level

you set is too low. Increase the voltage level by rotating

the voltage adjustment knob clockwise.

Caution

The electronic balance is a precision instrument. Sub-

jecting it to impact could cause it to fail. Treat it with care.

Note

For zeroing the balance, press [영점] and stand by until

the [ZERO] mark lights up on the display.

When the initial value is relatively high, [영점] button will

not work. In this case, you can zero it using [용기] button.

([TARE] mark will light up on the display.)

If you have any problem zeroing the balance, turn the

power off and then on again.

Note

If any of [CT], [%], [PCS], [CHK], or [ANI] symbols lights

up on the display, press [모드] repeatedly until all sym-

bols disappear.


Lab Manual




(4) Increase the current and record your results.

Increase the current to 0.5 A, read the mass, and record the

force value in the ‘Force’ column of the table below. Note the

actual force is proportional to the mass reading, 𝐹𝐹 = 𝑚𝑚𝘨𝘨.

Increase the current in 0.5 A increments to 5.0 A, each time

recording the new force value.

Current (A) Mass (kg) Force (N)

0.0

0.5

1.0

…

5.0

(5) Analyze your results.

Plot a graph of the magnetic force as a function of the cur-

rent through the conductor.

Q

What is the nature of the relationship between these two

variables? What does this tell us about how changes in the

current will affect the force acting on a wire that is inside a

magnetic field?

A

Experiment 2. Force vs. Length of current-carrying wire


Follow the setup of experiment 1.

(2) Plug the shortest Conductor Board.

Select the Conductor Board with the shortest conductive foil,

and plug it into the ends of the Conductor Holder.

(3) Set the current to 2.0 A, and record the measured value of

mass.

(4) Repeat the measurements for other length of conductor.

Before removing the Conductor Board, you should TURN

OFF the power supply.

Length (m) Mass (kg) Force (N)

Note

You can swing the arm of the Conductor Holder up to

raise the present Conductor Board out of the magnetic

field gap.


Lab Manual





Plot a graph of the magnetic force as a function of the length

of current-carrying wire.

Q

What is the nature of the relationship between these two

variables? What does this tell us about how changes in the

length of a current-carrying wire will affect the force that it

feels when it is in a magnetic field?

A

Experiment 3. Force vs. Magnetic Field


Follow the setup of experiment 1.

(2) Plug the shortest Conductor Board.

Select the Conductor Board with the shortest conductive foil,

and plug it into the ends of the Conductor Holder.

(3) Mount a single magnet in the Magnet Assembly.

Disassemble the Magnet Assembly and attach a single

magnet in the center of the assembly.

(4) Zero the balance.

Place the Magnet Assembly on the balance. With no current

flowing, zero the balance by pressing [영점] or [용기] button.

(5) Set the current to 2.0 A, and record the force.

(6) Repeat the measurements for other number of magnets.

Add additional magnets, one at a time. Make sure the north

poles of the magnets are all on the same side of the Magnet

Assembly. Each time you add a magnet, repeat steps (4)-(5).

Number of magnets Mass (kg) Force (N)

1

2

3

4

5

6


Plot a graph of the force as a function of number of magnets.

Q

What is the relationship between these two variables?

How does the number of magnets affect the force between

a current carrying wire and a magnetic field? Is it reasona-

ble to assume that the strength of the magnetic field is

directly proportional to the number of magnets?

A

Note

The magnetic field is varied by changing the number of

magnets that are mounted on the Magnet Assembly. The

field may not be exactly proportional to the number of the

magnets, but it is reasonably close.


Lab Manual




Experiment 4. Force vs. Angle

Determine how the angle between the current carrying wire

and the magnetic field affects the force between them.


① Plug the Rotatable Conductor Assembly into the ends of

the Conductor Holder.

② Set the goniometer dial on the Rotatable Conductor to 0°.

③ Place the Magnet Assembly #2 on the balance, and align

it so that the magnetic field is parallel with the wires of the coil

at the end of the Rotatable Conductor Assembly.

(2) Zero the balance.

With the current turned off, zero the balance.

(3) Set the current to 1.0 A, and record the force.

(4) Repeat the measurements.

Increase the angle in 5° increments up to 90° and then in

−5° increments to −90°, by rotating the goniometer dial of

the Rotatable Conductor Assembly. At each angle, record the

force.

Angle

(°)

Mass

(kg)

Force

(N)

Angle

(°)

Mass

(kg)

Force

(N)

0 0

5 -5

10 -10

15 -15

… …

90 -90


Plot a graph of the force as a function of the angle.

Q

What is the relationship between these two variables?

How do changes in the angle between the current and the

magnetic field affect the force acting between them? What

angle produces the greatest force? What angle produces

the least force?

A

Caution

The current through the wire of the Rotatable Conductor

Assembly should never exceed 2.0 A.


Lab Manual




Your TA will inform you of the guidelines for writing the laboratory report during the lecture.

Please put your equipment in order as shown below.

□ Delete all your data files from your lab computer.

□ Turn off your lab Computer.

□ With the voltage and current adjustment knob set at zero, turn off the Power Supply and unplug the power cable.

□ Turn off the Electronic Balance and unplug the dc adaptor.

□ Reassemble the Magnet Assembly #1. (Be careful not to lose any parts of it.)

Make sure the north poles of the magnets are all on the same side of the magnet assembly.

Result & Discussion

End of LAB Checklist

Objective Theory - Yonsei Universityphylab.yonsei.ac.kr/exp_ref/204_MagneticForce_ENG.pdf ·...

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