16. FORCES AND FIELDS
16.1. INTRODUCTION
We showed that electrical charges can exert forces on other electrical charges. Magnetic poles also
exert forces on other magnetic poles. We didn’t calculate the magnitudes of these forces because the
mathematics is a bit more complicated.
These forces are different than those you may previously have encountered because they don’t occur
from one object coming into contact with other objects. Neither charges nor poles have to touch to
exert a force. We describe this phenomenon in terms of fields, which describe the effect of a source
at all regions of space.
16.2. GOALS
• Draw the electric field lines between any two charged objects
• Calculate the strength of the electric field at any distance from an object and realize that the
electric field strength decreases as the distance from the charge increases.
• Evaluate whether electric field strengths you calculate are reasonable.
• Understand the relationship between electric fields and forces.
• Draw the magnetic field lines between a pair of poles.
• Know that the Earth has a magnetic field that can be used for navigation. Understand the
orientation of the Earth’s magnetic field with respect to the geographic poles.
16.3. ELECTRIC FIELDS
Electric charges exert forces on each other. We
use the idea of a ‘field’ to explain how a charge
changes the characteristics of the space around
it. The field produced by an electrical charge is
+
called an electric field, or an ‘E-field’ for short.
Figure 18.1 shows the electric field lines from
isolated positive and negative charges. When
these pictures were taken, the charges were
isolated so that their electric fields do not affect
Figure 16.1: Electric field lines from positive
each other.)
(left) and negative (right) charges.
16.3.1. Rules for Drawing Electric Field
Lines:
1. Electric field lines point in the direction a positive charge would move if you placed it in the
field. (i.e. outward for a positive charge, inward for a negative charge.)
2. The denser the lines, the stronger the field. The electric field strength is larger when you are
closer to a charge.
3. Field lines always begin on positive charges and end on negative charges. The electric field
lines for a point charge continue to infinity.
4. Field lines never cross.
It is relatively straightforward to deal with very simple arrangements of charges. Figure 16.2 and
Figure 16.3 show examples of the electric field lines for complex arrangements of charges.
Compare the figures with the rules given above. Other configurations can be examples using the
applets at:
•
•
http://physics.weber.edu/amiri/director/DCRfiles/Electricity/efiel24s.dcr
http://www.colorado.edu/physics/2000/applets/forcefield.html (which also has a nice discussion
of fields and forces).
Figure 16.2: Electric field lines associated with Figure 16.3: Electric field lines associated with
two positive charges with the same magnitude.
one positive and one negative charge with the
same magnitude.
++
+
+
+
++
A
+
+
+ q
+
++
+
+
++
+
+
+
B
q
You can tell from the directions of the lines in
Figure 16.3 that the positive charge is on the left
and the negative charge is on the right.
Figure 16.4 shows the electric field lines between
two plates with equal and opposite charges. The red
charge q (which is positive) moves from A to B if
placed between the plates.
Although only a few electric field lines are drawn, it
is important to realize that the electric field has a
value everywhere.
Figure 16.4: Electric field lines between two
plates with opposite charge.
16.3.2. Calculating Electric Field Strength. E is the electric field strength at a distance r away
from a charge of magnitude q.
E=k
The units of electric field strength are:
q
r2
(16.3.1)
q
r2
Nm 2 ⎛ C ⎞
= 2 ⎜⎜ 2 ⎟⎟⎟
C ⎜⎝ m ⎠
E=k
=
N
C
Equation (16.3.1) indicates that the magnitude of the electric field strength decreases as you move
away from the charge. The electric field strength has an inverse-square dependence on r: if you go
two times as far away from the point charge, the electric field strength is 22 = 4 times weaker.
Electric field strength can be expressed in other units. A coulomb may be written in terms the volt
(V), which is a unit of electric potential that will investigate in the second part of this unit. The
relationship is:
1 C=
Nm
V
We can then write
N N ⎛⎜ 1 ⎞⎟
= ⎜ ⎟⎟
⎜⎝ C ⎠
C
N ⎛ V ⎞⎟
= ⎜⎜
⎜⎝ Nm ⎠⎟⎟
=
V
m
So volts per meter and newtons per coulomb are equivalent to each. Either may be used to express
electric field strength.
Electric field strength is a vector; however, (with the exception of the parallel charged plates), the
direction of the electric field is not as simple as the direction of the force between two charges. The
electric field lines provide a more specific description of the direction of the electric field at all
points around the charge.
EXAMPLE 16.1: a) What is the magnitude and direction of the electric field of a positive point charge of
magnitude 4.50 μC at a distance of 5.00 cm to the right of the charge? b) What is the electric field at a
distance of 10.00 cm away from the charge? c) How are the two values related?
+4.50 μC
Draw a picture for part a
0.0500 m
known:
q = + 4.50 μC
r = 0.0500 m
Need to find:
E =the electric field due to charge 1 at a distance of
5.00 cm from the charge.
E=k
Equation to use:
q
r2
−6
Nm 2 (4.50×10 C)
E = 9.00 ×10
C 2 (0.0500 m)2
9
Plug in numbers.
= 1.62×107
N
C
E at 5.00 cm = 1.62×107
Answer:
Check your answer
N
C
right
We know because the charge is positive, that the electric field lines go outward –
a positive test charge would move away from it. The direction is to the right.
+4.50 μC
Draw a picture for part b
0.1000 m
known:
q = + 4.50 μC
r = 0.1000 m
Equation to use:
Need to find:
E =the electric field due to charge 1 at a distance of
10.00 cm from the charge.
E=k
E = 9.00 ×109
Plug in numbers.
= 4.05×107
q
r2
−6
Nm 2 (4.50×10 C)
C 2 (0.1000 m)2
N
C
E at 10.00 cm = 4.05×107
Answer:
right
We know because the charge is positive, that the electric field lines go outward –
a positive test charge would move away from it. The direction is to the right.
Check your answer
7
E at 10.00 cm 4.05×10
=
E at 5.00 cm 1.62×107
How are they related to
each other? Take the
ratio.
Check your answer
N
C
=
N
C
N
C
1
4
The ratio of the distances is 2; however, the electric field is an inverse square
law, so the decrease in field is 212 or 14
16.3.3. Typical Electric Field Strengths. Table 16.1 shows the electric field strengths of some
representative sources.
Source
Field strength
⎛ N ⎞⎟
⎜⎜ ⎟
⎜⎝ C ⎠⎟
Wires in house
10-2
Center of typical living room
~1
In a fluorescent tube
10
30 cm from electric clock
15
30 cm from stereo
90
30 cm from electric blanket
250
Atmosphere during a thunderstorm
104
At cell membrane
107
Surface of a uranium nucleus
Table 16.1: Electric field strengths.
2 x 1021
16.4. THE RELATIONSHIP BETWEEN ELECTRIC FIELD AND FORCE.
The force between two charges q1 and q2 separated by a distance r is given by Equation (15.6.1).
F12 = k
q1q2
r2
(15.6.1)
Re-write this by moving q2 out front
⎛ q ⎞
F12 = q2 ⎜⎜k 12 ⎟⎟⎟
⎜⎝ r ⎠
The quantity in parentheses is the electric field due to the charge q1, as given by Equation (16.3.1).
We can thus write that the force between two charges q1 and q2 separated by a distance r is given by
F12 = q2 E
where E is the electric field due to charge 1. This relationship holds true in general. The force F
exerted by an electric field E on a charge q is
F = qE
(16.4.1)
EXAMPLE 16.2: A charge of magnitude 4.50 μC is in an electric field. If the force it experiences is
6.975 N, what is the strength of the electric field? If the force is to the right, what direction is the electric
field pointing? Draw the field lines.
+4.50 μC
Draw a picture
F=6.975 N
+
known:
q = + 4.50 μC
F = 6.975 N
Need to find:
Electric field strength E.
Equation to use:
F = qE
Solve for the unknown
E=
E=
Plug in numbers.
F
q
6.975 N
4.50×10-6 C
= 1.55×106
N
C
Direction: This is a positive charge and we know that the direction of the electric field line is the direction a
positive charge would travel. If the force is to the right, the electric field also must be to the right
Answer:
E = 1.55×106
N
C
to the right
16.5. MAGNETIC FIELDS
Just as electric charges create electric
fields, magnetic charges create magnetic
fields. You can visualize a magnetic field
by placing a compass near the magnet and
watching for a deflection at different
points. Iron filings also could be used.
16.5.1. Rules for Magnetic Field Lines.
S
N
• Magnetic field lines always are
continuous. They run all the way
through the magnet. The lines that go
off the picture in Figure 16.5 go out to
infinity and back again.
• Magnetic field lines run from North to
South outside the magnet. The field
pattern for a bar magnet is shown in Figure 16.5: Magnetic field lines. Note the direction
Figure 16.53
of the arrows.
• Magnetic field lines – unlike electric
field lines – don’t terminate at a pole. Magnetic field lines continue inside the magnet.
• The closer the magnetic field lines are to each other, the stronger the magnetic field.
16.5.2. Units of Magnetic Field Strength. The magnetic field strength has units tesla (T).
N
Am
kg
= 2
As
Ns
=
Cm
T=
The ‘A’ stands for ‘ampere’, which is a unit of current we will use later. We will not calculate
magnetic field strengths.
16.5.3. Properties of the Magnetic Field. Like the electric field strength, the magnetic field
strength decreases as you move away from the source of the field. Interestingly, the magnetic field
decreases as the inverse cube of the distance from the source, compared to the electric field, which
decreases as the inverse square of the distance from the source.
The magnetic field lines point in the direction that the north pole of a magnetic dipole would point if
placed at that spot. You can use compasses around a bar magnet to illustrate this.
3
See also http://www.walter-fendt.de/ph14e/mfbar.htm. This applet shows the magnetic field lines
running through the magnet.
16.6. MAGNETIZING MATERIALS
All atoms have magnetic moments. Magnetic
moment is another name for magnetic dipole.
In the drawing on the left-hand side of
Figure 16.6, the magnetic moments of the
atoms are shown as small bar magnets with
the north pole red and the south pole blue. In
most materials, the domains are randomly
oriented.
If you place a magnet near the atoms, the
magnetic moments line up in opposition to the
field. The north poles of the atomic magnetic
moments are attracted toward the south pole
of the magnet. You can do this experiment Figure 16.6: The magnetic moments of atoms
using compasses and a bar magnet.
shown in a random state (left) and after a magnetic
16.6.4. Magnetic Polarization. In most field is applied via a bar magnet.
materials that haven’t been exposed to a
magnetic field, the magnetic moments are
randomly ordered, as shown in the left-hand
picture of Figure 18.8. Magnetic moments are
vectors, so if you were to add up all magnetic
moments, taking their directions into account,
you would find that there would be no net
magnetic moment.
If the material is placed in a magnetic field,
the moments line up so that the north ends are
pointing toward the south end of the magnet. The magnetic moments are randomly oriented
When the moments in a magnet align, the (left). When a magnet is brought near the material,
material has a magnetic polarization, which the magnetic moments line up (middle). If the
also is called magnetization. The sample now material is a paramagnet, the magnetic moments
return to a random orientation when the magnet is
has an overall net magnetic moment.
16.6.5. Paramagnets. In a paramagnet, the removed.
magnetic polarization remains only as long as the external magnetic field is applied. When the
magnetic field is removed, the moments go back to a random orientation, as shown in Figure 18.8.
Paramagnets include materials such as stainless steel and tin.
Some paramagnets retain their magnetic polarization after the field is removed because the process
that scrambles the directions of the moments is not complete. For examples, if you put a paper clip
near a strong magnet, you can make the paper clip act like a magnet itself. The magnetically
polarized paper clip will retain a net magnetic moment, even if the magnet is removed. After some
period of time, however, the magnetic polarization will disappear because there is nothing internal to
the material that makes the magnetic moments want to line up with each other. Temperature and
force also destroy magnetic polarization.
16.6.6.
Ferromagnets.
Ferromagnets or
permanent magnets, as shown in Figure 16.7,
behave very much like paramagnets, but they
retain their magnetic polarization when the
external magnetic field is removed. Only 3 of
the naturally occurring elements (iron, nickel and
cobalt) are ferromagnets at room temperature.
All magnetic materials contain one or more of
these elements, although other elements also are
present; however, many non-ferromagnetic
materials also contain iron, cobalt or nickel along Figure 16.7: In a ferromagnet, the atoms can
with other elements.
Bar magnets and initially be randomly oriented (left). When a
magnet is brought near the material, the spins in
horseshoes magnets are ferromagnets.
the material line up (middle), and when the
16.6.7. Destroying Magnetic Polarization. A
magnet is removed, the spins remain lined up.
ferromagnetic material also can be demagnetized by heating or physical damage. This is why classroom magnets must be re-magnetized,
which usually is accomplished by placing them near a very large magnet.
S
N
16.7. MAGNETIC DOMAINS
S N S N
Materials contain many, many atoms, and
S N
they usually don’t behave as a single unit.
Small areas within a magnet have the atomic
north and south poles lined up with each
other. These regions, which usually have
dimensions ranging from 0.01 to 1 mm are
called magnetic domains (or sometimes just
domains). The directions of the domains are
indicated by a single arrow, as shown in the
leftmost drawing of Figure 16.8. You can
think of it as bar magnets grouped together
N S
pointing in different directions, as shown in
N S N S
Figure 16.8. The head of the arrow points
north.
Figure 16.8: Two ways to think of domains.
All of the smaller bar magnets are pointing the
same way in some regions, but each region has an overall magnetization in a different direction. The
net magnetization – what you get when you add up all the moments of each small bar magnet – is
not very large. Ferromagnetic materials don’t generally come out of the ground already magnetized.
In many cases, the domains are randomly oriented. The size of domains is large enough that you can
see them using a magnetic field visualizer or iron filings.
S N
N
S
S
N
S
N
S
S N
N
N
S
S
S N
S N
N
S N
N
S
S
S
S N
S N
N
N
N
S
N
S
S
S N
N
N
S
S
N
S
N
N
S
S
N
16.8. APPLICATIONS
16.8.1. Information Storage. The ability of ferromagnets to retain their magnetic polarization is
exploited to store information on magnetic tapes and hard disks. Computer information is stored in
0s and 1s. Each of these corresponds to a section of the tape or disk where the domains point in one
direction or the other.
16.8.2. Metal Detectors. Many metals, when placed in a magnetic field, start generating their own
field. A metal detector works by sending brief pulses of magnetic field (about 100 pulses per
second). Any metal on a person change how quickly the magnetic pulse dies out and alerts the
inspectors to a potential problem. This works for metals that are attracted to normal magnets, as
well as metals like aluminum and gold that are not attracted to magnets.
16.9. MAGNET SAFETY
Magnetic fields extend over all space, so they affect things even when far away. Safety procedures
for magnets include:
• Magnets can change the functioning of pacemakers. Never bring a magnet near a pacemaker.
• Don’t put magnets near floppy disks, zip disks, credit cards, student IDs, or anything that stores
information.
• Don’t place a magnet near a television or computer monitor. You can permanently damage the
monitor because you will ruin the cathode ray tube inside (see Chapter 18 for cathode-ray tubes).
• The materials from which magnets are made have some structural properties that require extra
care.
• The strongest permanent magnets (that are made from materials like neodymium iron boride
and samarium cobalt and also are called ‘ceramic magnets’) can be very, very strong. When
bringing two strong magnets together, do not let fingers, ears, etc. get in between the
magnets. Some can snap together strongly enough that they can take off an ear. Keep a
piece of cardboard between the magnets.
• Ceramic magnets are very brittle. If they snap together, they can produce very sharp
fragments of metal that will become airborne if a magnet gets anywhere near them. It is a
good idea to buy magnets with a plastic coating, and to keep very strong magnets in their
own boxes so that if they do snap together, they don’t crack.
• When doing experiments with magnets, keep in mind that anything metal might be magnetic and
thus affect the results of your experiment. Computer monitors also can generate magnetic fields
that may disturb sensitive experiments.
16.10. THE EARTH’S MAGNETIC FIELD
A compass can be used for navigation because the Earth has its own (small) magnetic field. A
compass needle has a north and a south pole. The magnetic field from the needle interacts with the
magnetic field of the Earth, causing the needle to rotate. The Earth’s magnetic field is about
5.5 × 10-5 T. For comparison, the field near a magnet that might be used in a classroom can be on
the order of 0.1 T. The Earth is a magnetic dipole – it has the same field line pattern as in
Figure 16.5.
16.10.1. Orientation of the Earth’s Magnetic Field. What we call the North Pole of the Earth is
not a magnetic north pole. The north pole of a compass needle points toward geographic North;
however, since opposites attract, this means that the geographic North Pole is actually a south
magnetic pole. To make the situation even more complex, the geographic poles are not lined up
exactly with the magnetic poles – they are canted somewhat. Figure 16.10 shows the orientation of
the magnetic poles (shown by the magnet inside the Earth) and the location of the geographic poles
(shown by the N and S outside the Earth).
Figure 16.9: Internal composition of the Earth.4
Figure 16.10: The Earth as a bar magnet.
16.10.2. Why Does the Earth Have a Magnetic Field? The Earth’s magnetic moment is a direct
result of its composition. The Earth is made up of a solid crust, which overlays a semi-solid mantle.
Inside both of those is the Earth’s core, which has a liquid iron outer core and a solid iron inner core,
as shown schematically in Figure 16.9. The magnetic field near the Earth is from a combination of
three sources:
• 97 - 99 % of the magnetic field is due to electric currents in the outer core. We will see in
the next chapter that electric currents (which are moving charges) create magnetic fields.
• 1 - 2 % of the field is due to magnetized rock in the crust.
• 1 - 2 % of the field is due to charged particles above the Earth.
16.11. SOURCES, FIELD AND FORCES
It is important to have each of these items separately in your head.
• A source is an object that can create a field
o Charges are the sources for electric fields
o Pairs of poles are the source for magnetic fields
A field is a way for us to picture the effects that charges or poles have on each other. The electric
field of Figure 18.4 is a representation that tells us that if we place a positive charge anywhere near
these charges, exactly what will happen to it – what effect the force between the two objects will
have. One object can generate a field; however, forces always are calculated between two objects.
16.12. SUMMARIZE
16.12.1. Definitions: Define the following in your own words. Write the symbol used to represent
the quantity where appropriate.
1. Electric Field
4
http://denali.gsfc.nasa.gov/research/mag_field/conrad/explain.html
2.
Magnetic Field
3.
Source
4.
Permanent magnet
5.
Magnetic polarization
6.
Paramagnet
7.
Ferromagnet
A
8.
Magnetic domain
9.
Magnetic moment
16.12.2. Equations: For each question: a) Write the
equation that relates to the quantity b) Define each
variable by stating what the variable stands for and the
units in which it should be expressed, and c) State whether
there are any limitations on using the equation.
1. The electric field strength as a function of distance
from the charge.
2.
The relationship between the force exerted by an
electric field on a charge and the electric field
strength.
16.12.3. Concepts: Answer the following briefly in your
own words.
1. In Figure 16.11, is the electric field stronger at point
A or point B? Justify your answer. Which direction
would a negative charge move if placed at point A?
Which direction would a positive charge move if
placed at point B? Answer the same questions for
Figure 16.12.
B
Figure 16.11: Electric field lines
associated with two positive charges
with the same magnitude.
B
A
Figure 16.12: Electric field lines
associated with one positive and one
negative charge with the same
magnitude.
2.
Can you tell whether Figure 16.12 shows a magnetic or an electric field? Can you tell whether
Figure 16.11 shows a magnetic or an electric field?
3.
What are the differences between magnetic field lines and electric field lines?
4.
You can’t see electric or magnetic fields. How might you demonstrate the existence of an
electric or magnetic field?
5.
What is an electric field?
6.
Explain what domain wall motion is and how it relates to magnetic polarization.
7.
Explain the difference between a source, a field and a force.
8.
If you move three times further from a charge, by how much does the electric field strength
change? Does it matter whether the
charge is positive or negative?
9.
Which of the two charges shown in
Figure 16.13 is stronger? Are they both
positive, both negative or is one positive
and one negative?
10. The Earth’s North magnetic pole is a)
located at the geographic North Pole, b) a
magnetic south pole, c) a magnetic north
pole, d) located at the geographic South
Pole.
Figure 16.13: Electric field lines from positive
(left) and negative (right) charges.
11. How do the number of magnetic flux lines coming from one side of a permanent magnet
compare with the number of magnetic flux lines coming into the other side?
12. You touch the north pole of a permanent magnet to the end of a paper clip. What pole will the
end of the paper clip nearest the permanent magnet acquire and why?
13. Why is a piece of iron attracted to either pole of a magnet?
14. Hammering on a permanent magnet can demagnetize it. Explain why.
15. How does a magnetized piece of iron differ from an unmagnetized piece of iron?
16.12.4. Your Understanding
1. What are the three most important points in this chapter?
2.
Write three questions you have about the material in this chapter.
16.12.5. Questions to Think About
1. A thin magnet, like the type you put on
your refrigerator, sticks only on one side.
Explain how this can be. Hint: think
about it using domains. What would the
domains look like?
2. What do you think the magnetic field
would look like for a horseshoe magnet?
3. If you bring two magnetic compasses
near each other, they can attract each
other. Can they ever repel? Explain.
16.12.6. Problems
1. The electric field at a distance of 30.0 cm
from an electric blanket is 90.0 NC . If we
assume that the charge responsible for the
electric field is a point charge, what is the
magnitude of the charge producing this
field?
2. A positive charge produces an electric
field of 205 NC at a distance of 31.5 m
directly to the right. What force
(magnitude and direction) would a
negative
charge
of
magnitude
-5.05 × 10-6 C feel if it were located
31.5 m directly to the right of the positive
charge?
3. The arrows were removed from
Figure 16.14. Is it possible for you two
determine whether the charges are
positive or negative, and their relative
magnitudes. Explain your reasoning in
detail.
Figure 16.14: Electric field lines..
Figure 16.15: Magnetic field lines
4.
In Figure 16.15 which end of the magnet (A or B) is the north pole?
PHYS 261 Spring 2007
HW 17
HW Covers Class 16 and is due February 16, 2007
1.
2.
3.
The electric field at a distance of 30.0 cm from an electric blanket is 90.0 NC . If we assume that
the charge responsible for the electric field is a point charge, what is the magnitude of the
charge producing this field?
Explain why a piece of iron will be attracted to either side of a magnet in terms of the magnetic
moments in the iron.
A positive charge produces an electric field of 205 NC at a distance of 31.5 m directly to the
right. What force (magnitude and direction) would a negative charge of magnitude -5.05 μC
feel if it were located 31.5 m directly to the right of the positive charge?