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 each other.) 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:

+ -

Figure 16.1: Electric field lines from positive (left) and negative (right) charges.

• 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 two positive charges with the same magnitude.

Figure 16.3: Electric field lines associated with one positive and one negative charge with the same magnitude.

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.

16.3.2. Calculating Electric Field Strength. E is the electric field strength at a distance r away from a charge of magnitude q.

2

qE kr

= (16.3.1)

The units of electric field strength are:

++

+++

++++++

+ +++++

++

+

q

Aq

B

Figure 16.4: Electric field lines between two plates with opposite charge.

2

2

2 2

Nm CC m

N=C

qE kr

=

⎛ ⎞⎟⎜= ⎟⎜ ⎟⎜⎝ ⎠

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:

N m1C=V

We can then write

N 1C

NC

N VNm

Vm

⎛ ⎞⎟⎜= ⎟⎜ ⎟⎟⎜⎝ ⎠⎛ ⎞⎟⎜= ⎟⎜ ⎟⎟⎜⎝ ⎠

=

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?

Draw a picture for part a +4.50 μC

0.0500 m

known: 4.50 μCq =+

0.0500 mr = Need to find: E =the electric field due to charge 1 at a distance of

5.00 cm from the charge.

Equation to use: 2

qE kr

=

Plug in numbers.

( )( )

629

22

7

4.50 10 C9.00 10

0.0500 m

1.62 10 NC

NmEC

−×= ×

= ×

Answer: 7at 5.00 cm 1.62 10 rightNC

E = ×

Check your answer 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.

Draw a picture for part b +4.50 μC

0.1000 m

known: 4.50 μCq =+

0.1000 mr = Need to find: E =the electric field due to charge 1 at a distance of

10.00 cm from the charge.

Equation to use: 2

qE kr

=

Plug in numbers.

( )( )

629

22

7

4.50 10 C9.00 10

0.1000 m

4.05 10 NC

NmEC

−×= ×

= ×

Answer: 7at 10.00 cm 4.05 10 rightNC

E = ×

Check your answer 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.

How are they related to each other? Take the ratio.

7

7

4.05 10at 10.00 cmat 5.00 cm 1.62 10

14

NCNC

EE

×=

×

=

Check your answer The ratio of the distances is 2; however, the electric field is an inverse square law, so the decrease in field is 2

12

or 14

16.3.3. Typical Electric Field Strengths. Table 16.1 shows the electric field strengths of some representative sources.

Source Field strength

NC

⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜⎝ ⎠

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 2 x 1021

Table 16.1: Electric field strengths.

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).

1 212 2

q qF kr

= (15.6.1)

Re-write this by moving q2 out front

112 2 2

qF q kr

⎛ ⎞⎟⎜= ⎟⎜ ⎟⎟⎜⎝ ⎠

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

12 2F q 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.

Draw a picture +4.50 μC

F=6.975 N

+

known: 4.50 μCq =+

6.975 NF = Need to find: Electric field strength E.

Equation to use: F qE=

Solve for the unknown FEq

=

Plug in numbers. -6

6 NC

6.975 N4.50×10 C1.55×10

E =

=

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: 6 NC

1.55×10 to the rightE =

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. • 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 16.53

• 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).

2

NT=Amkg=AsN s=C m

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.

S N

Figure 16.5: Magnetic field lines. Note the direction of the arrows.

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 using compasses and a bar magnet. 16.6.4. Magnetic Polarization. In most 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. When the moments in a magnet align, the material has a magnetic polarization, which also is called magnetization. The sample now has an overall net magnetic moment. 16.6.5. Paramagnets. In a paramagnet, the 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.

Figure 16.6: The magnetic moments of atoms shown in a random state (left) and after a magnetic field is applied via a bar magnet.

The magnetic moments are randomly oriented (left). When a magnet is brought near the material, the magnetic moments line up (middle). If the material is a paramagnet, the magnetic moments return to a random orientation when the magnet is removed.

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 with other elements. Bar magnets and horseshoes magnets are ferromagnets. 16.6.7. Destroying Magnetic Polarization. A ferromagnetic material also can be de-magnetized 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.

16.7. MAGNETIC DOMAINS Materials contain many, many atoms, and 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 pointing in different directions, as shown in Figure 16.8. The head of the arrow points north. 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.

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

Figure 16.7: In a ferromagnet, the atoms can initially be randomly oriented (left). When a magnet is brought near the material, the spins in the material line up (middle), and when the magnet is removed, the spins remain lined up.

N S

S NS NS N

N SN S

S

NS

N

S

N

S

NS

N

S

NS

N

S

NS

N

S

NS

N

S

NS

N

S NS N

S N

S NS N

S NS N

S NS N

S NS N

S NS N

Figure 16.8: Two ways to think of domains.

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).

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

Figure 16.9: Internal composition of the Earth.4 Figure 16.10: The Earth as a bar magnet.

2. Magnetic Field

3. Source

4. Permanent magnet

5. Magnetic polarization

6. Paramagnet

7. Ferromagnet

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

A

Figure 16.11: Electric field lines associated with two positive charges with the same magnitude.

A

B

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.

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?

Figure 16.13: Electric field lines from positive (left) and negative (right) charges.

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 N

C 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. 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. Explain why a piece of iron will be attracted to either side of a magnet in terms of the magnetic moments in the iron.

3. 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?