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CLASS 17. : THE ELECTRICITY AND MAGNETISM CONNECTION
17.1. INTRODUCTION
We have investigated interactions between electric charges and interactions between magnetic poles;
however, electric fields can influence magnetic poles and magnetic fields can influence electric
charges. These interactions become important, however, only when one of the elements is moving
relative to the other. (i.e. a moving charge in a stationary magnetic field.)
17.2. GOALS
• Understand that current is composed of moving charges
• Understand that moving charges produce magnetic field and be able to determine the
direction of the field produced by positive or negative charges.
• Understand how a magnetic field exerts a force on a charged particle and calculate the
resulting motion.
17.3. CURRENT
Most of the initial studies of electricity were focused on static electricity –electrical charges that are
not moving. Du Fay built the first device to generate an electric spark about 1650 and many other
scientists explored the phenomenon; however, electricity appeared to be a curiosity with no practical
applications. If electricity were to be harnessed, there had to be a way to store and transport it. In
1729, Stephen Gray showed that electricity can be transferred from place to place via conducting
wires. The flow of charges is called a current. In 1745, Pieter van Musschenbroek invented the
Leyden jar, which allows the storage (and transportation) of charge. In 1793, Volta demonstrated
the first battery, which used chemical reactions to generate current. We now had the ability to
generate, store and transport charge. These inventions led to the ‘golden age’ of electricity and
magnetism between 1750 and 1850. In addition to fundamental understanding of these phenomena,
the telegraph (1837), the telephone (1876), the electric light bulb (1880) and many other tools were
invented during this period.
17.4. MOVING CHARGES PRODUCE MAGNETIC FIELDS.
17.4.1. Discovery. Hans Christian Oersted
Hans Christian Oersted (1777-1851) was born
discovered that moving charges produce a
just before the French Revolution, as people
magnetic field during a lecture demonstration for
started focusing on practicality in science. He
a group of students in 1820. He had set up a
was trained as a philosopher, but became an
wire near a compass needle. When he flipped
experimental physicist. His first work was on
the switch that started the charges moving
the identity of electrical and chemical forces; his
through the wire, the compass needle moved.
work formed the foundation for Berzelius’ later
The only thing that would have made the
work. Oersted worked on a number of practical
compass needle move would be a magnetic field.
projects, including the compressibility of water
Since he had no magnets nearby, Oersted
and the use of electric currents to explore mines.
repeated the experiment and realized that the
only possible source of the magnetic field was the wire. A series of experiments demonstrated that
moving electrical charges creates a magnetic field. You can duplicate his experiment. Open the
hood of a car (while the car is not running). Locate the battery and the large cables that carry current
to and from the battery. Close the hood and hold a compass just over the position of one of the
cables. Have a friend start the car and watch what happens to the compass.
17.4.2. The Direction of the Magnetic Field Due to a
Moving Charge: Right-Hand Rule #1. The direction of the
magnetic field produced by a moving charge is found using
the first of two Right-Hand Rules we will apply in this
chapter. We will call this rule RHR #1. RHR #1 is used to
find the magnetic field generated by a moving charge.
To use RHR #1: place your hand in a ‘thumbs up’ sign, as
shown in Figure 17.1. If the charge is positive, point your
thumb in the direction the charge is moving. If the charge is
negative, point your finger in the direction opposite the
motion of the charge. The magnetic field direction at every
point is given by the direction of your fingers. This gets a
little tricky because we have to work in three dimensions.
Consider the case in which a positive charge is moving out of
the page toward you, as shown in Figure 17.2. The magnetic
field is in a circle around the wire that is defined by the
direction of your fingers. The field direction changes, so the
field direction is different at different points around the wire.
At point A, the magnetic field is up. At point B, the
magnetic field is to the left. At point C, the magnetic field is
down and at point D, the magnetic field is to the right.
17.4.3. Notation. Working in three dimensions requires us
to specify whether a magnetic field is going into or out of the
page. By convention, we use a ‘×’ to indicate that the field is
going INTO the paper. We use a dot to indicate that the field
is coming out of the paper. This convention originated
because we represent vector quantities using an arrow, with
the tip indicating direction. If an arrow is coming toward
you, you see a dot (the tip) and if it is headed away from you,
you see the crossed feathers (which form a ‘×’). Figure 17.3
shows the directions.
Figure 17.1: RHR #1: If the thumb
points in the direction of positive
charge motion, then the magnetic
field points in the direction of the
curved fingers.
B
C
A
D
Figure 17.2: Looking at a positive
charge moving out of the paper
toward you, the direction of the
magnetic field is counterclockwise.
X
Into the Page
·
Out of the Page
Up
Left
Down
Right
Figure 17.3:
Conventions
indicating directions.
for
EXAMPLE 17.1: A positive charge moves from right to left. What is the direction of the field above the
path of the charge? What is the direction of the field below the path of the charge?
Above
+
Draw a picture
Below
known:
the charge is positive
need to find:
The direction of the magnetic field above and
below the path of the charge.
Again, no equation is necessary here. Put the thumb of the right hand pointing to the left. Above the path,
your fingers point away from you, or into the page. Below the path, your fingers point toward you or out of
the page.
Answer:
Above the path, the magnetic field points into the page. Below the path, the magnetic
field points out of the page. We indicate this by an X above the path and a dot below
Check:
Look at your fingers when they are cupped. Anytime you are looking at a path, the
magnetic field direction above and below will be in opposite directions – it’s
impossible to make your hand do anything else. The same holds for the left and the
right.
17.5. LORENTZ’S LAW: A MAGNETIC FIELD EXERTS A FORCE ON ELECTRICAL CHARGES.
Two magnets will either repel or attract one another when their magnetic fields interact. If a moving
electrical charge creates a magnetic field, that magnetic field will interact with any other magnetic
field in the region. This means that a magnetic field will exert a force on an electrical charge.
17.5.1. Magnitude of the Force on a Moving Charge. Lorentz’s Law says that a charge of
magnitude q, moving at velocity v perpendicular to a magnetic field B will experience a force F
equal to
F = qvB
(17.5.1)
Note that this formula is ONLY valid when the velocity of the charge is perpendicular to the
direction of the magnetic field. A more complicated expression is required if the angle between v
and B is other than 90 degrees. When you apply Equation (17.5.1), you only use the absolute values
of the charge, velocity, and magnetic field. You will use the right-hand-rule described below (RHR
#2) to determine the direction. The force will always turn out to be positive number.
17.5.2. Units for the Force on a Moving Charged Particle in a Field. The units for Equation
(17.5.1) will be:
F = qvB
⎛m⎞
=C ⎜⎜ ⎟⎟⎟ T
⎜⎝ s ⎠
CmT
s
To unravel this, we need to use the definition of tesla and a piece of new information, which is that
the unit of current, the ampere (A), is a coulomb over a second.
A=
C
s
The tesla is:
N
Am
kg m s
= 2
s Cm
kg
=
sC
T=
Plug this into the units from the equation F = qvB.
CmT
s
C m kg
=
sCs
m kg
= 2
s
=N
Units of F =
The units of force do indeed turn out to be N.
17.5.3. The Direction of the Force on a Charged
Particle – Right-Hand-Rule #2. Determining the
direction of the force on a charged particle requires
using a second ‘Right-Hand Rule’ that is shown in
Figure 17.4. Extend your hand so that your fingers
point in the direction of v and then rotate your fingers
so they point in the direction of B. There is only one
way to do this without pain. Your thumb now points
in the direction of the force that would be exerted on a
positive charge. If the charge is negative, then the
force is in the direction 180 degrees opposite the
direction your thumb is pointing.
Figure 17.4: The right-hand rule for
finding the direction of the force. (From
Hecht’s Physics/Algebra book).
There is a second version of RHR #2. Place your hand as if you were making a ‘stop’ sign with it.
Point your thumb in the direction of the velocity and your fingers in the direction of the magnetic
field. Again, there is only one painless way to do this. When your hand is in this position, the force
on a positive charge will be in the direction your palm faces. The force on a negative charge will be
in the opposite direction. Both versions of RHR #2 only apply when the velocity of the particle and
the magnetic field are perpendicular to each other.
v and B always define a plane and F always is perpendicular to that plane. If you are working a
problem and find any two of these quantities lying in the same plane, you have made a mistake. For
any configuration of v and B, there are only two possible answers for the direction of F.
EXAMPLE 17.2: A positive charge moves toward the right in a magnetic field that points downward. What
is the direction of the force on the charge?
B
Draw a picture
+
known:
the charge is positive
the magnetic field is down
need to find:
v
The direction of the force felt by the positive
charge.
No equation is necessary here. Apply RHR #2. You should find that the direction of the force is into the
page.
Answer:
Into the page
Check:
v, B and F are each perpendicular to each other and none are in the same plane.
EXAMPLE 17.3: A positive charge of magnitude 1.60 × 10-19 C moves in the +x direction (to the right) with
velocity 2.10 x105 m/s. If a magnetic field of magnitude 3.00 T is applied in the +y direction, what force is
experienced by the charge?
q
v
Draw a picture
B
known:
v = 2.10×105 ms right = velocity of the
charge
need to find:
q = 1.60 × 10-19 C = value of the charge.
B = 3.00 T up = magnetic field
The magnitude and direction of the
force felt by the positive charge.
Equation to use
(already solved for
unknown)
F = qvB
F = (1.60×10-19 C )( 2.10×105
m
s
) ( 3.00T )
Cm Ns
s Cm
-13
=1.01×10 N
=1.01×10-13
Plug in numbers.
Find the direction using RHR #2
F = 1.01×10-13 N out of the page
Answer:
Check:
v, B and F are each perpendicular to each other and none are in the same plane.
17.6. STEERING CHARGES WITH A CONSTANT MAGNETIC FIELD
Since a magnetic field exerts a force on a moving charged particle, it makes sense that one should be
able to use a magnetic field to ‘steer’ a charged particle. Figure 17.5 shows a positively charged
particle moving into a uniform (constant) magnetic field. When the charge enters the magnetic field
at position 1, it will feel a force (which is downward in this case). This force changes the direction
the particle is moving. At some later time, the particle is in position 2.
The moving charge still feels a force from the magnetic field; however, since the velocity is in a
different direction now, the force will be in a different direction as well. Every time the particle
moves, the direction of the force changes. The net result is that a charged particle with charge q and
mass m moving at speed v in a constant magnetic field B moves in a circle of radius R, as shown in
Figure 17.6. The relationship between the speed, magnetic field and radius of the circle is given by
Equation (17.6.1).
mv = qBR
(17.6.1)
The absolute value of the charge is used in Equation (17.6.1) because the radius can never be
negative. The direction of the arc must be determined using right-hand rule #2.
F
1
q
1
F
q
2
R
3
F
Figure 17.5: The motion of a positive charge Figure 17.6: The radius of the circle in which
as it moves through a magnetic field
the charged particle moves due to the magnetic
field.
EXAMPLE 17.4: A negative charge q = -1.60 × 10-19 C moves into a constant magnetic field. The mass of
the charge is 9.11 x 10-31 kg, the magnetic field is 7.50 mT out of the page, and the velocity is 2.05 x 107 ms to
the right. a) What is the radius of the circle the charge will travel in? b) What is the direction in which the
charge will travel if the magnetic field points out of the paper?
1
q
Draw a picture
v = 2.05×107 ms right = velocity of the charge
known:
q = -1.60 × 10-19 C is the value of the charge.
m = 9.11×10-31 kg is the mass of the charge
B = 7.50 mT out of the page = magnetic field
mv = qBR
Equation to use
R=
Solve for the unknown
R=
Plug in numbers.
R = The radius of the circle the
charge will travel in and the
direction of travel
need to find:
mv
qB
9.11×10-31kg ( 2.05×107
m
s
1.60×10-19 C ( 7.5×10-3T )
= 0.0155629167
= 1.56 × 10−2
)
kg m
sCT
kg m 2 C
sCNs
= 1.56 × 10−2 m
Using the right-hand rule, point your fingers in the +x direction (to your right as
you look at the page), in such as way that you can bend your fingers out of the
Find the direction using paper to be in the direction of B. This shows that the force is down; BUT don’t
RHR #2
forget that the charge is negative, so the direction is actually the opposite, and
the force on the negative charge is UP. This means that the charge will move in
a circle of radius R upward from its initial position.
Answer:
Check:
R = 1.56 × 10−2 m (upward circle)
The examples in the text were done with the same velocity and field directions,
but a positive charge. Those circles were downward. We have a positive charge
here, so it makes sense that the direction of the arc is upward in this case.
17.7. SUMMARIZE
17.7.1. Definitions: Define the following in your own words. Write the symbol used to represent
the quantity where appropriate.
1. Current
2.
Static Electricity
17.7.2. 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 expression for the magnitude
of the force exerted by a magnetic
field on a moving charge
2.
The expression that determines the
movement of a charged particle in
a semicircle in a constant magnetic
field.
B
I
A
17.7.3.
Concepts:
Answer the
following briefly in your own words.
Figure 17.7: Charges flowing through a wire.
1. Under what circumstances can an
electrical charge produce a magnetic field?
2.
Explain the circumstances in which you use RHR #1 and in which you use RHR #2.
3.
A horizontal wire has negative charges moving to the right, as shown in Figure 17.7. What is
the direction of the field at the points marked “A” and ‘B’?
4.
Which of the following are true? a) a moving electric charge creates a current; b) a moving
electric charge creates a magnetic field; c) a moving magnetic field creates an electric charge;
d) a stationary magnetic field creates a moving charge.
5.
If you reverse the direction a current runs in a wire (i.e. the charge moves the other way), the
magnetic field around the wire a) is oriented in the same way as it was before; b) is oriented
oppositely to the way it was before; c) flips to become aligned parallel to the length of the wire;
d) ceases to exist because the new current cancels the old.
6.
Which of the following are true? a) A positively charged particle moving to your right, in a
magnetic field going up will move out of the paper; b) A positively charged particle moving to
your right, in a magnetic field going up will move into the paper; c) A positively charged
particle moving up, in a magnetic field pointing down will move right; d) A positively charged
particle moving up, in a magnetic field pointing down will not be deflected; e) Only positively
charged particles can be deflected by a magnetic field.
17.7.4. 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.
17.7.5. Questions to Think About
1. Can an electric field cause a charged particle to move in a curved path?
2. In a television, a beam of negative charges is responsible for form the image on the screen. If
the beam of negative charges is coming toward you, in what direction is the magnetic field?
17.7.6. Problems
1. A negative charge of 5.60 μC moves in the -y direction with velocity 5.20 × 105 ms . If a
magnetic field of magnitude 7.00 T is applied in the +x direction (to the right), what is the force
experienced by the charge? (magnitude and direction)
2. A charge of 5.654 x 10-10 C moves in the +y direction with velocity 7.25 × 105 m/s. If the force
experienced by the charge is 1.25 × 10-5 N down, what is the magnitude and direction of the
magnetic field?
3. A charge of -5.61 μC moves into a constant magnetic field of 21.4 mT. The velocity is
6.50 × 107 ms down. a) If the radius of the circle the charge travels in is 5.78 m, what is the
mass of the charge? b) What direction does the charge will travel if the magnetic field points
out of the paper?
PHYS 261 Spring 2007
HW 18
HW Covers Class 17 and is due February 19, 2007
1.
2.
3.
Explain the circumstances in which you use RHR #1 and in which you use RHR #2. (In other
words, how do you know when to use one rule and not the other one?)
A charge of 5.654 x 10-10 C moves to the right with velocity 7.25 × 105 m/s. If the force
experienced by the charge is 1.25 × 10-5 N down, what is the magnitude and direction of the
magnetic field?
A charge of -5.61 μC moves into a constant magnetic field of 21.4 mT. The velocity is
6.50 × 107 ms down. a) If the radius of the circle the charge travels in is 5.78 m, what is the
mass of the charge? b) What direction does the charge will travel if the magnetic field points
out of the paper?