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Honors Physics
Unit 10: Magnetism
Name:____________
Permanent Magnets
Certain types of stones were found in ancient times to be able to attract
metal over short distances. These were called lodestones, and in modern
terms we would say that the stones have been magnetized naturally. We
have been able to make powerful permanent magnets by mimicking the
formation of lodestones. We find that permanent magnets have two poles,
which we label N and S.
Play with the magnets you have and determine the type of behavior
the magnets exhibit by drawing arrows indicating the force you
feel when you put like poles near each other vs. when you place
unlike poles near each other.
N
N
S
N
S
S
N
S
Like poles: ___________ Unlike poles: ___________
In ancient Greece, stones were discovered that behaved like the bars above. They were eventually called magnets after
the Greek region called Magnesia. The ends of the bars behave in a way that reminds us of electric charges; in this sense,
bar magnets like the ones above act as dipoles.
Discussion question: If one were to cut a bar magnet in half, as shown to the right, would you
expect those halves to attract or repel? Explain, and indicate the appropriate letter at each
end of the new bars.
S
?
N
If we could isolate an N or S pole by itself, it would be called a monopole; so we say that as far as we know, magnetic
monopoles do not exist. There is a great deal of speculation, however, about the existence of magnetic monopoles early in
the history of the universe.
N
One of the common uses of permanent magnets is as compass needles; a permanent magnet is suspended at
the middle and allowed to rotate to an equilibrium position.
S
Magnetic Field: Direction
Compass
We have defined several fields so far, such as the electric field E. We will define the magnetic field B in a
similar fashion. It is a vector field, so we have to define both its direction and the magnitude. We define the direction of
the magnetic field at any point as simply:
The direction of the magnetic field at any point is the direction that a compass needle points
when placed there.
Discuss with your group and draw predicted vector arrows to indicate
the direction of the magnetic field at points A-E on the diagram at
right.
Download the PhET simulation Magnet and Compass from the web
site. Move the simulated compass to the corresponding points to check
your predictions.
B
•
A
•
S
N
D
•
C
•
E
•
With your group, predict what the magnetic field lines will look like
inside the magnet. Draw your predicted lines. On the PhET simulator,
check the “See Inside Magnet” checkbox. Verify that your prediction about the strength and direction of the field inside
the magnet is correct.
Honors Physics
Unit 10: Magnetism
On the diagram to the right, sketch the complete magnetic field lines through the
magnet and all around. The field lines should be closer together where the field
is stronger, and farther apart where it is weaker. We arbitrarily choose the
direction in which the N pole of a compass points to be the direction of the
magnetic field. Draw arrowheads on your field lines to indicate the direction of
the field.
S
N
Compasses have been used for centuries to guide travelers on land and sea. On
Magnetic Dipole
the PhET simulation, check the “Show planet Earth” checkbox. Move the
compass along the surface of the Earth, describe what you find, explain why the letters “S” and “N” were chosen to label
the poles of magnetic dipoles and why the Earth’s magnetic poles are labeled as they are.
Magnetic dipoles are similar in many ways to electric dipoles, but there is a major
difference. On the diagram to the right, sketch a few electric field lines in the region of
the electric dipole. Look in particular at the region between the poles.
–q
+q
a) What is the major difference between the fields?
b) What is the source of electric field lines?
c) What is the source of magnetic field lines?
The way we represent the magnetic field is similar to how we represent the electric field. The magnetic field lines always
go from one pole (N) to the opposite pole (S), similar to how electric field lines always go from one charge (+) to the
opposite charge (-). Also, the strength of the magnetic field can be seen by the density of the field lines: where the lines
are closer together, the magnetic field is stronger. Both magnetic and electric field lines never cross, and they are not
actual or physical lines, they are representations of the force created by the presence of something magnetic.
d) What is the biggest difference between drawing electric field lines and magnetic field lines?
The earth is not the only solar system object with a strong magnetic field; the sun has a very powerful and complex
magnetic field, some planets (Jupiter, Saturn) have strong fields while other planets (Mercury, Venus) don't.
Magnetic Force
Both electric and magnetic forces act on charges. Electric force depends only on the charge, and not on its state of
motion. Magnetic force, however, depends not only on the charge but also on its state of motion; this means that if a
charged particle is NOT moving, it will not feel any magnetic force. In order for a charged particle to feel a magnetic
force, it must not only be moving but it must have a component of its velocity perpendicular to the magnetic field. The
interaction between velocity, magnetic field and force create a three-dimensional relationship.
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Honors Physics
Unit 10: Magnetism
The magnitude of the magnetic force depends directly on the size of the charge qo and on the component of the velocity
perpendicular to the magnetic field. If we call the angle between the velocity vector and the magnetic field θ, then the
force is proportional to qo v sin θ:
F ∝ qov sin θ
Magnetic Field: Magnitude
We define the magnitude of the magnetic field B as the proportionality constant between the magnetic force and qv sin θ.
a) Rewrite the proportionality above as an equation:
b) When θ = 90°, what does the equation simplify to?
c) Solve the equation from b) for the magnetic field B:
We now have our three dimensional relationship among three
vectors: B, F, and v.
F
In order to remember how they relate, we use the Right-Hand Rule, which applies to
positive charges:
B
Align the fingers of your right hand with the magnetic field B, and point your thumb
in the direction of the component of v that is perpendicular to B. The palm of your
hand will push in the direction of the magnetic force on the positive charge.
Negative charges obey the similar Left-Hand Rule.
Since we often have to draw vectors in three dimensions on two-dimensional paper,
we have a convention for drawing vectors perpendicular to the page. We think of the
v
vector as an arrow. “•” (the tip of the arrow) represents a vector coming out of the
page, and “ × ” (the tail feathers of the arrow) represents a vector going into the page.
Practice Problems
1. For each of the following, state the direction of the magnetic force on the given moving charge. Note that vectors
going into the page are symbolized by × and vectors coming out of the page are symbolized by •.
!
B into the page
!
v
+q
• ! • • • •
B out of the page
• • • • •
• • • v! • •
• • • • •
–q
• • • • •
•
•
!
B
!
B
!
v
•
•
+q
!
v
–q
•
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Honors Physics
Unit 10: Magnetism
2. The MKS unit of magnetic field is called the Tesla. What is the Tesla equivalent to in SI units?
3. The photograph to the right shows an electron
gun, which shoots a beam of electrons to the right.
What is the direction of the magnetic field to the
right of the gun that deflects the electrons?
Explain how you determine this.
4. Discussion Questions:
a. Where does the maximum magnetic force on a moving charge occur? At what angle between the velocity
and the magnetic field?
b. How does the force due to a magnetic field affect the speed of the moving charged particle?
c. How does the force due to a magnetic field affect the direction of the moving charged particle?
d. If no other forces are present, what will be the path of a charged particle moving perpendicularly to the
direction of the magnetic field?
5. Four particles follow the paths shown to the right. What are the signs of the charges?
1:
2:
3:
4:
6. Imagine that you are in a room with your back to one wall, and that an electron beam,
traveling horizontally from the back wall to the front wall, is deflected to your right.
What is the direction of the magnetic field that exists in the room?
2
1
3
4
7. Imagine the room in which you are seated to be filled with a uniform magnetic field B pointing vertically
downward. At the center of the room two electrons are suddenly projected horizontally with the same initial
speed but in opposite directions. Describe and sketch their motions.
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Honors Physics
Unit 10: Magnetism
8. Two particles with mass m and charge +q move with a
velocity v as shown to the right. One is in a magnetic field
B and the other is in an electric field E.
a. What are the mathematical names for their two
different paths?
ur
B
m
ur
E
r
v
r
v
m
+q
+q
b. Draw the different shapes on the images.
c. Explain why these shapes are different.
9. A magnetic field B points into the page. A charge +q with mass m moves to the right at
velocity v.
a. What is the direction of the electric field E that will cancel the magnetic force and
allow the charge to move in a straight line?
!
B
m
!
v
+q
b. What is the magnitude of this electric field E in terms of B, +q, m and v?
10. A particle of mass m and charge +q moves in a circle path, perpendicular to a
magnetic field B at a velocity v as shown to the right.
a. What is the radius r of the path of the charge in terms of these quantities?
B
r
+q
m
b. How much work is done by the magnetic force as the charge moves in a
v
quarter circle? Explain your answer.
11. An electric field E of 1,500 V/m and a magnetic field B of 0.5 T act on an electron that is moving perpendicular
to both fields but is experiencing no net force.
a. Calculate the speed v of the electron
b. Sketch the vectors E, B and v, showing their relationship to each other.
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Honors Physics
Unit 10: Magnetism
Magnetic Force on a Current
L
We have seen that a charge moving in a magnetic field experiences a force. We
can think of a wire carrying current I as a hollow tube of length L in which
positive charges are moving at a certain velocity v. These charges feel a
magnetic force given by the equation you determined on page 3.
I
a) What is the equation describing the force felt by a moving charge in
a magnetic field?
q
b) What does that equation simplify to when θ = 90°?
v
ur
B
c) What are the dimensions of qv in units?
d) Rearrange the units so that you can replace with the unit for current.
e) Replace the variables qv in your force equation with the variables that use the new units.
f) For a wire of length L, given in units of meters and with a current of strength I, given in units of A, what is
the force this wire feels when it is placed in a magnetic field?
g) Since you simplified the setup in b), what would this equation be if θ was not 90°?
Discussion Question: Current is downward in a wire that is in a magnetic field directed horizontally to the south. In
what direction is the force on the wire? Explain how you use Right Hand Rule to get your answer.
Practice Problems
1. A current segment 3 cm long is placed broadside to a magnetic field of 1.2 T. The current in the wire is 2000 A.
a) What is the magnetic force on the wire?
b)
Could a strong person supply enough force to hold the wire in equilibrium? Explain why or why not.
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Honors Physics
Unit 10: Magnetism
2. A current segment 10 cm long is placed perpendicular to the magnetic field of the earth, which is about 50 µT.
a)
What is the current in the wire if it experiences a force of 1 N?
b)
The currents you measured in the circuit lab were on the order of a few hundred mA. How much force would
this wire feel if the current were 500 mA?
Magnetic Fields Produced by Currents
That which feels the force of a gravitational field (namely mass) is also what causes the gravitational field to exist.
Electric charge plays the same dual role for electric field; it both feels the force of an existing electric field and
causes electric fields to exist.
The same is true of the magnetic field; that which feels the force also causes the field. We have seen that moving
charges, and currents in particular, feel the magnetic force. We now look at the other side of the coin: currents
produce magnetic fields. Whenever a charge oscillates, it will create an electric field and a magnetic field. As this is
difficult to conceptualize, take a look at the following animations (you can access the links from fizzixprof.com).
http://www.learnerstv.com/animation/animation.php?ani=89&cat=physics
http://physics.gac.edu/~chuck/PRENHALL/Chapter%2025/AABXTEN0.html
wir
e
See how this relates to the airport walk through metal detectors here.
Note that the magnetic field lines loop around the wire in a
counterclockwise direction when viewed from above, with the current
coming toward the observer. In order to remember this relationship,
we use the second Right-Hand Rule. Imagine grasping the wire with
the thumb of your right hand pointing in the direction of the current.
Your fingers wrap around the wire in the direction of the field.
I
!
F
#2
Right-Hand Rules
Compasses
Wire
As a reminder, we how have two RightHand Rules:
!
B
!
B
Pla
ne
per
pen
The simplest current is just a straight wire. If we investigate a
current-carrying wire with a compass to map out the magnetic field,
we would find that it loops around the wire in concentric circles, as
shown to the right.
I
I
dic
ula
r to
The Field of a Long, Straight Wire
#1
RHR1 is with an open hand, where your palm
indicates the direction of the magnetic force.
RHR2 is with a closed hand, so force isn't
involved. In both cases your fingers point in the
direction of the magnetic field, and your thumb
points in the direction of current (or moving
charge / velocity).
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Honors Physics
Unit 10: Magnetism
The Magnitude of the Field Near a Wire
If we measure the strength of the field near a long, straight wire, we find that it gets weaker as we move away from
the wire, varying inversely with the distance from the wire. If we call this distance r, then
a) how is B related to r?
If we stay at the same distance from the wire and vary the current I, we find that the strength of the field is directly
proportional,
b) how is B related to I?
c) Combining these relationships, how is B related to both I and r?
µo
µ = 4π × 10−7 T•m
A and it is known
For historical reasons, the proportionality constant is written as 2π , where o
as the magnetic permeability constant.
d) Using µ0 write the expression for the magnitude of the magnetic field near a long straight wire:
Practice Problems
1. Lightning strikes a vertical metal flagpole, and there is a momentary current up the pole. What is the direction of
the magnetic field due to this current at a point just east of the center of the pole? Explain how you use RHR #2
for your answer.
2. What is the strength of the magnetic field at a point 10 cm away from a long wire in which the current is 20 A?
3.
Two wires carry currents I1 and I2 as shown to the right. They are each of length
L and are held a distance d apart.
a) What is the direction of the magnetic field due to the top wire if you are
anywhere above (toward the top of the page) the top wire?
L
I1
d
b) What is the direction of the magnetic field due to the top wire if you are
anywhere below (toward the bottom of the page) the top wire?
I2
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Honors Physics
Unit 10: Magnetism
L
c) What is the direction of the magnetic field B1 at the bottom wire due to the
top wire?
I1
d
d) What is the magnitude of the magnetic field B1 at the bottom wire?
I2
e) What is the direction of the magnetic force F1 on the bottom wire?
f) What is the magnitude of the magnetic force F1 on the bottom wire?
g) Go through the same steps to find the magnitude and direction of the force on the top wire.
L
I1
h) Draw the forces on both wires due to the presence of the magnetic field the other
creates on the image at right. Draw the currents in the wires.
d
I2
i)
a)
What would be different if one of the currents were reversed? Answer the
previous questions for this situation.
b)
c)
d)
L
I1
d
e)
f)
j)
Draw the forces on both wires due to the presence of the magnetic field the other
creates on the image at right. Draw the currents in the wires.
I2
L
I1
k) When would the wires be attracted to each other and when would they be
repulsed?
d
I2
l)
What would happen if you switched the current in both wires? Would they be attracted or repulsed? Explain your
answer.
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Honors Physics
Unit 10: Magnetism
Discussion Question: A bar magnet is held vertically and charges or current segments are held at or moved past a pole as
shown below. For each situation:
a) draw in the magnetic field lines and then
b) state the direction (if any) of the force on the charge or current segment as none, left, right, up (toward the top of
the page), down (toward the bottom of the page), into (the page) or out (of the page).
N
S
N
S
A positive
A positive
A positive charge A segment
charge is hung charge moves to
carries current
moves down
by a thread
the right
left
Direction:
N
S
A segment carries
A segment
carries current up current into the
page
Magnetic Field of a Current Loop
If a wire is bent into a circular loop, the field lines on the inside of the loop become
concentrated, creating a strong field in the middle. Outside the loop, the field lines are
spread out, creating a much weaker field.
1. Assume the circle at right is a loop of wire in the plane of the page. Draw an arrow
indicating that the current through this wire is counterclockwise.
a. What would be the direction of the magnetic field outside the loop, to the
right?
b. What would be the direction of the magnetic field outside the loop, to the left?
c. What would be the direction of the magnetic field inside the loop, on the right hand side?
d. What would be the direction of the magnetic field inside the loop, on the right hand side?
e. Discuss where you think the magnetic field is stronger: inside or outside the loop. Explain your answer.
f.
If you increased the current through the loop, what would happen to the strength of the magnetic field
inside the loop? Why?
g. If you increased the radius of the loop, what would happen to the strength of the magnetic field inside the
loop? Why?
10
Honors Physics
Unit 10: Magnetism
µo I
It can be shown that the strength of the magnetic field in the center of the loop of radius R is: 2R
If the wire wraps around the loop N times, then the field is the sum of the fields from each turn of the loop, so in
general,
µ I
Bcenter = N o
2R
Practice Problem
1. There is a current in a circular path through a horizontal loop, clockwise as viewed from above. The coil has 10
turns, a radius of 3 cm, and the current is 210 mA.
a) Sketch the loop, label the radius and indicate the direction of the current through the loop as well as the
magnetic at the center of the loop.
b) What is the magnitude of the field at the center in µT?
Magnetic Field of a Solenoid
A solenoid can be thought of as a stack of current loops, with the current the same in each loop. The field in the
center of the solenoid is very close to being uniform, and depends only on two things: the current I in the solenoid,
and the number of turns per unit length n. The field in the center of the solenoid is directly proportional to these
quantities, and the proportionality constant is simply µo.
a) The equation for the magnetic field of a solenoid is:
L
I
A solenoid is a very useful electric device, because it acts
like a bar magnet when it is energized with current. If we
place an iron bar inside a solenoid and switch the current
on, the bar will be pushed one way or the other by the
ur
magnetic field of the solenoid. This action can be made to
ur
B
B
open or close a valve or switch.
Solenoid
For example, when a dishwasher or washing machine runs
through its cycle, you hear "clicks" when the water starts or
stops. You are hearing a solenoid controlling the water valve.
Discussion question: Given the direction of the current in the diagram above, determine the direction of the magnetic
field through the solenoid and sketch corresponding arrowheads on the field lines on the image above. Draw in
the N and S poles. Explain how you use RHR#2 to find the direction of the field.
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Honors Physics
Unit 10: Magnetism
Practice Problems
1. A solenoid with 400 loops has a current of 5A running through it. If the length of the solenoid is 400 mm, what
is the magnetic field it is producing in mT?
2. A solenoid with 4 loops produces a magnetic field of 1.2 x 10-5 T. If the length of the solenoid is 0.8 m, find the
current through the solenoid.
3. A circuit is set up through a solenoid as seen at right. Initially the
switch is open and there is no current through the circuit. There is
current through the solenoid, in the direction shown.
a. On the image, draw the direction of the magnetic field due to
the solenoid.
The switch is closed and the circuit is complete, with current going
from high to low potential.
b. On the image, draw the direction of the current in the circuit.
c. The current is moving charge and it moves through the
solenoid. Describe how the current is affected by the
solenoid's magnetic field. Explain your answer using the
equations for force due to a magnetic field and the right hand rules.
I
+
V
switch
4. A solenoid is seen here with current I, length L and
number of turns N. The current flows into the right and
out of the left as shown, creating a B-field with the
north pole at left.
a. How would this magnetic field change if the
solenoid were stretched out?
b. How would this field change if the current were reversed?
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